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in this article , we consider a class of homogeneous partial differential operators on a finite dimensional vector space and study their associated heat kernels . these operators , which we call nondegenerate - homogeneous operators , are seen to generalize the well - studied classes of semi - elliptic operators introduced by f. browder @xcite , also known as quasi - elliptic operators @xcite , and a special  positive " subclass of semi - elliptic operators which appear as the spatial part of s. d. eidelman s @xmath0-parabolic operators @xcite . in particular , this class of operators contains all integer powers of the laplacian .      to motivate the definition of nondegenerate - homogeneous operators , given in the next section , we first introduce the class of semi - elliptic operators . semi - elliptic operators are seen to be prototypical examples of nondegenerate - homogeneous operators ; in fact , the definition of nondegenerate - homogeneous operators is given to formulate the following construction in a basis - independent way . given @xmath1-tuple of positive integers @xmath2 and a multi - index @xmath3 , set @xmath4 . consider the constant coefficient partial differential operator @xmath5 with principal part ( relative to @xmath6 ) @xmath7 where @xmath8 and @xmath9 for each multi - index @xmath10 . such an operator @xmath11 is said to be _ semi - elliptic _ if the symbol of @xmath12 , defined by @xmath13 for @xmath14 , is non - vanishing away from the origin . if @xmath11 satisfies the stronger condition that @xmath15 is strictly positive away from the origin , we say that it is _ positive - semi - elliptic_. what seems to be the most important property of semi - elliptic operators is that their principal part @xmath12 is homogeneous in the following sense : if given any smooth function @xmath16 we put @xmath17 for all @xmath18 and @xmath19 , then @xmath20 for all @xmath18 . this homogeneous structure was used explicitly in the work of f. browder and l. hrmander and , in this article , we generalize this notion . we note that our definition for the differential operators @xmath21 is given to ensure a straightforward relationship between operators and symbols under our convention for the fourier transform ( defined in subsection [ subsec : preliminaries ] ) ; this definition differs only slightly from the standard references @xcite in which @xmath22 is replaced by @xmath23 . in both conventions , the symbol of the operator @xmath24 is the positive polynomial @xmath25 . in fact , the principal symbols of all positive - semi - elliptic operators agree in both conventions . + as mentioned above , the class of semi - elliptic operators was introduced by f. browder in @xcite who studied spectral asymptotics for a related class of variable - coefficient operators ( operators of constant strength ) . elliptic operators appeared later in l. hrmander s text @xcite as model examples of hypoelliptic operators on @xmath26 beyond the class of elliptic operators . around the same time , l. r. volevich @xcite independently introduced the same class of operators but instead called them `` quasi - elliptic '' . since then , the theory of semi - elliptic operators , and hence quasi - elliptic operators , has reached a high level of sophistication and we refer the reader to the articles @xcite , which use the term semi - elliptic , and the articles @xcite , which use the term quasi - elliptic , for an account of this theory . we would also like to point to the 1971 paper of m. troisi @xcite which gives a more complete list of references ( pertaining to quasi - elliptic operators ) . + shortly after f. browder s paper @xcite appeared , s. d. eidelman considered a subclass of semi - elliptic operators on @xmath27 ( and systems thereof ) of the form @xmath28 where @xmath29 and the coefficients @xmath30 are functions of @xmath31 and @xmath32 . such an operator is said to be _ @xmath33-parabolic _ if its spatial part , @xmath34 , is ( uniformly ) positive - semi - elliptic . we note however that eidelman s work and the existing literature refer exclusively to @xmath0-parabolic operators , i.e. , where @xmath35 , and for consistency we write @xmath0-parabolic henceforth @xcite . the relationship between positive - semi - elliptic operators and @xmath0-parabolic operators is analogous to the relationship between the laplacian and the heat operator and , in the context of this article , the relationship between nondegenerate - homogeneous and positive - homogeneous operators described by proposition [ prop : dichotomy ] . the theory of @xmath0-parabolic operators , which generalizes the theory of parabolic partial differential equations ( and systems ) , has seen significant advancement by a number of mathematicians since eidelman s original work . we encourage the reader to see the recent text @xcite which provides an account of this theory and an exhaustive list of references . it should be noted however that the literature encompassing semi - elliptic operators and quasi - elliptic operators , as far as we can tell , has very few cross - references to the literature on @xmath0-parabolic operators beyond the 1960s . we suspect that the absence of cross - references is due to the distinctness of vocabulary . +      we motivate the study of homogeneous operators by first demonstrating the natural appearance of their heat kernels in the study of convolution powers of complex - valued functions . to this end , consider a finitely supported function @xmath37 and define its convolution powers iteratively by @xmath38 for @xmath39 where @xmath40 . in the special case that @xmath41 is a probability distribution , i.e. , @xmath41 is non - negative and has unit mass , @xmath41 drives a random walk on @xmath36 whose nth - step transition kernels are given by @xmath42 . under certain mild conditions on the random walk , @xmath43 is well - approximated by a single gaussian density ; this is the classical local limit theorem . specifically , for a symmetric , aperiodic and irreducible random walk , the theorem states that @xmath44 uniformly for @xmath39 , where @xmath45 is the generalized gaussian density @xmath46 here , @xmath47 is the positive definite covariance matrix associated to @xmath41 and @xmath48 denotes the dot product @xcite . the canonical example is that in which @xmath49 ( e.g. simple random walk ) and in this case @xmath43 is approximated by the so - called heat kernel @xmath50 defined by @xmath51 for @xmath18 and @xmath52 . indeed , we observe that @xmath53 for each positive integer @xmath54 and @xmath39 and so the local limit theorem is written equivalently as @xmath55 uniformly for @xmath39 . in addition to its natural appearance as the _ attractor _ in the local limit theorem above , @xmath56 is a fundamental solution to the heat equation @xmath57 on @xmath58 . in fact , this connection to random walk underlies the heat equation s probabilistic / diffusive interpretation . beyond the probabilistic setting , this link between convolution powers and fundamental solutions to partial differential equations persists as can be seen in the examples below . in what follows , the heat kernels @xmath59 are fundamental solutions to the corresponding heat - type equations of the form @xmath60 the appearance of @xmath61 in local limit theorems ( for @xmath43 ) is then found by evaluating @xmath62 at integer time @xmath63 and lattice point @xmath39 . [ ex : intro1 ] consider @xmath64 defined by @xmath65    analogous to the probabilistic setting , the large @xmath54 behavior of @xmath43 is described by a generalized local limit theorem in which the attractor is a fundamental solution to a heat - type equation . specifically , the following local limit theorem holds ( see @xcite for details ) : @xmath66 uniformly for @xmath67 where @xmath59 is the `` heat '' kernel for the heat - type equation @xmath68 where @xmath69 this local limit theorem is illustrated in figure [ fig : ex1_intro ] which shows @xmath70 and the approximation @xmath71 when @xmath72 . [ ex : intro2 ] consider @xmath73 defined by @xmath74 , where @xmath75    in this example , the following local limit theorem , which is illustrated by figure [ fig : ex2_intro ] , describes the limiting behavior of @xmath43 . we have @xmath76 uniformly for @xmath67 where @xmath61 is again a fundamental solution to @xmath68 where , in this case , @xmath77    [ ex : intro3 ] consider @xmath78 defined by @xmath79 here , the following local limit theorem is valid : @xmath80 uniformly for @xmath67 . here again , the attractor @xmath61 is the fundamental solution to @xmath68 where @xmath81    looking back at preceding examples , we note that the operators appearing in examples [ ex : intro1 ] and [ ex : intro2 ] are both positive - semi - elliptic and consist only of their principal parts . this is easily verified , for @xmath82 in example [ ex : intro1 ] and @xmath83 in example [ ex : intro2 ] . in contrast to examples [ ex : intro1 ] and [ ex : intro2 ] , the operator @xmath11 which appears in example [ ex : intro3 ] is not semi - elliptic in the given coordinate system . after careful study , the @xmath11 appearing in example [ ex : intro3 ] can be written equivalently as @xmath84 where @xmath85 is the directional derivative in the @xmath86 direction and @xmath87 is the directional derivative in the @xmath88 direction . in this way , @xmath11 is seen to be semi - elliptic with respect to some basis @xmath89 of @xmath90 and , with respect to this basis , we have @xmath91 . for this reason , our formulation of nondegenerate - homogeneous operators ( and positive - homogeneous operators ) , given in the next section , is made in a basis - independent way . + all of the operators appearing in examples [ ex : intro1 ] , [ ex : intro2 ] and [ ex : intro3 ] share two important properties : homogeneity and positivity ( in the sense of symbols ) . while we make these notions precise in the next section , loosely speaking , homogeneity is the property that @xmath11 `` plays well '' with some dilation structure on @xmath26 , though this structure is different in each example . further , homogeneity for @xmath11 is reflected by an analogous one for the corresponding heat kernel @xmath61 ; in fact , the specific dilation structure is , in some sense , selected by @xmath43 as @xmath92 and leads to the corresponding local limit theorem . in further discussion of these examples , a very natural question arises : given @xmath37 , how does one compute the operator @xmath11 whose heat kernel @xmath61 appears as the attractor in the local limit theorem for @xmath43 ? in the examples we have looked at , one studies the taylor expansion of the fourier transform @xmath93 of @xmath41 near its local extrema and , here , the symbol of the relevant operator @xmath11 appears as certain scaled limit of this taylor expansion . in general , however , this is a very delicate business and , at present , there is no known algorithm to determine these operators . in fact , it is possible that multiple ( distinct ) operators can appear by looking at the taylor expansions about distinct local extrema of @xmath93 ( when they exist ) and , in such cases , the corresponding local limit theorems involve sums of of heat kernels  each corresponding to a distinct @xmath11 . this study is carried out in the article @xcite wherein local limit theorems involve the heat kernels of the positive - homoegeneous operators studied in the present article . we note that the theory presented in @xcite is not complete , for there are cases in which the associated taylor approximations yield symbols corresponding to operators @xmath11 which fail to be positive - homogeneous ( and hence fail to be positive - semi - elliptic ) and further , the heat kernels of these ( degenerate ) operators appear as limits of oscillatory integrals which correspond to the presence of  odd " terms in @xmath11 , e.g. , the airy function . in one dimension , a complete theory of local limit theorems is known for the class of finitely supported functions @xmath94 . beyond one dimension , a theory for local limit theorems of complex - valued functions , in which the results of @xcite will fit , remains open . + the subject of this paper is an account of positive - homogeneous operators and their corresponding heat equations . in section [ sec : homogeneousoperators ] , we introduce positive - homogeneous operators and study their basic properties ; therein , we show that each positive - homogeneous operator is semi - elliptic in some coordinate system . section [ sec : holder ] develops the necessary background to introduce the class of variable - coefficient operators studied in this article ; this is the class of @xmath95-positive - semi - elliptic operators introduced in section [ sec : uniformlysemielliptic]each of which is comparable to a constant - coefficient positive - homogeneous operator . in section [ sec : fundamentalsolution ] , we study the heat equations corresponding to uniformly @xmath95-positive - semi - elliptic operators with hlder continuous coefficients . specifically , we use the famous method of e. e. levi , adapted to parabolic systems by a. friedman and s. d. eidelman , to construct a fundamental solution to the corresponding heat equation . our results in this direction are captured by those of s. d. eidelman @xcite and the works of his collaborators , notably s. d. ivashyshen and a. n. kochubei @xcite , concerning @xmath0-parabolic systems . our focus in this presentation is to highlight the essential role played by the legendre - fenchel transform in heat kernel estimates which , to our knowledge , has not been pointed out in the context of semi - elliptic operators . in a forthcoming work , we study an analogous class of operators , written in divergence form , with measurable - coefficients and their corresponding heat kernels . this class of measurable - coefficient operators does not appear to have been previously studied . the results presented here , using the legendre - fenchel transform , provides the background and context for our work there . * fourier analysis : * our setting is a real @xmath1-dimensional vector space @xmath96 equipped with haar ( lebesgue ) measure @xmath97 and the standard smooth structure ; we do not affix @xmath96 with a norm or basis . the dual space of @xmath96 is denoted by @xmath98 and the dual pairing is denoted by @xmath99 for @xmath100 and @xmath101 . let @xmath102 be the haar measure on @xmath98 which we take to be normalized so that our convention for the fourier transform and inverse fourier transform , given below , makes each unitary . throughout this article , all functions on @xmath96 and @xmath98 are understood to be complex - valued . the usual lebesgue spaces are denoted by @xmath103 and equipped with their usual norms @xmath104 for @xmath105 . in the case that @xmath106 , the corresponding inner product on @xmath107 is denoted by @xmath108 . of course , we will also work with @xmath109 ; here the @xmath110-norm and inner product will be denoted by @xmath111 and @xmath112 respectively . the fourier transform @xmath113 and inverse fourier transform @xmath114 are initially defined for schwartz functions @xmath115 and @xmath116 by @xmath117 for @xmath101 and @xmath100 respectively .    for the remainder of this article ( mainly when duality is nt of interest ) , @xmath118 stands for any real @xmath1-dimensional vector space ( and so is interchangeable with @xmath96 or @xmath98 ) . for a non - empty open set @xmath119 , we denote by @xmath120 and @xmath121 the set of continuous functions on @xmath122 and bounded continuous functions on @xmath122 , respectively . the set of smooth functions on @xmath122 is denoted by @xmath123 and the set of compactly supported smooth functions on @xmath122 is denoted by @xmath124 . we denote by @xmath125 the space of distributions on @xmath122 ; this is dual to the space @xmath124 equipped with its usual topology given by seminorms . a partial differential operator @xmath126 on @xmath118 is said to be _ hypoelliptic _ if it satisfies the following property : given any open set @xmath119 and any distribution @xmath127 which satisfies @xmath128 in @xmath122 , then necessarily @xmath129 . + * dilation structure : * denote by @xmath130 and @xmath131 the set of endomorphisms and isomorphisms of @xmath118 respectively . given @xmath132 , we consider the one - parameter group @xmath133 defined by @xmath134 for @xmath18 . these one - parameter subgroups of @xmath131 allow us to define continuous one - parameter groups of operators on the space of distributions as follows : given @xmath132 and @xmath18 , first define @xmath135 for @xmath136 by @xmath137 for @xmath138 . extending this to the space of distribution on @xmath118 in the usual way , the collection @xmath139 is a continuous one - parameter group of operators on @xmath140 ; it will allow us to define homogeneity for partial differential operators in the next section . + * linear algebra , polynomials and the rest : * given a basis @xmath141 of @xmath118 , we define the map @xmath142 by setting @xmath143 whenever @xmath144 . this map defines a global coordinate system on @xmath118 ; any such coordinate system is said to be a linear coordinate system on @xmath118 . by definition , a polynomial on @xmath118 is a function @xmath145 that is a polynomial function in every ( and hence any ) linear coordinate system on @xmath118 . a polynomial @xmath146 on @xmath118 is called a nondegenerate polynomial if @xmath147 for all @xmath148 . further , @xmath146 is called a positive - definite polynomial if its real part , @xmath149 , is non - negative and has @xmath150 only when @xmath151 . the symbols @xmath152 mean what they usually do , @xmath153 denotes the set of non - negative integers and @xmath154\subseteq\mathbb{r}$ ] . the symbols @xmath155 , @xmath156 and @xmath157 denote the set of strictly positive elements of @xmath158 , @xmath153 and @xmath159 respectively . likewise , @xmath160 , @xmath161 and @xmath162 respectively denote the set of @xmath1-tuples of these aforementioned sets . given @xmath163 and a basis @xmath141 of @xmath118 , we denote by @xmath164 the isomorphism of @xmath118 defined by @xmath165 for @xmath166 . we say that two real - valued functions @xmath16 and @xmath167 on a set @xmath168 are comparable if , for some positive constant @xmath169 , @xmath170 for all @xmath171 ; in this case we write @xmath172 . adopting the summation notation for semi - elliptic operators of l. hrmander s treatise @xcite , for a fixed @xmath2 , we write @xmath173 for all multi - indices @xmath3 . finally , throughout the estimates made in this article , constants denoted by @xmath169 will change from line to line without explicit mention . in this section we introduce two important classes of homogeneous constant - coefficient on @xmath96 . these operators will serve as `` model '' operators in our theory in the way that integer powers of the laplacian serves a model operators in the elliptic theory of partial differential equations . to this end , let @xmath11 be a constant - coefficient partial differential operator on @xmath96 and let @xmath174 be its symbol . specifically , @xmath146 is the polynomial on @xmath98 defined by @xmath175 for @xmath101 ( this is independent of @xmath100 precisely because @xmath11 is a constant - coefficient operator ) . we first introduce the following notion of homogeneity of operators ; it is mirrored by an analogous notion for symbols which we define shortly . given @xmath176 , we say that a constant - coefficient partial differential operator @xmath11 is homogeneous with respect to the one - parameter group @xmath177 if @xmath178 for all @xmath18 ; in this case we say that @xmath179 is a member of the exponent set of @xmath11 and write @xmath180 . a constant - coefficient partial differential operator @xmath11 need not be homogeneous with respect to a unique one - parameter group @xmath177 , i.e. , @xmath181 is not necessarily a singleton . for instance , it is easily verified that , for the laplacian @xmath182 on @xmath26 , @xmath183 where @xmath184 is the identity and @xmath185 is the lie algebra of the orthogonal group , i.e. , is given by the set of skew - symmetric matrices . despite this lack of uniqueness , when @xmath11 is equipped with a nondegenerateness condition ( see definition [ def : homogeneousoperators ] ) , we will find that trace is the same for each member of @xmath181 and this allows us to uniquely define an  order " for @xmath11 ; this is lemma [ lem : trace ] . + given a constant coefficient operator @xmath11 with symbol @xmath146 , one can quickly verify that @xmath180 if and only if @xmath186 for all @xmath18 and @xmath101 where @xmath187 is the adjoint of @xmath179 . more generally , if @xmath146 is any continuous function on @xmath118 and is satisfied for some @xmath188 , we say that @xmath146 _ is homogeneous with respect to _ @xmath189 and write @xmath190 . this admitted slight abuse of notation should not cause confusion . in this language , we see that @xmath191 if and only if @xmath192 . + we remark that the notion of homogeneity defined above is similar to that put forth for homogeneous operators on homogeneous ( lie ) groups , e.g. , rockland operators @xcite . the difference is mostly a matter of perspective : a homogeneous group @xmath193 is equipped with a fixed dilation structure , i.e. , it comes with a one - parameter group @xmath194 , and homogeneity of operators is defined with respect to this fixed dilation structure . by contrast , we fix no dilation structure on @xmath96 and formulate homogeneity in terms of an operator @xmath11 and the existence of a one - parameter group @xmath177 that `` plays '' well with @xmath11 in sense defined above . as seen in the study of convolution powers on the square lattice ( see @xcite ) , it useful to have this freedom . [ def : homogeneousoperators ] let @xmath11 be constant - coefficient partial differential operator on @xmath96 with symbol @xmath146 . we say that @xmath11 is a nondegenerate - homogeneous operator if @xmath146 is a nondegenerate polynomial and @xmath181 contains a diagonalizable endomorphism . we say that @xmath11 is a positive - homogeneous operator if @xmath146 is a positive - definite polynomial and @xmath181 contains a diagonalizable endomorphism .    for any polynomial @xmath146 on a finite - dimensional vector space @xmath118 , @xmath146 is said to be _ nondegenerate - homogeneous _ if @xmath146 is nondegenerate and @xmath195 , defined as the set of @xmath196 for which holds , contains a diagonalizable endomorphism . we say that @xmath146 is _ positive - homogeneous _ if it is a positive - definite polynomial and @xmath195 contains a diagonalizable endomorphism . in this language , we have the following proposition . [ prop : operatorsymbolequivalence ] let @xmath11 be a positive homogeneous operator on @xmath96 with symbol @xmath146 . then @xmath11 is a nondegenerate - homogeneous operator if and only if @xmath146 is a nondegenerate - homogeneous polynomial . further , @xmath11 is a positive - homogeneous operator if and only if @xmath146 is a positive - homogeneous polynomial .    since the adjectives `` nondegenerate '' and `` positive '' , in the sense of both operators and polynomials , are defined in terms of the symbol @xmath146 , all that needs to be verified is that @xmath181 contains a diagonalizable endomorphism if and only if @xmath195 contains a diagonalizable endomorphism . upon recalling that @xmath180 if and only if @xmath192 , this equivalence is verified by simply noting that diagonalizability is preserved under taking adjoints .    to capture the class of nondegenerate - homogeneous operators ( or positive - homogeneous operators ) , in addition to requiring that that the symbol @xmath146 of an operator @xmath11 be nondegenerate ( or positive - definite ) , one can instead demand only that @xmath181 contains an endomorphism whose characteristic polynomial factors over @xmath158 or , equivalently , whose spectrum is real . this a priori weaker condition is seen to be sufficient by an argument which makes use of the jordan - chevalley decomposition . in the positive - homogeneous case , this argument is carried out in @xcite ( specifically proposition 2.2 ) wherein positive - homogeneous operators are first defined by this ( a priori weaker ) condition . for the nondegenerate case , the same argument pushes through with very little modification . we observe easily that all positive - homogeneous operators are nondegenerate - homogeneous . it is the `` heat '' kernels corresponding to positive - homogeneous operators that naturally appear in @xcite as the attractors of convolution powers of complex - valued functions . the following proposition highlights the interplay between positive - homogeneity and nondegenerate - homogeneity for an operator @xmath11 on @xmath96 and its corresponding `` heat '' operator @xmath197 on @xmath198 .    [ prop : dichotomy ] let @xmath11 be a constant - coefficient partial differential operator on @xmath96 whose exponent set @xmath181 contains a diagonalizable endomorphism . let @xmath146 be the symbol of @xmath11 , set @xmath149 , and assume that there exists @xmath101 for which @xmath199 . we have the following dichotomy : @xmath11 is a positive - homogeneous operator on @xmath96 if and only if @xmath197 is a nondegenerate - homogeneous operator on @xmath198 . given a diagonalizable endomorphism @xmath180 , set @xmath200 where @xmath184 is the identity on @xmath158 . obviously , @xmath201 is diagonalizable . further , for any @xmath202 , @xmath203 for all @xmath204 and @xmath205 . hence @xmath206 for all @xmath204 and therefore @xmath207 . it remains to show that @xmath146 is positive - definite if and only if the symbol of @xmath197 is nondegenerate . to this end , we first compute the symbol of @xmath197 which we denote by @xmath208 . since the dual space of @xmath198 is isomorphic to @xmath209 , the characters of @xmath198 are represented by the collection of maps @xmath210 where @xmath211 . consequently , @xmath212 for @xmath211 . we note that @xmath213 because @xmath192 ; in fact , this happens whenever @xmath195 is non - empty . now if @xmath146 is a positive - definite polynomial , @xmath214 whenever @xmath215 . thus to verify that @xmath208 is a nondegenerate polynomial , we simply must verify that @xmath216 for all non - zero @xmath217 . this is easy to see because , in light of the above fact , @xmath218 whenever @xmath219 and hence @xmath208 is nondegenerate . for the other direction , we demonstrate the validity of the contrapositive statement . assuming that @xmath146 is not positive - definite , an application of the intermediate value theorem , using the condition that @xmath199 for some @xmath101 , guarantees that @xmath220 for some non - zero @xmath221 . here , we observe that @xmath222 when @xmath223 and hence @xmath208 is not nondegenerate . we will soon return to the discussion surrounding a positive - homogeneous operator @xmath11 and its heat operator @xmath197 . it is useful to first provide representation formulas for nondegenerate - homogeneous and positive - homogeneous operators . such representations connect our homogeneous operators to the class of semi - elliptic operators discussed in the introduction . to this end , we define the `` base '' operators on @xmath96 . first , for any element @xmath224 , we consider the differential operator @xmath225 defined originally for @xmath226 by @xmath227 for @xmath100 . fixing a basis @xmath228 of @xmath96 , we introduce , for each multi - index @xmath10 , @xmath229 . [ prop : operatorrepresentation ] let @xmath11 be a nondegenerate - homogeneous operator on @xmath96 . then there exist a basis @xmath228 of @xmath96 and @xmath2 for which @xmath230 where @xmath231 . the isomorphism @xmath232 , defined by , is a member of @xmath181 . further , if @xmath11 is positive - homogeneous , then @xmath233 for @xmath234 and hence @xmath235    we will sometimes refer to the @xmath6 and @xmath236 of the proposition as _ weights_. before addressing the proposition , we first prove the following mirrored result for symbols .    [ lem : polynomialrepresentation ] let @xmath146 be a nondegenerate - homogeneous polynomial on a @xmath1-dimensional real vector space @xmath237 then there exists a basis @xmath141 of @xmath118 and @xmath2 for which @xmath238 for all @xmath239 where @xmath240 and @xmath241 . the isomorphism @xmath242 , defined by , is a member of @xmath195 . further , if @xmath146 is a positive - definite polynomial , i.e. , it is positive - homogeneous , then @xmath233 for @xmath234 and hence @xmath243 for @xmath244 . let @xmath245 be diagonalizable and select a basis @xmath141 which diagonalizes @xmath179 , i.e. , @xmath246 where @xmath247 for @xmath248 . because @xmath146 is a polynomial , there exists a finite collection @xmath231 for which @xmath249 for @xmath244 . by invoking the homogeneity of @xmath146 with respect to @xmath179 and using the fact that @xmath250 for @xmath166 , we have @xmath251 for all @xmath244 and @xmath18 where @xmath252 . in view of the nondegenerateness of @xmath146 , the linear independence of distinct powers of @xmath32 and the polynomial functions @xmath253 , for distinct multi - indices @xmath254 , as @xmath255 functions ensures that @xmath256 unless @xmath257 . we can therefore write @xmath258 for @xmath244 . we now determine @xmath259 by evaluating this polynomial along the coordinate axes . to this end , by fixing @xmath166 and setting @xmath260 for @xmath261 , it is easy to see that the summation above collapses into a single term @xmath262 where @xmath263 ( here @xmath264 denotes the usual @xmath265th - euclidean basis vector in @xmath26 ) . consequently , @xmath266 for @xmath248 and thus , upon setting @xmath267 , yields @xmath238 for all @xmath244 as was asserted . in this notation , it is also evident that @xmath268 . under the additional assumption that @xmath146 is positive - definite , we again evaluate @xmath146 at the coordinate axes to see that @xmath269 for @xmath261 . in this case , the positive - definiteness of @xmath146 requires @xmath270 and @xmath271 for each @xmath248 . consequently , @xmath233 for @xmath234 as desired . given a nondegenerate - homogeneous @xmath11 on @xmath96 with symbol @xmath146 , @xmath146 is necessarily a nondegenerate - homogeneous polynomial on @xmath98 in view of proposition [ prop : operatorsymbolequivalence ] . we can therefore apply lemma [ lem : polynomialrepresentation ] to select a basis @xmath272 of @xmath98 and @xmath2 for which @xmath238 for all @xmath273 where @xmath231 . we will denote by @xmath274 , the dual basis to @xmath275 , i.e. , @xmath228 is the unique basis of @xmath96 for which @xmath276 when @xmath277 and @xmath278 otherwise . in view of the duality of the bases @xmath274 and @xmath275 , it is straightforward to verify that , for each multi - index @xmath254 , the symbol of @xmath279 is @xmath280 in the notation of lemma [ lem : polynomialrepresentation ] . consequently , the constant - coefficient partial differential operator defined by the right hand side of also has symbol @xmath146 and so it must be equal to @xmath11 because operators and symbols are in one - to - one correspondence . using , it is now straightforward to verify that @xmath281 . the assertion that @xmath233 when @xmath11 is positive - homogeneous follows from the analogous conclusion of lemma [ lem : polynomialrepresentation ] by the same line of reasoning . in view of proposition [ prop : operatorrepresentation ] , we see that all nondegenerate - homogeneous operators are semi - elliptic in some linear coordinate system ( that which is defined by @xmath274 ) . an appeal to theorem 11.1.11 of @xcite immediately yields the following corollary .    every nondegenerate - homogeneous operator @xmath11 on @xmath96 is hypoelliptic . our next goal is to associate an `` order '' to each nondegenerate - homogeneous operator . for a positive - homogeneous operator @xmath11 , this order will be seen to govern the on - diagonal decay of its heat kernel @xmath61 and so , equivalently , the ultracontractivity of the semigroup @xmath282 . with the help of lemma [ lem : polynomialrepresentation ] , the few lemmas in this direction come easily . [ lem : ptoinfty ] let @xmath146 be a nondegenerate - homogeneous polynomial on a @xmath1-dimensional real vector space @xmath118 . then @xmath283 ; here @xmath284 means that @xmath285 in any ( and hence every ) norm on @xmath118 . the idea of the proof is to construct a function which bounds @xmath286 from below and obviously blows up at infinity . to this end , let @xmath287 be a basis for @xmath118 and take @xmath288 as guaranteed by lemma [ lem : polynomialrepresentation ] ; we have @xmath289 where @xmath290 for @xmath166 . define @xmath291 by @xmath292 where @xmath293 . we observe immediately @xmath294 because @xmath295 for @xmath166 . an application of proposition [ prop : comparepoly ] ( a basic result appearing in our background section , section [ sec : holder ] ) , which uses the nondegenerateness of @xmath146 , gives a positive constant @xmath169 for which @xmath296 for all @xmath244 . the lemma now follows by simply noting that @xmath297 as @xmath298 . let @xmath146 be a polynomial on @xmath118 and denote by @xmath299 the set of @xmath300 for which @xmath301 for all @xmath244 . if @xmath146 is a nondegenerate - homogeneous polynomial , then @xmath299 , called the symmetry group of @xmath146 , is a compact subgroup of @xmath131 . our supposition that @xmath146 is a nondegenerate polynomial ensures that , for each @xmath302 , @xmath303 is empty and hence @xmath304 . consequently , given @xmath305 and @xmath306 , we observe that @xmath307 and @xmath308 for all @xmath244 ; therefore @xmath299 is a subgroup of @xmath131 .    to see that @xmath299 is compact , in view of the finite - dimensionality of @xmath131 and the heine - borel theorem , it suffices to show that @xmath299 is closed and bounded . first , for any sequence @xmath309 for which @xmath310 as @xmath92 , the continuity of @xmath146 ensures that @xmath311 for each @xmath244 and therefore @xmath299 is closed . it remains to show that @xmath299 is bounded ; this is the only piece of the proof that makes use of the fact that @xmath146 is nondegenerate - homogeneous and not simply homogeneous . assume that , to reach a contradiction , that there exists an unbounded sequence @xmath309 . choosing a norm @xmath312 on @xmath118 , let @xmath313 be the corresponding unit sphere in @xmath118 . then there exists a sequence @xmath314 for which @xmath315 for all @xmath316 but @xmath317 . in view of lemma [ lem : ptoinfty ] , @xmath318 which can not be true for @xmath146 is necessarily bounded on @xmath313 because it is continuous . [ lem : trace ] let @xmath11 be a nondegenerate - homogeneous operator . for any @xmath319 , @xmath320    let @xmath146 be the symbol of @xmath11 and take @xmath319 . since @xmath321 , @xmath322 for all @xmath18 . as @xmath299 is a compact group in view of the previous lemma , the determinant map @xmath323 , a lie group homomorphism , necessarily maps @xmath299 into the unit circle . consequently , @xmath324 for all @xmath18 . therefore , @xmath325 as desired .    by the above lemma , to each nondegenerate - homogenerous operator @xmath11 , we define the _ homogeneous order _ of @xmath11 to be the number @xmath326 for any @xmath180 . by an appeal to proposition [ prop : operatorrepresentation ] , @xmath281 for some @xmath327 and so we observe that @xmath328 in particular , @xmath329 is a positive rational number . we note that the term `` homogeneous - order '' does not coincide with the usual `` order '' for a partial differential operator . for instance , the laplacian @xmath182 on @xmath26 is a second order operator ; however , because @xmath330 , its homogeneous order is @xmath331 . we now restrict our attention to the study of positive - homogeneous operators and their associated heat kernels . to this end , let @xmath11 be a positive - homogeneous operator on @xmath96 with symbol @xmath146 and homogeneous order @xmath329 . the heat kernel for @xmath11 arises naturally from the study of the following cauchy problem for the corresponding heat equation @xmath68 : given initial data @xmath332 which is , say , bounded and continuous , find @xmath333 satisfying @xmath334 the initial value problem is solved by putting @xmath335 where @xmath336 is defined by @xmath337 for @xmath18 and @xmath100 ; we call @xmath61 _ the heat kernel _ associated to @xmath11 . equivalently , @xmath61 is the integral ( convolution ) kernel of the continuous semigroup @xmath338 of bounded operators on @xmath107 with infinitesimal generator @xmath339 . that is , for each @xmath340 , @xmath341 for @xmath18 and @xmath100 . let us make some simple observations about @xmath61 . first , by virtue of lemma [ lem : ptoinfty ] , it follows that @xmath342 for each @xmath18 . further , for any @xmath180 , @xmath343 for @xmath18 and @xmath100 . this computation immediately yields the so - called on - diagonal estimate for @xmath61 , @xmath344 for @xmath18 ; this is equivalently a statement of ultracontractivity for the semigroup @xmath282 . as it turns out , we can say something much stronger . [ prop : ccestimates ] let @xmath11 be a positive - homogeneous operator with symbol @xmath146 and homogeneous order @xmath329 . let @xmath345 be the legendre - fenchel transform of @xmath149 defined by @xmath346 for @xmath100 . also , let @xmath274 and @xmath29 be as guaranteed by proposition [ prop : operatorrepresentation ] . then , there exit positive constants @xmath347 and @xmath348 and , for each multi - index @xmath254 , a positive constant @xmath349 such that , for all @xmath350 , @xmath351 for all @xmath352 and @xmath18 . in particular , @xmath353 for all @xmath352 and @xmath18 .    in view of , the exponent on the prefactor in can be equivalently written , for any multi - index @xmath254 and @xmath350 , as @xmath354 where @xmath355 . we note that the estimates of proposition [ prop : ccestimates ] are written in terms of the difference @xmath356 and can ( trivially ) be expressed in terms of a single spatial variable @xmath31 . the estimates are written in this way to emphasize the role that @xmath357 plays as an integral kernel . we will later replace @xmath11 in by a comparable variable - coefficient operator @xmath126 and , in that setting , the associated heat kernel is not a convolution kernel and so we seek estimates involving two spatial variables @xmath31 and @xmath358 . to that end , the estimates here form a template for estimates in the variable - coefficient setting . we prove the proposition above in the section [ sec : fundamentalsolution ] ; the remainder of this section is dedicated to discussing the result and connecting it to the existing theory . let us first note that the estimate is mirrored by an analogous space - time estimate , theorem 5.3 of @xcite , for the convolution powers of complex - valued functions on @xmath36 satisfying certain conditions ( see section 5 of @xcite ) . the relationship between these two results , theorem 5.3 of @xcite and proposition [ prop : ccestimates ] , parallels the relationship between gaussian off - diagonal estimates for random walks and the analogous off - diagonal estimates enjoyed by the classical heat kernel @xcite . + let us first show that the estimates and recapture the well - known estimates of the theory of parabolic equations and systems in @xmath26  a theory in which the laplacian operator @xmath359 and its integer powers play a central role . to place things into the context of this article , let us observe that , for each positive integer @xmath360 , the partial differential operator @xmath361 is a positive - homogeneous operator on @xmath26 with symbol @xmath362 ; here , we identify @xmath26 as its own dual equipped with the dot product and euclidean norm @xmath312 . indeed , one easily observes that @xmath363 is a positive - definite polynomial and @xmath364 where @xmath365 is the the identity . consequently , the homogeneous order of @xmath361 is @xmath366 and the legendre - fenchel transform of @xmath367 is easily computed to be @xmath368 where @xmath369 . hence , is the well - known estimate @xmath370 for @xmath371 and @xmath18 ; this so - called off - diagonal estimate is ubiquitous to the theory of  higher - order " elliptic and parabolic equations @xcite . to write the derivative estimate in this context , we first observe that the basis given by proposition [ prop : operatorrepresentation ] can be taken to be the standard euclidean basis , @xmath372 and further , @xmath373 is the ( isotropic ) weight given by the proposition . writing @xmath374 and @xmath375 for each multi - index @xmath254 , takes the form @xmath376 for @xmath371 and @xmath18 , c.f . , * property 4 , p. 93 ) . + the appearance of the @xmath377-dimensional legendre - fenchel transform in heat kernel estimates was previously recognized and exploited in @xcite and @xcite in the context of elliptic operators . due to the isotropic nature of elliptic operators , the @xmath377-dimensional transform is sufficient to capture the inherent isotropic decay of corresponding heat kernels . beyond the elliptic theory , the appearance of the full @xmath1-dimensional legendre - fenchel transform is remarkable because it sharply captures the general anisotropic decay of @xmath61 . consider , for instance , the particularly simple positive - homogeneous operator @xmath378 on @xmath90 with symbol @xmath379 . it is easily checked that the operator @xmath179 with matrix representation @xmath380 , in the standard euclidean basis , is a member of the @xmath181 and so the homogeneous order of @xmath11 is @xmath381 . here we can compute the legendre - fenchel transform of @xmath382 directly to obtain @xmath383 for @xmath384 where @xmath385 and @xmath386 are positive constants . in this case , proposition [ prop : ccestimates ] gives positive constants @xmath347 and @xmath348 for which @xmath387 and @xmath18 where @xmath388 and @xmath389 . we note however that @xmath11 is `` separable '' and so we can write @xmath390 where @xmath391 is the @xmath377-dimensional laplacian operator . in view of theorem 8 of @xcite and its subsequent remark , the estimate is seen to be sharp ( modulo the values of @xmath392 and @xmath169 ) . to further illustrate the proposition for a less simple positive - homogeneous operator , we consider the operator @xmath11 appearing in example [ ex : intro3 ] . in this case , @xmath393 and one can verify directly that the @xmath394 , with matrix representation @xmath395 in the standard euclidean basis , is a member of @xmath181 . from this , we immediately obtain @xmath396 and one can directly compute @xmath397 for @xmath384 where @xmath385 and @xmath386 are positive constants . an appeal to proposition [ prop : ccestimates ] gives positive constants @xmath347 and @xmath348 for which @xmath398 for @xmath399 and @xmath18 where @xmath388 and @xmath389 . furthermore , @xmath400 and the basis @xmath401 of @xmath90 given in discussion surrounding are precisely those guaranteed by proposition [ prop : operatorrepresentation ] . appealing to the full strength of proposition [ prop : ccestimates ] , we obtain positive constants @xmath402 , @xmath348 and , for each multi - index @xmath254 , a positive constant @xmath403 such that , for each @xmath350 , @xmath404 for @xmath399 and @xmath18 where @xmath388 and @xmath389 . + in the context of homogeneous groups , the off - diagonal behavior for the heat kernel of a positive rockland operator ( a positive self - adjoint operator which is homogeneous with respect to the fixed dilation structure ) has been studied in @xcite ( see also @xcite ) . given a positive rockland operator @xmath11 on homogeneous group @xmath193 , the best known estimate for the heat kernel @xmath61 , due to auscher , ter elst and robinson , is of the form @xmath405 where @xmath406 is a homogeneous norm on @xmath193 ( consistent with @xmath407 and @xmath408 is the highest order derivative appearing in @xmath11 . in the context of @xmath26 , given a symmetric and positive - homogeneous operator @xmath11 with symbol @xmath146 , the structure @xmath409 for @xmath410 where @xmath180 is a homogeneous group on which @xmath11 becomes a positive rockland operator . on @xmath411 , it is quickly verified that @xmath412 is a homogeneous norm ( consistent with @xmath11 ) and so the above estimate is given in terms of @xmath413 which is , in general , dominated by the legendre - fenchel transform of @xmath414 . to see this , we need not look further than our previous and simple example in which @xmath415 . here @xmath416 and so @xmath417 . in view of , the estimate gives the correct decay along the @xmath418-coordinate axis ; however , the bounds decay at markedly different rates along the @xmath419-coordinate axis . this illustrates that the estimate is suboptimal , at least in the context of @xmath26 , and thus leads to the natural question : for positive - homogeneous operators on a general homogeneous group @xmath193 , what is to replace the legendre - fenchel transform in heat kernel estimates ? + returning to the general picture , let @xmath11 be a positive - homogeneous operator on @xmath96 with symbol @xmath146 and homogeneous order @xmath329 . to highlight some remarkable properties about the estimates and in this general setting , the following proposition concerning @xmath420 is useful ; for a proof , see section 8.3 of @xcite . [ prop : legendretransformproperties ] let @xmath11 be a positive - homogeneous operator with symbol @xmath146 and let @xmath420 be the legendre - fenchel transform of @xmath149 . then , for any @xmath180 , @xmath421 . moreover @xmath420 is continuous , positive - definite in the sense that @xmath422 and @xmath423 only when @xmath424 . further , @xmath420 grows superlinearly in the sense that , for any norm @xmath425 on @xmath96 , @xmath426 in particular , @xmath427 as @xmath428 . let us first note that , in view of the proposition , we can easily rewrite , for any @xmath191 , as @xmath429 for @xmath352 and @xmath18 ; the analogous rewriting is true for . the fact that @xmath420 is positive - definite and grows superlinearly ensures that the convolution operator @xmath282 defined by for @xmath18 is a bounded operator from @xmath430 to @xmath431 for any @xmath432 . of course , we already knew this because @xmath433 is a schwartz function ; however , when replacing @xmath11 with a variable - coefficient operator @xmath126 , as we will do in the sections to follow , the validity of the estimate for the kernel of the semigroup @xmath434 initially defined on @xmath110 , guarantees that the semigroup extends to a strongly continuous semigroup @xmath435 on @xmath436 for all @xmath105 and , what s more , the respective infinitesimal generators @xmath437 have spectra independent of @xmath438 @xcite . further , the estimate is key to establishing the boundedness of the riesz transform , it is connected to the resolution of kato s square root problem and it provides the appropriate starting point for uniqueness classes of solutions to @xmath439 @xcite . with this motivation in mind , following some background in section [ sec : holder ] , we introduce a class of variable - coefficient operators in section [ sec : uniformlysemielliptic ] called @xmath95-positive - semi - elliptic operators , each such operator @xmath126 comparable to a fixed positive - homogeneous operator . in section [ sec : fundamentalsolution ] , under the assumption that @xmath126 has hlder continuous coefficients and this notion of comparability is uniform , we construct a fundamental solution to the heat equation @xmath439 and show the essential role played by the legendre - fenchel transform in this construction . as mentioned previously , in a forthcoming work we will study the semigroup @xmath434 where @xmath126 is a divergence - form operator , which is comparable to a fixed positive - homogeneous operator , whose coefficients are at worst measurable . as the legendre - fenchel transform appears here by a complex change of variables followed by a minimization argument , in the measurable coefficient setting it appears quite naturally by an application of the so - called davies method , suitably adapted to the positive - homogeneous setting . in this section , we provide the necessary background on one - parameter contracting groups , anisotropic hlder continuity , and the legendre - fenchel transform and its interplay with the two previous notions .      in what follows , @xmath118 is a @xmath1-dimensional real vector space with a norm @xmath312 ; the corresponding operator norm on @xmath131 is denoted by @xmath406 . of course , since everything is finite - dimensional , the usual topologies on @xmath118 and @xmath131 are insensitive to the specific choice of norms . let @xmath440 be a continuous one - parameter group . @xmath441 is said to be contracting if @xmath442    we easily observe that , for any diagonalizable @xmath132 with strictly positive spectrum , the corresponding one - parameter group @xmath443 is contracting . indeed , if there exists a basis @xmath141 of @xmath118 and a collection of positive numbers @xmath444 for which @xmath445 for @xmath248 , then the one parameter group @xmath443 has @xmath446 for @xmath248 and @xmath18 . it then follows immediately that @xmath447 is contracting . [ prop : comparepoly ] let @xmath208 and @xmath414 be continuous real - valued functions on @xmath118 . if @xmath448 for all @xmath148 and there exists @xmath449 for which @xmath447 is contracting , then , for some positive constant @xmath169 , @xmath450 for all @xmath451 . if additionally @xmath452 for all @xmath148 , then @xmath453 . let @xmath313 denote the unit sphere in @xmath118 and observe that @xmath454 because @xmath208 and @xmath414 are continuous and @xmath414 is non - zero on @xmath313 . now , for any non - zero @xmath451 , the fact that @xmath455 is contracting implies that @xmath456 for some @xmath18 by virtue of the intermediate value theorem . therefore , @xmath457 . in view of the continuity of @xmath208 and @xmath414 , this inequality must hold for all @xmath451 . when additionally @xmath452 for all non - zero @xmath458 , the conclusion that @xmath453 is obtained by reversing the roles of @xmath208 and @xmath414 in the preceding argument . [ cor : movingconstants ] let @xmath11 be a positive - homogeneous operator on @xmath96 with symbol @xmath146 and let @xmath420 be the legendre - fenchel transform of @xmath149 . then , for any positive constant @xmath348 , @xmath459 .    by virtue of proposition [ prop : operatorrepresentation ] , let @xmath29 and @xmath274 be a basis for @xmath96 and for which @xmath460 . in view of proposition [ prop : legendretransformproperties ] , @xmath420 and @xmath461 are both continuous , positive - definite and have @xmath462 . in view of , it is easily verified that @xmath463 where @xmath464 and so it follows that @xmath465 is contracting . the corollary now follows directly from proposition [ prop : comparepoly ] . [ lem : scaling ] let @xmath146 be a positive - homogeneous polynomial on @xmath118 and let @xmath466 and @xmath287 be a basis for @xmath118 for which the conclusion of lemma [ lem : polynomialrepresentation ] holds . let @xmath149 and let @xmath254 and @xmath467 be multi - indices such that @xmath468 ( in the standard partial ordering of multi - indices ) ; we shall assume the notation of lemma [ lem : polynomialrepresentation ] .    1 . [ item : scaling1 ] for any @xmath316 such that @xmath469 , there exist positive constants @xmath348 and @xmath470 for which @xmath471 for all @xmath472 . [ item : scaling2 ] if @xmath473 , there exist positive constants @xmath348 and @xmath470 for which @xmath474 for all @xmath475 . [ item : scaling3 ] if @xmath473 and @xmath476 , then for every @xmath477 there exists a positive constant @xmath348 for which @xmath478 for all @xmath475 .    assuming the notation of lemma [ lem : polynomialrepresentation ] , let @xmath479 and consider the contracting group @xmath480 on @xmath481 . because @xmath414 is a positive - definite polynomial , it immediately follows that @xmath482 is positive - definite . let @xmath312 be a norm on @xmath481 and respectively denote by @xmath483 and @xmath313 the corresponding unit ball and unit sphere in this norm .    to see item [ item : scaling1 ] , first observe that @xmath484 now , for any @xmath485 , because @xmath486 is contracting , it follows from the intermediate value theorem that , for some @xmath487 , @xmath488 . correspondingly , @xmath489 because @xmath490 . one obtains the constant @xmath470 and hence the desired inequality by simply noting that @xmath491 is bounded for all @xmath492 .    for item [ item : scaling2 ] , we use analogous reasoning to obtain a positive constant @xmath348 for which @xmath493 for all @xmath494 . now , for any non - zero @xmath495 , the intermediate value theorem gives @xmath18 for which @xmath496 and hence @xmath497 where we have used the fact that @xmath498 and that @xmath499 . as this inequality must also trivially hold at the origin , we can conclude that it holds for all @xmath472 , as desired . finally , we prove item [ item : scaling3 ] . by virtue of item [ item : scaling2 ] , for any @xmath472 and @xmath18 , @xmath500 noting that @xmath501 because @xmath502 , we can make the coefficient of @xmath503 arbitrarily small by choosing @xmath32 sufficiently large and thereby obtaining the desired result .      throughout the remainder of this article , @xmath274 will denote a fixed basis for @xmath96 and correspondingly we henceforth assume the notational conventions appearing in proposition [ prop : operatorrepresentation ] and @xmath233 is fixed . for @xmath504 , consider the homogeneous norm @xmath505 defined by @xmath506 for @xmath100 where @xmath507 . as one can easily check , @xmath508 for all @xmath18 and @xmath100 where @xmath509 is defined by . [ def : consistent ] let @xmath29 . we say that @xmath504 is consistent with @xmath236 if @xmath510 for some @xmath511 . as one can check , @xmath512 is consistent with @xmath236 if and only if @xmath513 where @xmath514 is defined by . let @xmath515 and let @xmath516 . we say that @xmath16 is @xmath274-hlder continuous on @xmath122 if for some @xmath517 and positive constant @xmath348 , @xmath518 . in this case we will say that @xmath512 is the @xmath274-hlder exponent of @xmath16 . if @xmath519 we will simply say that @xmath16 is @xmath274-hlder continuous with exponent @xmath512 . the following proposition essentially states that , for bounded functions , hlder continuity is a local property ; its proof is straightforward and is omitted . let @xmath520 be open and non - empty . if @xmath16 is bounded and @xmath274-hlder continuous of order @xmath517 , then , for any @xmath521 , @xmath16 is also @xmath274-hlder continuous of order @xmath254 .    in view of the proposition , we immediately obtain the following corollary . [ cor : holderconsistent ] let @xmath522 be open and non - empty and @xmath29 . if @xmath16 is bounded and @xmath274-hlder continuous on @xmath122 of order @xmath523 , there exists @xmath517 which is consistent with @xmath236 for which @xmath16 is also @xmath274-hlder continuous of order @xmath512 . the statement follows from the proposition by choosing any @xmath512 , consistent with @xmath236 , such that @xmath524 . the following definition captures the minimal regularity we will require of fundamental solutions to the heat equation . let @xmath288 , @xmath274 be a basis of @xmath96 and let @xmath525 be a non - empty open subset of @xmath526\times\mathbb{v}$ ] . a function @xmath333 is said to be @xmath527-regular on @xmath525 if on @xmath525 it is continuously differentiable in @xmath32 and has continuous ( spatial ) partial derivatives @xmath528 for all multi - indices @xmath254 for which @xmath529 .      throughout this section , @xmath414 is the real part of the symbol @xmath146 of a positive - homogeneous operator @xmath11 on @xmath96 . we assume the notation of proposition [ prop : legendretransformproperties ] ( and hence proposition [ prop : operatorrepresentation ] ) and write @xmath530 . let us first record two important results which follow essentially from proposition [ prop : legendretransformproperties ] . [ cor : legendrecomparehomogeneousnorm ] @xmath531 where @xmath514 was defined in .    in view of propositions [ prop : operatorrepresentation ] and [ prop : legendretransformproperties ] , @xmath532 . after recalling that @xmath465 is contracting , proposition [ prop : comparepoly ] yields the desired result immediately .    by virtue of proposition [ prop : legendretransformproperties ] , standard arguments immediately yield the following corollary . [ cor : exponentialoflegendretransformintegrable ] for any @xmath477 and polynomial @xmath533 , i.e. , @xmath208 is a polynomial in any coordinate system , then @xmath534    [ lem : prelegendresubscale ] let @xmath535 . then for any @xmath536 , there exists @xmath537 such that @xmath538 for all @xmath100 and @xmath539 .    in view of corollary [ cor : legendrecomparehomogeneousnorm ] , it suffices to prove the statement @xmath540 for all @xmath100 and @xmath539 where @xmath537 and @xmath514 is given by . but for any @xmath539 and @xmath100 , @xmath541 from which the result follows . [ lem : legendresubscale ] let @xmath517 be consistent with @xmath236 . then there exists positive constants @xmath542 and @xmath543 such that @xmath544 and for any @xmath536 there exists @xmath537 such that @xmath545 for all @xmath100 and @xmath539 .    by an appeal to corollary [ cor : legendrecomparehomogeneousnorm ] and lemma [ lem : prelegendresubscale ] , @xmath546 for all @xmath100 and @xmath539 . since @xmath512 is consistent with @xmath236 , @xmath513 where @xmath547 is that of definition [ def : consistent ] , the desired inequality follows by setting @xmath548 and @xmath549 . because @xmath550 , it is necessary that @xmath551 whence @xmath552    the following corollary is an immediate application of lemma [ lem : legendresubscale ] . [ cor : holderlegendreestimate ] let @xmath332 be @xmath274-hlder continuous with exponent @xmath517 and suppose that @xmath512 is consistent with @xmath236 . then there exist positive constants @xmath542 and @xmath543 such that @xmath544 and , for any @xmath536 , there exists @xmath537 such that @xmath553 for all @xmath352 and @xmath539 . in this section , we introduce a class of variable - coefficient operators on @xmath96 whose heat equations are studied in the next section . these operators , in view of proposition [ prop : operatorrepresentation ] , generalize the class of positive - homogeneous operators . fix a basis @xmath274 of @xmath96 , @xmath29 and , in the notation of the previous section , consider a differential operator @xmath126 of the form @xmath554 where the coefficients @xmath555 are bounded functions . the symbol of @xmath126 , @xmath556 , is defined by @xmath557 for @xmath558 and @xmath101 . we shall call @xmath559 the principal part of @xmath126 and correspondingly , @xmath560 is its principal symbol . let s also define @xmath561 by @xmath562 for @xmath101 . at times , we will freeze the coefficients of @xmath126 and @xmath559 at a point @xmath558 and consider the constant - coefficient operators they define , namely @xmath563 and @xmath564 ( defined in the obvious way ) . we note that , for each @xmath558 , @xmath564 is homogeneous with respect to the one - parameter group @xmath139 where @xmath565 is defined by . that is , @xmath559 is homogeneous with respect to the same one - parameter group of dilations at each point in space . this also allows us to uniquely define the _ homogeneous order of _ @xmath126 by @xmath566 we remark that this is consistent with our definition of homogeneous - order for constant - coefficient operators and we remind the reader that this notion differs from the usual order a partial differential operator ( see the discussion surrounding ) . as in the constant - coefficient setting , @xmath564 is not necessarily homogeneous with respect to a unique group of dilations , i.e. , it is possible that @xmath567 contains members of @xmath568 distinct from @xmath179 . however , we shall henceforth only work with the endomorphism @xmath179 , defined above , for worrying about this non - uniqueness of dilations does not aid our understanding nor will it sharpen our results . let us further observe that , for each @xmath558 , @xmath569 and @xmath414 are homogeneous with respect to @xmath570 where @xmath571 . the operator @xmath126 is called @xmath95-positive - semi - elliptic if for all @xmath558 , @xmath572 is a positive - definite polynomial . @xmath126 is called uniformly @xmath95-positive - semi - elliptic if it is @xmath95-positive - semi - elliptic and there exists @xmath573 for which @xmath574 for all @xmath558 and @xmath101 . when the context is clear , we will simply say that @xmath126 is positive - semi - elliptic and uniformly positive - semi - elliptic respectively .    in light of the above definition , a semi - elliptic operator @xmath126 is one that , at every point @xmath558 , its frozen - coefficient principal part @xmath564 , is a constant - coefficient positive - homogeneous operator which is homogeneous with respect to the same one - parameter group of dilations on @xmath96 . a uniformly positive - semi - elliptic operator is one that is semi - elliptic and is uniformly comparable to a constant - coefficient positive - homogeneous operator , namely @xmath575 . in this way , positive - homogeneous operators take a central role in this theory .    in view of proposition [ prop : operatorrepresentation ] , the definition of @xmath414 via agrees with that we have given for constant - coefficient positive - homogeneous operators .    for an @xmath95-positive - semi - elliptic operator @xmath126 , uniform semi - ellipticity can be formulated in terms of @xmath576 for any @xmath577 ; such a notion is equivalent in view of proposition [ prop : comparepoly ] . for a uniformly positive - semi - elliptic operator @xmath126 , we are interested in constructing a fundamental solution to the heat equation , @xmath578 on the cylinder @xmath526\times \mathbb{v}$ ] ; here and throughout @xmath536 is arbitrary but fixed . by definition , a fundamental solution to on @xmath526\times\mathbb{v}$ ] is a function @xmath579\times\mathbb{v}\times\mathbb{v}\rightarrow \mathbb{c}$ ] satisfying the following two properties :      given a fundamental solution @xmath584 to , one can easily solve the cauchy problem : given @xmath582 , find @xmath333 satisfying @xmath585 this is , of course , solved by putting @xmath586 for @xmath100 and @xmath539 and interpreting @xmath587 as that defined by the limit of @xmath333 as @xmath588 . the remainder of this paper is essentially dedicated to establishing the following result :    [ thm : fundamentalsolution ] let @xmath126 be uniformly @xmath95-positive - semi - elliptic with bounded @xmath274-hlder continuous coefficients . let @xmath414 and @xmath589 be defined by and respectively and denote by @xmath420 the legendre - fenchel transform of @xmath414 . then , for any @xmath536 , there exists a fundamental solution @xmath579\times\mathbb{v}\times\mathbb{v}\rightarrow \mathbb{c}$ ] to on @xmath526\times\mathbb{v}$ ] such that , for some positive constants @xmath169 and @xmath348 , latexmath:[\[\label{eq : fundamentalsolution }      we remark that , by definition , the fundamental solution @xmath584 given by theorem [ thm : fundamentalsolution ] is @xmath95-regular . thus @xmath584 is necessarily continuously differentiable in @xmath32 and has continuous spatial derivatives of all orders @xmath254 such that @xmath591 . + as we previously mentioned , the result above is implied by the work of s. d. eidelman for @xmath0-parabolic systems on @xmath26 ( where @xmath592 ) @xcite . eidelman s systems , of the form , are slightly more general than we have considered here , for their coefficients are also allowed to depend on @xmath32 ( but in a uniformly hlder continuous way ) . admitting this @xmath32-dependence is a relatively straightforward matter and , for simplicity of presentation , we have not included it ( see remark [ rmk : gp ] ) . in this slightly more general situation , stated in @xmath26 and in which @xmath593 is the standard euclidean basis , theorem 2.2 ( p.79 ) @xcite guarantees the existence of a fundamental solution @xmath594 to , which has the same regularity appearing in theorem [ thm : fundamentalsolution ] and satisfies latexmath:[\[\label{eq : eidelmansestimate }    for @xmath371 and @xmath539 where @xmath169 and @xmath348 are positive constants . by an appeal to corollary [ cor : legendrecomparehomogeneousnorm ] , we have @xmath596 and from this we see that the estimates and are comparable . + in view of corollary [ cor : holderconsistent ] , the hypothesis of theorem [ thm : fundamentalsolution ] concerning the coefficients of @xmath126 immediately imply the following a priori stronger condition :    [ hypoth : holdercoefficients ] there exists @xmath517 which is consistent with @xmath236 and for which the coefficients of @xmath126 are bounded and @xmath274-hlder continuous on @xmath96 of order @xmath512 .      in this subsection , we construct a fundamental solution to under only the assumption that @xmath126 , a uniformly @xmath95-positive - semi - elliptic operator , satisfies hypothesis [ hypoth : holdercoefficients ] . henceforth , all statements include hypothesis [ hypoth : holdercoefficients ] without explicit mention . we follow the famous method of e. e. levi , c.f . , @xcite as it was adopted for parabolic systems in @xcite and @xcite . although well - known , levi s method is lengthy and tedious and we will break it into three steps . let s motivate these steps by first discussing the heuristics of the method . + we start by considering the auxiliary equation      where @xmath558 is treated as a parameter . this is the so - called frozen - coefficient heat equation . as one easily checks , for each @xmath558 , @xmath598 solves . by the uniform semi - ellipticity of @xmath126 , it is clear that @xmath599 for @xmath18 and @xmath558 . as we shall see , more is true : @xmath600 is an approximate identity in the sense that @xmath601 for all @xmath582 . thus , it is reasonable to seek a fundamental solution to of the form @xmath602 where @xmath41 is to be chosen to ensure that the correction term @xmath118 is @xmath95-regular , accounts for the fact that @xmath600 solves but not , and is `` small enough '' as @xmath603 so that the approximate identity aspect of @xmath584 is inherited directly from @xmath600 . + assuming for the moment that @xmath118 is sufficiently regular , let s apply the heat operator to with the goal of finding an appropriate @xmath41 to ensure that @xmath584 is a solution to . putting @xmath604 we have formally , @xmath605 where we have made use of leibniz rule and our assertion that @xmath600 is an approximate identity . thus , for @xmath584 to satisfy , @xmath41 must satisfy the integral equation @xmath606 viewing @xmath607 as a linear integral operator , is the equation @xmath608 which has the solution @xmath609 provided the series converges in an appropriate sense . + taking the above as purely formal , our construction will proceed as follows : we first establish estimates for @xmath600 and show that @xmath600 is an approximate identity ; this is step 1 . in step 2 , we will define @xmath41 by and , after deducing some subtle estimates , show that @xmath41 s defining series converges whence is satisfied . finally in step @xmath610 , we will make use of the estimates from steps 1 and 2 to validate the formal calculation made in . everything will be then pieced together to show that @xmath584 , defined by , is a fundamental solution to . our entire construction depends on obtaining precise estimates for @xmath600 and for this we will rely heavily on the homogeneity of @xmath560 and the legendre - fenchel transform of @xmath414 . +    [ rmk : gp ] one can allow the coefficients of @xmath126 to also depend on @xmath32 in a uniformly continuous way , and levi s method pushes though by instead taking @xmath600 as the solution to a frozen - coefficient initial value problem @xcite . +      the lemma below is a basic building block used in our construction of a fundamental solution to via levi s method and it makes essential use of the uniform semi - ellipticity of @xmath126 . we note however that the precise form of the constants obtained , as they depend on @xmath265 and @xmath254 , are more detailed than needed for the method to work . also , the partial differential operators @xmath279 of the lemma are understood to act of the @xmath31 variable of @xmath611 . [ lem : legendreestimate ] there exist positive constants @xmath348 and @xmath347 and , for each multi - index @xmath254 , a positive constant @xmath349 such that , for any @xmath350 , latexmath:[\[\label{eq : legendreestimate1 }      before proving the lemma , let us note that @xmath613 for all @xmath18 and @xmath100 in view of proposition [ prop : legendretransformproperties ] . thus the estimate can be written equivalently as latexmath:[\[\label{eq : legendreestimate11 }      let us first observe that , for each @xmath352 and @xmath18 , @xmath615 where we have used the homogeneity of @xmath560 with respect to @xmath616 and the fact that @xmath617 . therefore @xmath618 for all @xmath352 and @xmath18 . thus , to establish ( equivalently ) it suffices to estimate the right hand side of which is independent of @xmath32 . the proof of the desired estimate requires making a complex change of variables and for this reason we will work with the complexification of @xmath98 , whose members are denoted by @xmath619 for @xmath620 ; this space is isomorphic to @xmath621 . we claim that there are positive constants @xmath622 and , for each multi - index @xmath254 , a positive constant @xmath349 such that , for each @xmath350 , latexmath:[\[\label{eq : legendreestimate3 }    for all @xmath620 and @xmath558 . let us first observe that @xmath624 for all @xmath625 and @xmath558 , where @xmath626 are bounded functions of @xmath358 arising from the coefficients of @xmath126 and the coefficients of the multinomial expansion . by virtue of the uniform semi - ellipticity of @xmath126 and the boundedness of the coefficients , we have @xmath627 for all @xmath620 and @xmath558 where @xmath169 is a positive constant . by applying lemma [ lem : scaling ] to each term @xmath491 in the summation , we can find a positive constant @xmath348 for which the entire summation is bounded above by @xmath628 for all @xmath620 . by setting @xmath629 , we have @xmath630 for all @xmath620 and @xmath558 . by analogous reasoning ( making use of item 1 of lemma [ lem : scaling ] ) , there exists a positive constant @xmath169 for which @xmath631 for all @xmath620 and @xmath558 . thus , for any @xmath350 , latexmath:[\[\label{eq : legendreestimate5 }    for all @xmath620 and @xmath558 where @xmath633 . finally , for each multi - index @xmath254 , another application of lemma [ lem : scaling ] gives @xmath634 for which @xmath635 for all @xmath620 where @xmath636 has been chosen to satisfy @xmath637 . consequently , there is a positive constant @xmath349 for which latexmath:[\[\label{eq : legendreestimate6 }    @xmath620 . upon combining , and , we obtain the inequality @xmath639 which holds for all @xmath620 and @xmath558 . upon paying careful attention to the way in which our constants were chosen , we observe the claim is established by setting @xmath640 . from the claim above , it follows that , for any @xmath641 and @xmath558 , the following change of coordinates by means of a @xmath621 contour integral is justified : @xmath642 thus , by virtue of the estimate , @xmath643 for all @xmath352 and @xmath641 where we have absorbed the integral of @xmath644 into @xmath349 . upon minimizing with respect to @xmath641 , we have @xmath645 for all @xmath31 and @xmath558 because @xmath646 in this we see the natural appearance of the legendre - fenchel transform . the replacement of @xmath647 by @xmath648 is done using corollary [ cor : movingconstants ] and , as required , the constant @xmath348 is independent of @xmath265 and @xmath254 . upon combining and , we obtain the desired estimate .      given a positive - homogeneous operator @xmath11 , we invoke proposition [ prop : operatorrepresentation ] to obtain @xmath274 and @xmath236 for which @xmath649 . in other words , @xmath11 is an @xmath95-positive - semi - elliptic operator which consists only of its principal part . consequently , the heat kernel @xmath61 satisfies @xmath650 for all @xmath100 and @xmath18 and so we immediately obtain the estimate from the lemma . [ lem : legendreholderestimate ] there is a positive constant @xmath348 and , to each multi - index @xmath254 , a positive constant @xmath349 such that @xmath651|\leq c_{\beta } t^{-(\mu_h+|\beta:2\mathbf{m}|)}|h|_{\mathbf{v}}^{\alpha}\exp(-tmr^{\#}(x / t))\ ] ] for all @xmath18 , @xmath652 . here , in view of hypothesis [ hypoth : holdercoefficients ] , @xmath512 is the @xmath274-hlder continuity exponent for the coefficients of @xmath126 . [ lem : approximateidentity ] suppose that @xmath653\times\mathbb{v})$ ] where @xmath654 . then , on any compact set @xmath655\times\mathbb{v}$ ] , @xmath656 uniformly on @xmath208 as @xmath603 . in particular , for any @xmath582 , @xmath657 uniformly on all compact subsets of @xmath96 as @xmath603 .    let @xmath208 be a compact subset of @xmath658\times\mathbb{v}$ ] and write @xmath659g(s - t , y)\,dy\\ & : = & i_t^{(1)}(s , x)+i_t^{(2)}(s , x).\end{aligned}\ ] ] let @xmath477 and , in view of corollary [ cor : exponentialoflegendretransformintegrable ] , let @xmath357 be a compact subset of @xmath96 for which @xmath660 where the constant @xmath348 is that given in of lemma [ lem : legendreestimate ] . using the continuity of @xmath167 , we have for sufficiently small @xmath18 , @xmath661 we note that , for any @xmath18 and @xmath100 , @xmath662 appealing to lemma [ lem : legendreestimate ] we have , for any @xmath663 , @xmath664 here we have made the change of variables : @xmath665 and used the homogeneity of @xmath560 to see that @xmath666 . therefore @xmath667 uniformly on @xmath208 as @xmath603 . let us now consider @xmath668 . with the help of lemmas [ lem : legendresubscale ] and [ lem : legendreholderestimate ] and by making similar arguments to those above we have @xmath669 for all @xmath670 $ ] , @xmath671 and @xmath100 ; here @xmath544 . @xmath672 uniformly on @xmath208 as @xmath603 and the lemma is proved .        * step 2 . construction of @xmath41 and the integral equation * + for @xmath18 and @xmath352 , put @xmath674 and iteratively define @xmath675 where @xmath676 . in the sense of , note that @xmath677 . + we claim that for some @xmath678 and positive constants @xmath169 and @xmath348 , latexmath:[\[\label{eq : k1bound }    all @xmath352 and @xmath539 . indeed , observe that @xmath680 for all @xmath352 and @xmath18 where we have used the fact that the coefficients of @xmath126 are bounded . in view of lemma [ lem : legendreestimate ] , we have @xmath681 for all @xmath352 and @xmath539 where @xmath682 using hypothesis [ hypoth : holdercoefficients ] , an appeal to corollary [ cor : holderlegendreestimate ] gives @xmath544 and @xmath683 for which @xmath684 for all @xmath352 and @xmath539 . our claim is then justified by setting @xmath685 and adjusting the constants @xmath169 and @xmath348 appropriately to absorb the prefactor @xmath686 into the exponent . it should be noted that the constant @xmath687 is inherently dependent on @xmath126 . for it is clear that @xmath688 depends on @xmath126 . the constants @xmath542 and @xmath543 are specified in lemma [ lem : legendresubscale ] and are defined in terms of the hlder exponent of the coefficients of @xmath126 and the weight @xmath236 . + taking cues from our heuristic discussion , we will soon form a series whose summands are the functions @xmath689 for @xmath690 in order to talk about the convergence of this series , our next task is to estimate these functions and in doing this we will observe two separate behaviors : a finite number of terms will exhibit singularities in @xmath32 at the origin ; the remainder of the terms will be absent of such singularities and will be estimated with the help of the gamma function . we first address the terms with the singularities . [ lem : kfiniteterms ] let @xmath678 be given by and @xmath537 be any constant for which is satisfied . for any positive natural number @xmath54 such that @xmath691 and @xmath477 for which @xmath692 , there is a constant @xmath693 such that @xmath694 for all @xmath352 and @xmath539 .    in view of , it is clear that the estimate holds when @xmath695 . let us assume the estimate holds for @xmath696 such that @xmath697 and @xmath477 for which @xmath698 . then @xmath699 for @xmath352 and @xmath539 where we have set @xmath700 . observe that @xmath701 for all @xmath702 and @xmath703 . using the fact that @xmath704 , guarantees that @xmath705 or equivalently @xmath706 for all @xmath702 and @xmath707 . combining and yields @xmath708\end{aligned}\ ] ] for all @xmath352 and @xmath539 , where we have used the fact that @xmath420 is non - negative . let us focus our attention on the first term above . for @xmath709 , @xmath710 because @xmath711 . consequently , @xmath712 for all @xmath558 and @xmath18 . we note that the second inequality is justified by making the change of variables @xmath713 ( thus canceling the term @xmath714 in the integral over @xmath715 ) and the final inequality is valid because @xmath716 . by similar reasoning , we obtain @xmath717 for all @xmath100 and @xmath18 . upon combining the estimates , and , we have @xmath718 for all @xmath352 and @xmath539 where we have put @xmath719 and made use of corollary [ cor : exponentialoflegendretransformintegrable ] . the estimate is an important one and will be used again . in the context of elliptic operators , i.e. , where @xmath368 , the analogous result is captured in lemma 5.1 of @xcite . it is interesting to note that s. d. eidelman worked somewhat harder to prove it . perhaps this is because the appearance of the legendre - fenchel transform was nt noticed . it is clear from the previous lemma that for sufficiently large @xmath54 , @xmath689 is bounded by a positive power of @xmath32 . the first such @xmath54 is @xmath720 . in view of the previous lemma , @xmath721 for all @xmath352 and @xmath539 where we have adjusted @xmath722 to account for this positive power of @xmath32 . let @xmath723 and set @xmath724 upon combining preceding estimate with the estimates and , we have @xmath725 for all @xmath352 and @xmath539 where @xmath726 let us take this a little further .        the euler - beta function @xmath730 satisfies the well - known identity @xmath731 . using this identity , one quickly obtains the estimate @xmath732 it therefore suffices to prove that latexmath:[\[\label{eq : kseriesboundeulerbeta }      we first note that @xmath734 and so , for @xmath735 , follows directly from the calculation proceeding the lemma . we shall induct on @xmath265 . by another application of and , we have @xmath736^{-1}|k_{\bar n+k+1}(t , x , y)|\\ & \leq & \int_{0}^t\int_{\mathbb{v}}(t - s)^{-(\mu_h+(1-\rho))}s^{-k\rho}\exp(-mr^{\#}((t - s)^{-e}(x - z)))\\ & & \hspace{6cm}\times\exp(-m_1r^{\#}(s^{-e}(z - y)))\,dz\,ds\\ & \leq & \exp(-m_1r^{\#}(t^{-e}(x - y)))\\ & & \hspace{1cm}\times\int_{0}^t\int_{\mathbb{v}}(t - s)^{-(\mu_h+(1-\rho))}s^{-k\rho}\exp(-m\delta r^{\#}((t - s)^{-e}(x - z)))\,dz\,ds\end{aligned}\ ] ] for all @xmath352 and @xmath539 . upon making the changes of variables @xmath737 followed by @xmath738 , we have @xmath739 for all @xmath352 and @xmath539 . therefore holds for @xmath740 as required . [ prop : integralidentity ] let @xmath741\times\mathbb{v}\times\mathbb{v}\rightarrow\mathbb{c}$ ] be defined by @xmath742 this series converges uniformly for @xmath352 and @xmath743 where @xmath744 is any positive constant . there exists @xmath745 for which latexmath:[\[\label{eq : phiestimate }      using lemmas [ lem : kfiniteterms ] and [ lem : kseries ] we see that @xmath749\exp(-m_1r^{\#}(t^{-e}(x - y)))\ ] ] for all @xmath352 and @xmath539 from which and our assertion concerning uniform convergence follow . a similar calculation and an application of tonelli s theorem justify the following use of fubini s theorem : for @xmath352 and @xmath539 , @xmath750 as desired . the following hlder continuity estimate for @xmath41 is obtained by first showing the analogous estimate for @xmath357 and then deducing the desired result from the integral formula . as the proof is similar in character to those of the preceding two lemmas , we omit it . a full proof can be found in @xcite . we also note here that the result is stronger than is required for our purposes ( see its use in the proof of lemma [ lem : wproperties ] ) . all that is really required is that @xmath751 satisfies the hypotheses ( for @xmath16 ) in lemma [ lem : integraldifferentiation ] for each @xmath558 . [ lem : integraldifferentiation ] let @xmath517 be consistent with @xmath236 and , for @xmath754 , let @xmath755\times\mathbb{v}\rightarrow\mathbb{c}$ ] be bounded and continuous . moreover , suppose that @xmath16 is uniformly @xmath274-hlder continuous in @xmath31 on @xmath756\times\mathbb{v}$ ] of order @xmath512 , by which we mean that there is a constant @xmath757 such that @xmath758}|f(t , x)-f(t , y)|\leq c|x - y|_{\mathbf{v}}^{\alpha}\ ] ] for all @xmath352 . then @xmath759\times\mathbb{v}\rightarrow \mathbb{c}$ ] defined by @xmath760 is @xmath95-regular on @xmath761 . moreover , @xmath762 and for any @xmath254 such that @xmath591 , we have @xmath763 for @xmath100 and @xmath764 .    before starting the proof , let us observe that , for each multi - index @xmath254 , @xmath765 using the assumption that @xmath16 is bounded , we observe that @xmath766 for all @xmath743 and @xmath100 . when @xmath767 , @xmath768 converges and consequently @xmath769 for all @xmath743 and @xmath100 . from this it follows that @xmath528 is continuous on @xmath761 and moreover holds for such an @xmath254 in view of lebesgue s dominated convergence theorem . when @xmath473 , does not converge and hence the above argument fails . the main issue in the proof below centers around using @xmath274-hlder continuity to remove this obstacle . our argument proceeds in two steps . the fist step deals with the spatial derivatives of @xmath770 . therein , we prove the asserted @xmath31-regularity and show that the formula holds . in fact , we only need to consider the case where @xmath473 ; the case where @xmath767 was already treated in the paragraph proceeding the proof . in the second step , we address the time derivative of @xmath770 . as we will see , and the asserted @xmath32-regularity are partial consequences of the results proved in step 1 ; this is , in part , due to the fact that the time derivative of @xmath600 can be exchanged for spatial derivatives . the regularity shown in the two steps together will automatically ensure that @xmath770 is @xmath95-regular on @xmath761 . + _ step 1 . _ let @xmath254 be such that @xmath473 . for @xmath771 write @xmath772 and observe that @xmath773 for all @xmath774 and @xmath100 ; it is clear that @xmath775 is continuous in @xmath32 and @xmath31 . the fact that we can differentiate under the integral sign is justified because @xmath32 has been replaced by @xmath776 and hence the singularity in is avoided in the upper limit . we will show that @xmath775 converges uniformly on all compact subsets of @xmath761 as @xmath777 . this , of course , guarantees that @xmath778 exists , satisfies and is continuous on @xmath761 . to this end , let us write @xmath779 using our hypotheses , corollary [ cor : holderconsistent ] and lemma [ lem : legendresubscale ] , for some @xmath544 and @xmath683 , there is @xmath537 such that @xmath780 for all @xmath781 , @xmath782 $ ] and @xmath783 $ ] . in view of the preceding estimate and lemma [ lem : legendreestimate ] , we have @xmath784 for all @xmath781 , @xmath782 $ ] and @xmath783 $ ] , where @xmath169 and @xmath348 are positive constants . we then observe that @xmath785 for all @xmath782 $ ] and @xmath786 , where the validity of the second inequality is seen by making the change of variables @xmath787 and canceling the term @xmath788 . consequently , @xmath789 exists for each @xmath782 $ ] and @xmath100 . moreover , for all @xmath774 and @xmath100 , @xmath790 from this we see that @xmath791 converges uniformly on all compact subsets of @xmath761 . we claim that for some @xmath678 , there exists @xmath757 such that @xmath792 for all @xmath100 and @xmath783 $ ] . indeed , @xmath793\big\vert_{y = x}\,dz+\big [ d_{\mathbf{v}}^{\beta}\int_{\mathbb{v}}g_p(t - s , x - z;y)\,dz\big]\big\vert_{y = x}.\end{aligned}\ ] ] the first term above is estimated with the help of lemma [ lem : legendreholderestimate ] and by making arguments analogous to those in the previous paragraph ; the appearance of @xmath687 follows from an obvious application of lemma [ lem : legendresubscale ] . this term is bounded by @xmath794 . the second term is clearly zero and so our claim is justified .    by essentially repeating the arguments made for @xmath795 and making use of , we see that @xmath796 where this limit converges uniformly on all compact subsets of @xmath761 . it follows from leibnitz rule that @xmath797 for all @xmath798 and @xmath100 . now , in view of lemma [ lem : approximateidentity ] and our hypotheses concerning @xmath16 , @xmath799 where this limit converges uniformly on all compact subsets of @xmath761 .    using the fact that @xmath800 , we see that @xmath801 for all @xmath802 and @xmath100 . because the coefficients of @xmath126 are @xmath274-hlder continuous and bounded , for each @xmath254 , @xmath803 satisfies the same condition we have required for @xmath16 and so , in view of step 1 , it follows that @xmath804 converges uniformly on all compact subsets of @xmath761 as @xmath777 . we thus conclude that @xmath805 exists , is continuous on @xmath761 and satisfies . [ lem : wproperties ] let @xmath806\times\mathbb{v}\times\mathbb{v}\rightarrow \mathbb{c}$ ] be defined by @xmath807 for @xmath352 and @xmath539 . then , for each @xmath558 , @xmath808 is @xmath95-regular on @xmath581 and satisfies @xmath809 for all @xmath352 and @xmath810 . moreover , there are positive constants @xmath169 and @xmath348 for which latexmath:[\[\label{eq : wproperties2 }      the estimate follows from and by an analogous computation to that done in the proof of lemma [ lem : kfiniteterms ] . it remains to show that , for each @xmath558 , @xmath808 is @xmath95-regular and satisfies on @xmath581 . these are both local properties and , as such , it suffices to examine them on @xmath761 for an arbitrary but fixed @xmath812 . let us write @xmath813 for @xmath352 and @xmath814 . in view of lemmas [ holderphilemma ] and [ lem : integraldifferentiation ] , for each @xmath558 , @xmath815 is @xmath95-regular on @xmath761 and @xmath816 for all @xmath100 and @xmath814 ; here we have used the fact that @xmath817 treating @xmath818 is easier because @xmath819 and , for each multi - index @xmath254 , @xmath820 are , as functions of @xmath715 and @xmath821 , absolutely integrable on @xmath822\times \mathbb{v}$ ] for every @xmath823 $ ] and @xmath100 by virtue of lemma [ lem : legendreestimate ] . consequently , derivatives may be taken under the integral sign and so it follows that , for each @xmath558 , @xmath824 is @xmath95-regular on @xmath761 and @xmath825 for @xmath100 and @xmath826 . we can thus conclude that , for each @xmath558 , @xmath808 is @xmath95-regular on @xmath761 and , by combining and , @xmath827 for @xmath100 and @xmath826 . by , proposition [ prop : integralidentity ] and the dominated convergence theorem , @xmath828 and therefore @xmath829 for all @xmath352 and @xmath814 . the theorem below is our main result . it is a more refined version of theorem [ thm : fundamentalsolution ] because it gives an explicit formula for the fundamental solution @xmath584 ; in particular theorem [ thm : fundamentalsolution ] is an immediate consequence of the result below . let @xmath126 be a uniformly @xmath95-positive - semi - elliptic operator . if @xmath126 satisfies hypothesis [ hypoth : holdercoefficients ] then @xmath579\times\mathbb{v}\times\mathbb{v}\rightarrow \mathbb{c}$ ] , defined by @xmath830 for @xmath352 and @xmath539 , is a fundamental solution to . moreover , there are positive constants @xmath169 and @xmath348 for which latexmath:[\[\label{eq : main2 }      as @xmath678 , and lemma [ lem : legendreestimate ] imply the estimate . in view of lemma [ lem : wproperties ] and corollary [ frozenfundsolcor ] , for each @xmath558 , @xmath580 is @xmath95-regular on @xmath581 and @xmath832 for all @xmath100 and @xmath833 . it remains to show that for any @xmath582 , @xmath834 for all @xmath100 . indeed , let @xmath582 and , in view of , observe that @xmath835 for all @xmath100 and @xmath539 . an appeal to lemma [ lem : approximateidentity ] gives , for each @xmath100 , @xmath836 as required . in fact , the above argument guarantees that this convergence happens uniformly on all compact subsets of @xmath96 . we remind the reader that implicit in the definition of fundamental solution to is the condition that @xmath584 is @xmath95-regular . in fact , one can further deduce estimates for the spatial derivatives of @xmath584 , @xmath837 , of the form for all @xmath254 such that @xmath838 ( see @xcite ) . using the fact that @xmath584 satisfies and @xmath126 s coefficients are bounded , an analogous estimate is obtained for a single @xmath32 derivative of @xmath584 . evan randles : department of mathematics , university of california , los angeles , los angeles ca 90025 . e - mail : randles@math.ucla.edu + laurent saloff - coste : department of mathematics , cornell university , ithaca ny 14853 . e - mail : lsc@math.cornell.edu
we consider a class of homogeneous partial differential operators on a finite - dimensional vector space and study their associated heat kernels . 
 the heat kernels for this general class of operators are seen to arise naturally as the limiting objects of the convolution powers of complex - valued functions on the square lattice in the way that the classical heat kernel arises in the ( local ) central limit theorem . 
 these so - called positive - homogeneous operators generalize the class of semi - elliptic operators in the sense that the definition is coordinate - free . 
 more generally , we introduce a class of variable - coefficient operators , each of which is uniformly comparable to a positive - homogeneous operator , and we study the corresponding cauchy problem for the heat equation . under the assumption that such an operator has hlder continuous coefficients , we construct a fundamental solution to its heat equation by the method of e. e. levi , adapted to parabolic systems by a. friedman and s. d. eidelman . though our results in this direction are implied by the long - known results of s. d. eidelman for @xmath0-parabolic systems , 
 our focus is to highlight the role played by the legendre - fenchel transform in heat kernel estimates . 
 specifically , we show that the fundamental solution satisfies an off - diagonal estimate , i.e. , a heat kernel estimate , written in terms of the legendre - fenchel transform of the operator s principal symbol 
 an estimate which is seen to be sharp in many cases .    _ dedicated to professor rodrigo bauelos on the occasion of his 60th birthday . _ 
 * keywords : * semi - elliptic operators , quasi - elliptic operators , @xmath0-parabolic operators , heat kernel estimates , legendre - fenchel transform . + * mathematics subject classification : * primary 35h30 ; secondary 35k25 .