You are an expert at summarizing long articles. Proceed to summarize the following text: 
let @xmath2 be a linear partial differential operator on @xmath1 with constant coefficients . then a classical theorem of lions and titchmarsch states that , for every distribution @xmath6 on @xmath1 with compact support , the convex hulls of the supports of @xmath7 and @xmath6 are equal : @xmath8 by regularization , this is equivalent to stating that for every compactly supported smooth function @xmath4 on @xmath1 , the convex hulls of the supports of @xmath9 and @xmath4 are equal : @xmath10 in fact , lions @xcite proved a more general version of the support theorem , namely @xmath11 we refer to @xcite for an elementary proof of this theorem . the first version ( [ eq : supportconstantpdo ] ) obviously follows from the third one ( [ eq : suppconv ] ) by taking @xmath12 , the delta distribution at the origin . the comparison of the supports of @xmath4 and @xmath9 plays an important role at several places in analysis . one typical situation is the study of solvability of differential operators . recall that a linear partial differential operator @xmath13 with smooth coefficients @xmath14 is said to be _ solvable in @xmath1 _ , provided @xmath15 , that is , if for every @xmath16 the differential equation @xmath17 has a solution @xmath18 . the following theorem gives a necessary and sufficient condition for the solvability of @xmath2 ( see @xcite , theorem 3.3 ) . [ thm : solvable ] let @xmath2 be a linear partial differential operator with smooth coefficients in @xmath1 . then @xmath2 is solvable if and only if the following two condition are satisfied :    1 .   _ ( semi - global solvability ) _ for every compact subset @xmath19 of @xmath1 and for every @xmath16 there is a function @xmath18 so that @xmath17 on @xmath19 . ( @xmath2-convexity of @xmath1 ) _ for every compact subset @xmath19 of @xmath1 there is a compact set @xmath20 so that for every @xmath21 the inclusion @xmath22 implies @xmath23 . here @xmath24 denotes the formal transpose of @xmath2 . the support theorem of lions and titchmarsch implies that , for every linear partial differential operator @xmath2 with constant coefficients , @xmath1 is @xmath2-convex , and that we can in fact take @xmath25 . observe that , in this case , condition ( b ) also implies that the operator @xmath24 is injective on @xmath26 . the main result of this paper is theorem [ thm : maindistr ] , which provides an extension of the support theorem of lions and titchmarsch to specific ( but very large ) classes of invariant singular linear partial differential operators with real analytic coefficients and to distributions having a compact support with invariant convex hull . the invariance considered here is with respect to certain finite groups of orthogonal transformations generated by reflections . in the 2-dimensional case , examples of such groups are the groups of symmetries of regular @xmath27-agons . the principal symbols of the examined differential operators are allowed to vanish , but only in a precise way , along the reflecting hyperplanes . see formula ( [ eq : symbol ] ) below . some restrictions in generalizing the theorem of titchmarsch and lions are of course needed . the following easy example shows that the theorem can not hold for arbitrary linear differential operators with variable coefficients , even in the one - dimensional case and with real analytic coefficients . [ ex : xdx ] consider the differential operator @xmath28 on @xmath29 . let @xmath30}$ ] be the characteristic function of the interval @xmath31 $ ] . then @xmath32 , where @xmath33 denotes the delta measure at @xmath34 . therefore @xmath35 and @xmath36 $ ] are convex and different .    in the one - dimensional case , the differential operator @xmath28 of example [ ex : xdx ] belongs to the class of differential operators to which our support theorem applies . observe that @xmath2 is an even differential operator which is singular at @xmath37 . for this very special differential operator , our theorem states that @xmath38 if @xmath39 satisfies one of the following conditions :    1 . @xmath400,+\infty[$ ] , 2 . @xmath40-\infty,0[$ ] , 3 . @xmath41 is symmetric with respect to the origin @xmath42 . of course the distribution @xmath43}$ ] from example [ ex : xdx ] does not fulfill any of these conditions . the core of the proof of theorem [ thm : maindistr ] is to show that the considered situation allows us to apply holmgren s uniqueness theorem for comparing the size of the supports of @xmath4 and @xmath9 when @xmath4 is a compactly supported smooth function with the property that @xmath5 is invariant . the employ of holmgren s theorem is the reason for imposing to the coefficients of the considered differential operators to be real analytic . several other authors employed holmgren s uniqueness theorem for proving @xmath2-convexity properties ( see e.g. @xcite , @xcite , @xcite ) . remarkable is nevertheless , that the proof of our theorem is very elementary . it requires only basic facts from convex geometry and an application of equation ( [ eq : suppconv ] ) . the article consists of two parts . the first part contains the proof of theorem [ thm : maindistr ] , which does not require any knowledge of symmetric spaces . in the second part , we discuss several applications of our main results , including hypergeometric differential operators , bessel differential operators , shift operators , hamiltonian systems , and invariant differential operators on symmetric spaces . the solvability of @xmath44-invariant differential operators is one of the fundamental problems in the analysis on a symmetric space @xmath45 ( see @xcite , p. 275 ) . recall that theorem [ thm : solvable ] holds more generally if @xmath1 is replaced with a 2nd countable smooth manifold ( see @xcite , p. 14 ) . since @xmath45 is a second countable smooth manifold , one obtains the equivalence between global and semi - global solvability , provided one can prove that @xmath45 is @xmath2-convex . the solvability of invariant differential operators on riemannian symmetric space was proved by helgason @xcite . in the general pseudo - riemannian symmetric case , van den ban and schlichtkrull @xcite determined a sufficient condition for a @xmath44-invariant differential operator @xmath2 ensuring that @xmath45 is @xmath2-convex . this condition involves the degree of the polynomial which is the image of @xmath2 under the harish - chandra isomorphism . it is for instance always satisfied when @xmath45 is split , i.e. , has a vectorial cartan subspace . the riemannian symmetric spaces of the noncompact type are examples of split symmetric spaces and so are the @xmath46 space of oshima and sekiguchi , @xcite .    as a first application of theorem [ thm : maindistr ] , we deduce in section [ section : appl ] the @xmath2-convexity of riemannian symmetric spaces @xmath47 of noncompact type when @xmath2 is a @xmath44-invariant differential operator on @xmath47 . our method , which is based on taking the radial component of @xmath2 along the cartan subgroup , is different from those of used in @xcite and @xcite . support theorems play also an important role in harmonic analysis , in particular in paley - wiener type theorems . these theorems characterize the space of functions which are image , under a suitable generalization of the fourier transform , of the compactly supported smooth functions . applications in this setting appeared first in the work of van den ban and schlichtkrull on fourier transforms on pseudo - riemannian symmetric spaces @xcite . the basic idea , which we shall outline more precisely in section [ section : appl ] , is to cancel the singularities appearing in a wave packet @xmath4 by applying a suitable differential operator @xmath2 . the problem is to compare the size of the support of @xmath9 , which can be easily determined , with the hard to determine size of the support of the original wave packet @xmath4 . in fact , our need for support properties like those stated in the present paper turned up in the proof of a paley - wiener type theorem for the @xmath48-hypergeometric transform , which is a fourier type transform related to the theory of hypergeometric functions associated with root systems . see @xcite . however , we point out that theorem [ thm : maindistr ] is stated in a very general setting and applies to many different situations . see example [ ex : diffrankone ] , and the subsections [ ex : hyper ] , [ ex : bessel ] and [ ex : shift ] for several special cases . our paper is organized as follows . in section [ section : setting ] we introduce the general setting in which our extension of the theorem of lions and titchmarsch will be proved . the main results , theorems [ thm : maindistr ] and [ thm : mainfunction ] , will be stated and proved in section [ section : thmsupport ] . section [ section : examples ] presents several concrete situations where our support theorem applies . the presented examples are related to the theory of special functions associated with root systems . the last section is devoted to applications . we deduce the @xmath2-convexity of riemannian symmetric spaces of noncompact type when @xmath2 is an invariant differential operator . moreover , we describe how to employ theorem [ thm : mainfunction ] for proving of paley - wiener type theorems in the harmonic analysis on symmetric spaces and on root systems . in this section we introduce the notation and set up that will be used in this article . in particular we introduce the class of differential operators that will be considered in this article and give few examples .    in the following @xmath49 stands for an real euclidean vector space of dimension @xmath27 , i.e. , @xmath50 . furthermore @xmath2 will stand for a differential operator on @xmath49 with analytic coefficients . we denote by @xmath51 a ( positive definite ) inner product on @xmath52 . set @xmath53 . for @xmath54 we denote by @xmath55 the closed euclidean ball in @xmath52 with center @xmath42 and radius @xmath56 . let @xmath57 denote the real dual of @xmath52 . for each @xmath58 we denote by @xmath59 the unique element of @xmath52 satisfying @xmath60 for all @xmath61 . we set @xmath62 and notice that @xmath63 is independent of @xmath51 . with each @xmath64 we associate the reflection @xmath65 of @xmath52 across the hyperplane @xmath66 . thus @xmath67 a finite set @xmath68 is called a _ ( reduced ) root system _ if the following conditions holds for @xmath69 :    1 . if @xmath70 , then @xmath71 ; 2 .   if @xmath72 , then @xmath73 .    the elements of @xmath69 are called _ roots_. observe that _ we are not requiring _ that @xmath69 contains a basis of @xmath57 . in particular , our definition allows @xmath69 to be the empty set .    a subset @xmath74 of @xmath69 is called a _ simple system _ if @xmath74 is linearly independent and if any root in @xmath69 can be written as a linear combination of elements in @xmath74 in which all non - zero coefficients are either all positive or all negative . if @xmath75 , then we set @xmath76 . let a simple system @xmath74 of @xmath69 be fixed . set @xmath77 , where @xmath78 . the elements in @xmath79 are said to be positive . observe that @xmath80 for all @xmath81 . hence @xmath82 . let @xmath83 be the group of orthogonal transformations of @xmath52 generated by the reflections @xmath84 . it coincides with the group generated by @xmath85 . we set @xmath86 , if @xmath69 is empty . if @xmath87 and @xmath88 , then @xmath89 . the group @xmath0 is a _ finite coxeter group_. conversely , every finite coxeter group originates from a root system as above . see e.g. @xcite , chapter 1 . among the finite coxeter groups we find the _ weyl groups _ ( for instance the finite groups of permutations ) and the _ dihedral groups _ ( that is , the groups of symmetries of the regular @xmath27-gons ) .    the group @xmath0 acts on functions @xmath90 according to @xmath91 it also acts on compactly supported distributions @xmath92 and on differential operators @xmath2 on @xmath52 by : @xmath93 where we have written the pairing between distributions and functions by @xmath94 . the function @xmath4 ( resp . the compactly supported distribution @xmath6 or the differential operator @xmath2 ) is said to be _ @xmath0-invariant _ provided @xmath95 for all @xmath96 ( resp . @xmath97 or @xmath98 for all @xmath96 ) . for instance , if @xmath87 and @xmath99 , then @xmath0-invariant means even . more generally , let @xmath100 be a character of @xmath0 . then @xmath4 is said _ to transform under @xmath0 according to @xmath100 _ if @xmath101 for all @xmath96 . this definition extends similarly to distributions and differential operators . notice that if @xmath4 transforms under @xmath0 according to a character @xmath100 , then the support of @xmath4 is @xmath0-invariant . we will be studying differential operators with leading symbol of a specific form . we will therefore need the following definition :    let @xmath102 be a non - empty @xmath0-invariant convex open subset of @xmath52 , and @xmath103 a polynomial function . we say that @xmath104 is a _ homogeneous @xmath102-elliptic polynomial _ if the following holds :    1 . @xmath104 is a homogeneous polynomial in @xmath105 with real analytic coefficients on @xmath52 , i.e. there is @xmath106 so that @xmath107 where @xmath108 is real analytic , and @xmath109 , if @xmath110 is a multi - index . 2 .   if @xmath111 and @xmath112 , then @xmath113 .    let @xmath2 be a @xmath0-invariant linear partial differential operator on @xmath52 with real analytic coefficients . we say that @xmath2 is _ @xmath102-elliptic _ if its principal symbol is of the form @xmath114^{n({\alpha } ) } \right ) p(x,{\lambda } ) , \qquad ( x,{\lambda})\in \frak a \times \frak a^*,\ ] ] where @xmath115 is a homogeneous polynomial , @xmath116 for all @xmath70 , and @xmath117 is a homogeneous @xmath102-elliptic polynomial .    in this paper we consider @xmath102-elliptic linear partial differential operators as operators acting on functions or distributions on @xmath102 . note that the decomposition of @xmath118 in ( [ eq : symbol ] ) is in general not unique : if @xmath119 for all @xmath120 , then we can replace @xmath104 by @xmath121 , @xmath122 , and replace @xmath123 by @xmath124 . the class of @xmath102-ellipltic contains all elliptic partial differential operators on @xmath49 by taking @xmath125 and @xmath75 . but , more generally , the principal symbols of the considered operators are allowed to vanish at the zeros of @xmath115 as well as along the hyperplanes @xmath126 with @xmath70 . observe that the condition in ( [ eq : symbol ] ) imposes a restriction only on the principal part of the differential operators . in particular , suppose @xmath127 , @xmath128 are @xmath0-invariant linear partial differential operators with real analytic coefficients so that @xmath129 . if @xmath127 satisfies condition ( [ eq : symbol ] ) , then the same is true for @xmath130 .    any partial differential operator with constant coefficients @xmath131 satisfies ( [ eq : symbol ] ) when we choose @xmath132 and @xmath75 . indeed , in this case we do not impose any symmetry condition , and @xmath133 is of the form ( [ eq : symbol ] ) . [ ex : diffrankone ] suppose @xmath134 and @xmath88 . we shall identify @xmath135 with @xmath29 . the possible subsets @xmath102 are the open intervals of the form @xmath136-r , r[$ ] with @xmath137 . then the differential operators considered are the even ordinary differential operators with real analytic coefficients and with principal symbol @xmath138 here , @xmath139 , the polynomial @xmath115 is homogeneous , and @xmath117 is homogeneous @xmath102-elliptic . for @xmath140 , examples are @xmath141 and the euler operator @xmath142 where @xmath143 . other examples , which are relevant in harmonic analysis , are constructed from the jacobi and the bessel differential operators on @xmath1360,+\infty[$ ] . let @xmath144 and @xmath145 respectively denote the jacobi and the bessel differential operators , i.e. @xmath146 \frac{d}{dx}\ , , \\ l_0&=\frac{d^2}{dx^2}+ \big[a \frac{1}{x}+b \frac{1}{2x}\big ]   \frac{d}{dx}\ , . \end{aligned}\ ] ] then , for specific positive integral values of the constants @xmath147 and @xmath148 , the operator @xmath144 agrees with the radial part of the laplace - beltrami operator of a rank - one riemannian symmetric spaces @xmath47 of noncompact type , whereas @xmath145 is the radial part of the laplace - beltrami operator of the corresponding rank - one riemannian symmetric space @xmath149 of euclidean type . examples of such spaces @xmath47 are the ( real , complex or quaternionic ) hyperboloids ; examples of spaces @xmath149 are given by @xmath1 regarded as homogeneous space @xmath150 . we refer to @xcite , ch . ii , propositions 3.9 and 3.13 , for more information on radial parts of the laplace - beltrami operators . the operators @xmath151 and @xmath152 satisfy our assumptions since they are even operators on @xmath29 with real analytic coefficients , and @xmath153 and @xmath154 . generalizations in more variables of these examples will be treated in section [ section : examples ] . note that , in the first case , we can take @xmath155 or @xmath156 .    [ ex : calogero ] the calogero model is a non - relativistic quantum mechanical system of @xmath157 identical particles on a line interacting pairwise . such a system is described by the hamiltonian @xmath158 where the positive constant @xmath159 is the coupling coefficient . the associated schrdinger operator is @xmath160 see e.g. @xcite , ( 3.1.1 ) and ( 3.1.14 ) , i. see also @xcite . let @xmath161 be the standard basis of @xmath162 and let @xmath51 be the standard inner product on @xmath1 . for @xmath163 with @xmath164 set @xmath165 . then @xmath166 is a root system of type @xmath167 . we take @xmath168 as a set of positive roots . the corresponding system of simple roots is @xmath169 . the finite coxeter group @xmath0 associated to @xmath167 is the group @xmath170 of permutations of the set @xmath161 . we can now write @xmath171 where @xmath172 is the euclidean laplace operator on @xmath52 . it follows that @xmath173 is @xmath0-invariant . if @xmath174 , then the differential operator @xmath175 satisfies our requirements in ( [ eq : symbol ] ) . this example will be generalized in section [ section : examples ] . multiplication by @xmath176 and @xmath177 in example [ ex : diffrankone ] ( resp . by @xmath178 in example [ ex : calogero ] ) , which is used to obtain @xmath0-invariant differential operators with analytic coefficients in @xmath52 , is in fact inessential when dealing with even ( resp . @xmath0-invariant functions ) . indeed , despite the singularity of their coefficients , the operators @xmath144 and @xmath145 map even smooth functions on @xmath29 in smooth functions . likewise , the operator @xmath173 maps smooth @xmath0-invariant functions on @xmath52 into smooth functions . this remark will be stated , in a more general form , in section [ section : examples ] . we now introduce the class of functions that will be considered in this paper . as above , let @xmath102 be a fixed @xmath0-invariant open convex subset of @xmath52 . let @xmath179 denote the space consisting of distributions @xmath6 on @xmath52 so that @xmath41 is a @xmath0-invariant compact subset of @xmath102 , and let @xmath180 be the subspace of @xmath179 consisting of @xmath181 functions . important subspaces of @xmath180 are the spaces @xmath182 , formed by the smooth compactly supported functions @xmath183 which transform under @xmath0 according to the character @xmath100 of @xmath0 . for instance , if @xmath100 is the trivial character , then we obtain the space @xmath184 of @xmath0-invariant smooth functions on @xmath102 with compact support . similarly , inside @xmath179 we find the spaces @xmath185 of compactly supported distributions on @xmath102 which transform according to @xmath100 , and the space @xmath186 of @xmath0-invariant distributions on @xmath0 with compact support . if @xmath87 and @xmath132 , then we can identify @xmath187 with @xmath29 . suppose first @xmath75 . then @xmath0 is trivial , and @xmath188 ( resp . @xmath189 ) reduces to the space of @xmath181 functions ( resp . distributions ) on @xmath29 with compact support . if @xmath88 , then @xmath89 . in this case , @xmath188 ( resp . @xmath189 ) is the space of smooth functions ( resp . distributions ) @xmath4 on @xmath29 so that @xmath5 is a bounded interval of the form @xmath190 $ ] for some @xmath191 . the subspace @xmath192 consists of the compactly supported even smooth functions on @xmath29 ; with @xmath100 equal to the sign character , the subspace @xmath193 consists of the compactly supported odd smooth functions on @xmath29 . in this section we proof the main result of this article . this is the following version of the the support theorem of lions and titchmarsch :    [ thm : maindistr ] let @xmath194 be open , convex and @xmath0-invariant . suppose @xmath2 is a @xmath0-invariant linear partial differential operator on @xmath52 with real analytic coefficients and with principal symbol of the form ( [ eq : symbol ] ) . then @xmath195 for each @xmath196 . we shall prove below that , by regularization , theorem [ thm : maindistr ] is equivalent to the following smooth version :    [ thm : mainfunction ] let @xmath102 and @xmath2 be as in theorem [ thm : maindistr ] . then @xmath197 for each @xmath198 . equivalently , for each @xmath198 and for every compact convex @xmath0-invariant subset @xmath199 we have @xmath200    _ proof of the equivalence of theorems [ thm : maindistr ] and [ thm : mainfunction ] . _ it is clear that theorem [ thm : maindistr ] implies theorem [ thm : mainfunction ] . for the other direction , let @xmath196 . let @xmath201 be an approximate identity with @xmath202 for all @xmath54 . since @xmath102 is a @xmath0-invariant , we can , if necessary , replace @xmath203 by @xmath204 . we can therefore suppose that @xmath203 is @xmath0-invariant . it follows , in particular , that @xmath205 and hence @xmath206 are @xmath0-invariant . thus , by ( [ eq : suppconv ] ) , @xmath207 is a @xmath0-invariant compact subset of @xmath52 . since @xmath102 is open , we can choose @xmath208 so that @xmath209 for @xmath210 . the functions @xmath211 converge to @xmath6 in @xmath212 as @xmath213 , and @xmath214 . it is easy to check ( without invoking deeper support theorems ) that there exists @xmath215 , converging to @xmath42 as @xmath213 , so that @xmath216 replacing @xmath6 with @xmath7 , we obtain @xmath217 the same inclusions are preserved for the convex hulls . by theorem [ thm : mainfunction ] , we have @xmath218 hence , @xmath219 which implies that @xmath220 . similarly , @xmath221 . this proves theorem [ thm : maindistr ] .    before going into the details of the proof of theorem [ thm : mainfunction ] , let us briefly explain the main ideas involved . since @xmath222 , it suffices to show that if @xmath223 is a compact convex @xmath0-invariant subset of @xmath102 , then @xmath224 implies that @xmath225 . this will be proved by contradiction . the main tool will be holmgren s uniqueness theorem . [ thm : holmgren ] let @xmath226 be open , and let @xmath227 be a real valued function in @xmath228 . let @xmath2 be a linear partial differential operator with analytic coefficients defined in @xmath229 . let @xmath230 denote the principal symbol of @xmath2 . suppose that @xmath231 is a point in @xmath229 such that @xmath232 then there exists a neighborhood @xmath233 of @xmath231 with the following property : if the distribution @xmath234 is annihilated by @xmath2 , i.e. , @xmath235 , and @xmath6 vanishes on the set @xmath236 , then @xmath6 must also vanish in @xmath237 . this is theorem 5.3.1 in @xcite . for non - elliptic differential operators the delicate matter is to choose points @xmath231 for which condition ( [ eq : nonvanishingsymb ] ) is fulfilled . this is exactly the kind of difficulty one encounters in the proof of support theorems . coming back to the streamline ideas of the proof of theorem [ thm : mainfunction ] , let @xmath223 be as above . . then @xmath239 . to reach a contradiction , we assume that @xmath240 . let @xmath241 . as @xmath223 is convex and compact , there exists a hyperplane strictly separating @xmath231 and @xmath223 , i.e. , we can find @xmath242 such that @xmath243 without loss of generality we can also assume that @xmath244 otherwise we translate the hyperplane to the boundary of @xmath245 in the direction opposite to @xmath223 . in this way , the entire set @xmath245 lies inside a closed half - space supported by the hyperplane @xmath246 our plan is to apply holmgren s uniqueness theorem to @xmath247 , @xmath248 and @xmath249 . note that in this case @xmath250 is constant and non - zero . observe also that @xmath251 on @xmath229 and @xmath252 on the side of @xmath253 not containing @xmath223 ( which is described by the equation @xmath254 ) . if the principal symbol of @xmath2 were not zero at @xmath255 , then all assumptions would be satisfied , and we could conclude that @xmath252 in a neighborhood of @xmath231 . this would yield the required contradiction because @xmath256 .    since @xmath257 , the condition @xmath258 is equivalent to @xmath259^{n({\alpha})}\neq 0.\ ] ] this might not be satisfied by the chosen pair @xmath255 . it is even possible to have a situation where there is no choice of @xmath260 for which the above procedure could guarantee that @xmath261 . figure 1 sketches an example in which this problem arises because of the `` corner '' at the boundary of @xmath245 . note that the set @xmath245 in this example is also convex and invariant with respect to the group @xmath262 , where @xmath263 denotes the reflection with respect to the @xmath264-axis .    in case @xmath265^{n({\alpha})}=0 $ ] , the above procedure must be modified . the first step is to show that it suffices to consider the case of smooth functions @xmath4 with the property that the convex hull @xmath266 of @xmath238 is @xmath0-invariant and has @xmath267 boundary . the point is that the @xmath231 , selected as above , will always belong to the boundary @xmath268 of @xmath266 , and that @xmath253 is a supporting hyperplane for @xmath266 . for the modified procedure we need some preparations .    for @xmath54 let @xmath269 denote the closed ball centered at the origin and with radius @xmath56 . let @xmath270 be the ( closed ) @xmath56-neighborhood of @xmath223 . [ lemma : smoothbdry ] let @xmath223 be a proper convex subset of @xmath271 with nonempty interior , and let @xmath54 . then @xmath272 is an @xmath27-dimensional convex subset of @xmath52 and its boundary @xmath273 is a @xmath267 @xmath274-dimensional submanifold of @xmath52 . this is satz 17.6 in @xcite . [ lemma : withsmoothbdry ] let @xmath2 be a differential operator on @xmath52 . suppose there exists a smooth compactly supported function @xmath275 on @xmath49 and a compact convex subset @xmath276 with nonempty interior such that @xmath277 then there exists a function @xmath278 and a compact convex subset @xmath279 with nonempty interior such that @xmath280 moreover , the boundary @xmath281 of the convex hull of @xmath282 is a @xmath267-manifold . if @xmath283 , where @xmath102 is a @xmath0-invariant open convex subset of @xmath52 , and @xmath284 is @xmath0-invariant . then we can choose @xmath198 and @xmath285 to be @xmath0-invariant . let @xmath286 be an approximate identity with @xmath287 for all @xmath56 . then @xmath288 and @xmath289 notice that ( [ eq : suppconv ] ) applied to compactly supported smooth functions implies that for all @xmath290 we have @xmath291 hence , there exists @xmath208 so that @xmath292 . otherwise ( [ eq : supp ] ) would imply that @xmath293 for all @xmath54 , and hence @xmath294 . as in ( [ eq : supp ] ) , we have @xmath295 therefore , by lemma [ lemma : smoothbdry ] , @xmath296 has @xmath267-boundary . we can thus select @xmath297 and @xmath298 . finally , suppose that @xmath299 and @xmath300 are @xmath0-invariants subset of @xmath102 . since @xmath102 is open , then , by choosing a sufficiently small @xmath208 , so are also @xmath301 and @xmath302 .    in the following we shall suppose that @xmath4 and @xmath223 are chosen as in lemma [ lemma : withsmoothbdry ] . as before , we set @xmath238 and @xmath303 . we also fix @xmath304 . we now proceed to the selection of the pair @xmath255 . let @xmath305 be fixed , and let @xmath306 then @xmath307 is a nonempty open subset of @xmath57 . the separation properties of compact convex sets ensure that @xmath308 . observe that @xmath309 hence @xmath3100,+\infty[)$ ] where @xmath311 . this proves the lemma because @xmath312 is proper and convex , hence continuous ( see @xcite , theorems 5.3 and 10.1 ) . since @xmath307 is open and nonempty we can choose @xmath313 with the following properties :    1 . @xmath314 , 2 . @xmath315 for all @xmath70 ( i.e. @xmath316 for all @xmath70 ) . since @xmath317 ,    [ le : cofx0 ] let @xmath260 be as above . then there exists @xmath318 such that @xmath319 . furthermore @xmath320 and @xmath321 . finally @xmath322    this follow as @xmath266 is the convex hull of @xmath245 .    when @xmath88 , the @xmath0-invariance of the situation plays a role because of the following lemma . [ lemma : tangent ] suppose @xmath266 is a @xmath0-invariant compact convex subset of @xmath52 with nonempty interior and @xmath267-boundary . let @xmath323 and @xmath324 such that @xmath325 . let @xmath326 denote the tangent space to @xmath327 at @xmath328 ( regarded as subspace of @xmath52 ) and let @xmath63 be as in ( [ eq : xa ] ) . then @xmath329 . the reflection @xmath65 of @xmath52 across the hyperplane @xmath330 maps @xmath266 onto itself and fixes @xmath328 . hence it maps @xmath331 ( resp . @xmath327 ) onto itself . the derived involution @xmath65 is therefore an automorphism of @xmath326 . hence @xmath332 , where @xmath333 is the outer normal vector to @xmath327 at @xmath328 . since @xmath334 by invariance of @xmath266 under @xmath65 , we conclude that @xmath335 , i.e. @xmath336 . thus @xmath337 . [ prop : nonzerosym ] let @xmath231 and @xmath260 be as in lemma [ le : cofx0 ] . then @xmath255 satisfies ( [ eq : nonzerosym ] ) .    the element @xmath260 has been chosen so that @xmath314 . this suffices to prove ( [ eq : symbol ] ) when @xmath338 . if @xmath88 , then it remains to show that @xmath339 for all @xmath70 . since @xmath319 , the hyperplane @xmath340 is the supporting hyperplane for @xmath341 at @xmath231 . the set @xmath266 is @xmath0-invariant and its boundary @xmath327 is @xmath267 . hence @xmath342 . if @xmath343 for some @xmath70 , then @xmath344 by lemma [ lemma : tangent ] , i.e. @xmath345 . thus @xmath346 , which implies @xmath347 , against our choice of @xmath260 . thus @xmath339 for all @xmath81 . _ proof of theorem [ thm : mainfunction ] . arguing by contradiction , we assume that there exists @xmath348 and a @xmath0-invariant subset @xmath223 of @xmath102 as in lemma [ lemma : withsmoothbdry ] . we select @xmath255 as in proposition [ prop : nonzerosym ] . set @xmath247 and @xmath248 . hence @xmath349 is a supporting hyperplane for @xmath350 , and @xmath250 is constant and non - zero . moreover , @xmath351 on @xmath229 and @xmath252 on the side of @xmath253 not containing @xmath223 ( which is described by the equation @xmath352 ) . proposition [ prop : nonzerosym ] ensures that @xmath258 . holmgren s uniqueness theorem then implies that @xmath252 in a neighborhood of @xmath231 . this gives the the required contradiction because @xmath353 .    before concluding this section we prove some immediate consequences of theorem [ thm : maindistr ] . recall that the transpose of the partial differential operator @xmath2 given by ( [ eq : d ] ) is @xmath354 hence @xmath24 belongs to the class of invariant differential operators considered in this papers if so does @xmath2 . in fact , the principal symbols of @xmath2 and @xmath24 are linked by the relation @xmath355    [ cor : maindistr ] let @xmath2 be a @xmath0-invariant differential operators @xmath2 on @xmath52 with real analytic coefficients and with principal symbol of the form ( [ eq : symbol ] ) . then the following properties are true .    1 . @xmath2 is injective on @xmath180 . 2 .   for all @xmath198 we have @xmath356 in this section we present some examples linked to the theory of special functions associated with root systems . in these examples the group @xmath0 is a parabolic subgroup of a fixed coxeter group @xmath357 acting on @xmath358 , and the differential operator @xmath2 is in fact invariant under the larger finite coxeter group @xmath357 . the general situation corresponds to a ( not necessarily reduced ) root systems @xmath359 . a _ root system _ is a finite set @xmath360 satisfying condition ( r2 ) of section [ section : setting ] . in this section we will also assume that @xmath359 satisfies the following additional conditions :    1 . @xmath359 spans @xmath57 ; 2 . @xmath359 is _ crystallographic _ , that is @xmath361 for all @xmath362 . crystallographic root systems arise naturally in several places in algebra and analysis . in particular , they are relevant in the theory of real lie algebras , lie groups , and symmetric spaces .    if @xmath359 is a root system , then @xmath363 is a reduced root system according to the definition of section [ section : setting ] . the finite coxeter group associated to @xmath359 is the finite coxeter group @xmath357 associated to @xmath69 . it is also called the _ weyl group _ of @xmath359 . multiplicity function _ is a @xmath357-invariant function @xmath364 . for @xmath365 , we adopt the common notation @xmath366 to denote @xmath367 . the set @xmath368 of positive roots and the set @xmath74 of positive simple roots can be defined as in the case of reduced root systems . because of ( r0 ) , @xmath74 is a basis of @xmath57 . fix a @xmath74 set of simple roots in @xmath368 . for each subset @xmath48 of @xmath74 we define @xmath369 as the set of elements of @xmath359 which are linear combinations of elements from @xmath48 , i.e. , @xmath370 . it is itself a root system , but in general it does not satisfy ( r0 ) . we denote the corresponding finite coxeter group by @xmath371 . note that @xmath372 is generated by the reflections @xmath65 with @xmath373 . for instance , @xmath374 and @xmath375 . we also set @xmath376 for the set of positive roots inside @xmath369 . the coxeter group @xmath371 will play the role of the group @xmath0 of the previous sections . recall that a subgroup @xmath0 of @xmath357 is called a _ parabolic subgroup _ if it is of the form @xmath371 for some @xmath377 . the parabolic subgroups can also be characterized as those subgroups of @xmath357 that stabilize a subspace of @xmath52 . thus @xmath378 is a parabolic subgroup if and only if there exists a subspace @xmath379 such that @xmath380    a _ chamber _ in @xmath52 a connected component of @xmath381 ( see @xcite , ch . > from now on we fix the chamber @xmath382 . it is an open simplicial cone with vertex @xmath42 ( _ loc . v ,  3.9 , proposition 7 ( iii ) ) , and its closure @xmath383 is a fundamental domain for the action of @xmath357 on @xmath52 ( _ loc . v ,  3.3 , theorem 2 ) . we define @xmath384 where @xmath385 denotes the interior . for instance , @xmath386 and @xmath387 . in the following examples , the sets @xmath388 will play the role of the set @xmath102 appearing in theorem [ thm : maindistr ] . the set @xmath389 is the smallest cone in @xmath52 which is closed , @xmath371-invariant and contains @xmath390 . it is a union of closed chambers . the open polyhedral cone in @xmath57 @xmath391 has the dual cone @xmath392 @xmath393 is a closed convex cone . it is the intersection of the closed hyperplanes defined by roots , and hence a union of closed chambers in @xmath52 . [ lemma : cone ] we have @xmath394 consequently , @xmath395 is a closed @xmath371-invariant convex cone in @xmath52 and also its interior @xmath388 is convex . moreover , @xmath396    this was proven in @xcite , lemma 3.4 , when @xmath0 is a weyl group . the same proof applies also to the more general case of finite coxeter groups . specializing the notation of section [ section : setting ] to this context , we consider the space @xmath397 of smooth functions @xmath398 with the property that @xmath5 is a @xmath371-invariant compact subset of @xmath388 . its subspace of @xmath371-invariant functions on @xmath388 with compact support is @xmath399 . furthermore , @xmath400 is the space consisting of distributions @xmath6 on @xmath52 so that @xmath41 is a @xmath371-invariant compact subset of @xmath388 .      as before let @xmath358 be a finite dimensional euclidean space and let @xmath359 be a ( non - necessarily reduced ) root system in @xmath57 . further , let @xmath357 the be the corresponding weyl group and let @xmath401 be a multiplicity function on @xmath359 . heckman and opdam associated with such a triple @xmath402 a commutative family of @xmath357-invariant differential operators on @xmath52 , the _ hypergeometric differential operators _ , having meromorphic coefficients of a specific type . more precisely , their coefficients are meromorphic functions on the complexification @xmath403 of @xmath52 and their singularities are cancelled by multiplication by a suitable power of the weyl denominator @xmath404 we refer to @xcite , @xcite and @xcite for more information on heckman - opdam s theory of hypergeometric differential operators .    for special values of @xmath401 , the triple @xmath402 arises from a riemannian symmetric space @xmath47 of the noncompact type , i.e. , @xmath44 is a noncompact connected semisimple lie group with finite center and @xmath405 is a maximal compact subgroup , see section [ section : appl ] . in this case , the hypergeometric differential operators coincide with the radial parts ( with respect to the @xmath19-action ) on @xmath406 of the @xmath44-invariant differential operators on @xmath47 . here we identify @xmath406 with its diffeomorphic image in @xmath47 under the exponential map ( usually denoted @xmath407 in the literature on analysis on symmetric spaces ) . for instance , the hypergeometric differential operator @xmath408 coincides with the radial part of the laplace - beltrami operator of @xmath47 . notice that @xmath144 is singular on the hyperplanes @xmath409 , @xmath410 . furthermore , @xmath144 is the multivariable analog of the jacobi differential operator of example [ ex : diffrankone ] . in ( [ eq : l ] ) , @xmath411 denotes the directional derivative in the direction of @xmath412 , and @xmath172 is the laplace operator on the euclidean vector space @xmath52 , that is , @xmath413 where @xmath414 is an orthonormal basis of @xmath52 .    set @xmath415 with the usual convention that empty products are equal to @xmath34 . we write @xmath416 instead of @xmath417 . finally , define @xmath418 . the principal symbol of @xmath2 is @xmath419 where @xmath420 ^ 2   \big[\frac{{{\delta}_{\scriptscriptstyle{\theta}}}(x)}{{{\pi}_{\scriptscriptstyle{\theta}}}(x)}\big]^2.\ ] ] because of lemma [ lemma : cone ] , each @xmath421 is positive on @xmath388 . therefore @xmath422 is a homogeneous @xmath388-elliptic polynomial on @xmath423 ( of degree @xmath42 in @xmath424 ) . more generally , for each hypergeometric differential operator @xmath425 there is @xmath426 so that the linear partial differential operator @xmath427 is a @xmath357-invariant differential operator with real analytic coefficients and with principal symbol of the form ( [ eq : symbol ] ) where @xmath422 is a homogeneous @xmath388-elliptic polynomial . in this case , we say that @xmath428 is a _ regularization _ of @xmath425 . the nature of the principal symbol of @xmath2 can in fact be deduced by the explicit representation of the hypergeometric differential operators in terms of cherednik operators ( see e.g. @xcite ) . according to this representation , each hypergeometric differential operator can be constructed as the `` differential part '' of the differential - reflection operator associated with a @xmath357-invariant polynomial in the symmetric algebra over @xmath403 by means of cherednik operators . if @xmath425 is associated with the @xmath357-invariant polynomial @xmath429 , then the principal part of the regularized operator @xmath430 is @xmath431^{2k}\,\\                    & = p_h({\lambda } ) { { \pi}_{\scriptscriptstyle{\theta}}}(x)^{2k } p({\lambda},x)\end{aligned}\ ] ] where @xmath432 is the highest homogeneous part of @xmath115 and @xmath433^{2k }   \big[\frac{{{\delta}_{\scriptscriptstyle{\theta}}}(x)}{{{\pi}_{\scriptscriptstyle{\theta}}}(x)}\big]^{2k}.\ ] ]    in this setting , theorems [ thm : maindistr ] and [ thm : mainfunction ] yield the following result . [ cor : mainhyp ] let the notation be as above . let @xmath434 . for a hypergeometric differential operator @xmath425 let @xmath430 be a regularization of @xmath425 . then for every @xmath435 we have @xmath436 equivalently , for every @xmath437 , we have @xmath438    in the case of @xmath371-invariant functions , corollary [ cor : mainhyp ] can be stated directly for the hypergeometric operators . [ cor : mainhypw ] let the notation be as above . let @xmath425 be a hypergeometric differential operator and @xmath439 . then @xmath440 for all @xmath441 and @xmath442    as @xmath443 and @xmath444 is compact and @xmath371-invariant , it follows that there exists a unique @xmath445 so that @xmath446 . let @xmath447 be a regularization of @xmath425 . then @xmath448 suppose @xmath425 is the differential part of the cherednik operator @xmath449 ( see e.g. @xcite ) . then @xmath450 by @xmath357-invariance . cherednik operators map @xmath357-invariant smooth functions into @xmath357-invariant smooth functions . it follows that @xmath451 is smooth and @xmath357-invariant . > from ( [ eq : restrictions ] ) we therefore deduce that @xmath452 extends to be smooth and @xmath357-invariant on @xmath388 . if @xmath453 is a continuous function , then @xmath454 and @xmath453 have the same support . the theorem therefore follows . let @xmath44 be a connected noncompact semisimple lie group with fintie center and @xmath405 a maximal compact subgroup . then @xmath455 for some cartan involution @xmath456 . the lie algebra @xmath457 decomposes into eigenspaces of the derived homomorphism @xmath458 : @xmath459 where @xmath460 is the @xmath461 eigenspace and @xmath462 is the @xmath463 eigenspace . consider @xmath462 as a abelian lie group . the group @xmath19 acts linearly on @xmath462 by @xmath464 . we can therefore consider the semi - direct product @xmath465    let @xmath402 be as in example [ ex : hyper ] . the _ bessel differential operators _ associated with @xmath402 are the `` rational '' analogs of the hypergeometric differential operators . they are @xmath0-invariant and the singularities of their meromorphic coefficients are cancelled by multiplication be a power of the polynomial @xmath466 when the triple @xmath402 arises from the structure of riemannian symmetric space @xmath47 of the noncompact type , the bessel differential operators coincide with the radial parts on @xmath406 of the @xmath467-invariant differential operators on the corresponding riemannian symmetric space @xmath149 of euclidean type . for instance , the bessel differential operator @xmath468 coincides with the radial part of the laplace - beltrami operator on @xmath149 . the operator @xmath145 is the multivariable analog of the bessel differential operator of example [ ex : diffrankone ] . the bessel differential operators can be constructed as `` differential parts '' of differential - reflection operators associated with @xmath0-invariant polynomials functions @xmath429 on @xmath469 by means of the dunkl operators @xcite . suppose @xmath425 is the bessel differential operator associated with @xmath429 . let @xmath426 be chosen so that @xmath470 has real analytic coefficients . then the principal symbol of @xmath2 is @xmath471 where @xmath432 is the highest homogeneous part of @xmath115 and @xmath472 is a homogeneous @xmath388-elliptic polynomial of degree @xmath42 in @xmath473 . as for the hypergeometric differential operators , we obtain the following corollary of theorems [ thm : maindistr ] and [ thm : mainfunction ] .    [ cor : mainbessel ] let the notation be as above . let @xmath48 be a fixed set of positive simple roots in @xmath359 . for a bessel differential operator @xmath425 let @xmath474 be a regularization of @xmath425 . then , for @xmath435 we have @xmath436 equivalently , for every @xmath437 , we have @xmath438 if @xmath441 , then @xmath440 and @xmath442      for the study of hypergeometric differential operators corresponding to different multiplicity functions ( see example [ ex : hyper ] ) , opdam introduced certain @xmath0-invariant differential operators , called _ shift operators_. for simplicity , we treat here only the case of a reduced root system @xmath359 and we refer the reader to @xcite , part i , ch . 3 for the general case . as in the case of hypergeometric differential operators , opdam s shift operators have meromorphic coefficients with singularities along the root hyperplanes , and their singularities are cancelled by multiplication by powers of @xmath69 . a _ shift _ is a multiplicity function @xmath475 which is even ( i.e. such that @xmath476 for all @xmath365 ) . opdam s shift operators exists for any shift @xmath475 . they can be considered as generalizations of the hypergeometric differential operators , which form the space @xmath477 of shift operators of shift @xmath42 . for an arbitrary nonzero shift @xmath475 , the space @xmath478 of shift operators of shift @xmath475 is a free rank - one ( right ) @xmath477-module generated by a certain shift operator @xmath479 . see _ loc . _ , theorem 3.3.7 . suppose furthermore that either @xmath480 for all @xmath481 or @xmath482 for all @xmath481 . then @xmath479 is obtained by composition of _ fundamental shift operators _ @xmath483 , shifting of @xmath484 the multiplicities along each weyl group orbit ( _ loc . cit . _ , p. 42 ) . the principal symbol of each fundamental shift operator is @xmath485 where @xmath486 and @xmath487 . see _ loc . _ , remark 3.3.8 . the analysis done in example [ ex : hyper ] for the hypergeometric differential operators , in particular corollary [ cor : mainhyp ] , easily extends also to the fundamental shift operators , and hence to all shift operators . a similar argument can be also applied to the shift operators associated with the bessel differential operators . for more information on the latter we refer to @xcite , where they are studied in the general case in which @xmath0 is an arbitrary finite coxeter group . a wide class of integrable hamiltonian systems associated with root systems were introduced by olshanetsky and perelemov in @xcite . let @xmath488 be a triple consisting of a finite dimensional euclidean space @xmath52 , a root system @xmath359 in @xmath57 , and a real - valued multiplicity function @xmath453 on @xmath359 . these integrable systems are described by a hamiltonian of the form @xmath489 with potential energy @xmath490 where the function @xmath264 has five possible forms . here we consider the cases i , ii and v as listed in @xcite , ( 3.1.14 ) and ( 3.8.3 ) : @xmath491 the calogero hamiltonian of example [ ex : calogero ] is a special instance of case i. the associated schrdinger operators are the linear partial differential operators @xmath492 they are invariant with respect to the weyl group @xmath357 of @xmath359 . the operator @xmath245 can be regularized by multiplication by the polynomial @xmath416 of ( [ eq : pi ] ) in the cases i and v , and by multiplication by @xmath493 as in ( [ eq : delta ] ) . the possibility of applying theorems [ thm : maindistr ] and [ thm : mainfunction ] to these regularized differential operators can be proven as in examples [ ex : hyper ] and [ ex : bessel ] . in fact , there is a close relation between the operator @xmath245 of case ii , resp . of case i , and the operator @xmath144 of ( [ eq : l ] ) , resp . to the operator @xmath145 of ( [ eq : lo ] ) . see e.g. @xcite , theorem 2.1.1 , and @xcite . for an overview on the role of the hamiltonian systems treated in this example in different areas of theoretical physics and mathematics , we refer the reader to @xcite . in this section we apply theorem [ thm : mainfunction ] to differential operators on symmetric spaces . in particular we give a new proof of the @xmath2-convexity of the riemannian symmetric space @xmath47 , c.f . our proof uses theorem [ thm : mainfunction ] , applied to the radial part of invariant differential operators and is , as far as we know , new . our notation is the same as in subsection [ ex : bessel ] . let us start by recalling the notation from subsection [ ex : bessel ] . here @xmath44 is a connected noncompact semisimple lie group with finite center . furthermore @xmath494 is a cartan involution and @xmath455 the corresponding maximal compact subgroup . denote by @xmath495 , the natural projection @xmath496 . we denote also by @xmath456 the derived involution on @xmath457 . then @xmath497 , where @xmath460 is the @xmath461-eigenspace of @xmath456 , and @xmath462 the @xmath463-eigenspace . then @xmath460 is the lie algebra of @xmath19 . fix a cartan subspace @xmath498 , that is a maximal abelian subspace of @xmath499 . the killing form on @xmath457 defines an euclidean inner product on @xmath52 . for @xmath500 set @xmath501={\alpha}(x ) y$ for all $ x \in \frak a$}\,\}$ ] . then the set @xmath359 consisting of all @xmath502 for which @xmath503 is a ( generally non - reduced ) root system as defined in section [ section : examples ] . it is called the _ ( restricted ) root system _ of @xmath504 . the multiplicity @xmath366 of @xmath365 is defined as the dimension of @xmath505 . the map @xmath401 given by @xmath506 is a multiplicity function on @xmath359 . this construction associates with the riemannian symmetric space @xmath47 a triple @xmath507 . as before we denote by @xmath357 the corresponding weyl group . let @xmath368 be a choice of positive roots and let @xmath508 be the corresponding positive chamber . denote by @xmath509 the exponential map . then @xmath510 is an abelian subgroup of @xmath44 diffeomorphic to @xmath52 . we set @xmath511 . the map @xmath512 is surjective and the @xmath513-component is unique up to conjugation by an element of @xmath357 . hence every @xmath19-bi - invariant subset of @xmath44 is of the form @xmath514 where @xmath515 is a @xmath357-invariant subset of @xmath52 . moreover , @xmath515 is compact if and only if @xmath516 is . let @xmath517 denote the space of compactly supported smooth functions on @xmath47 . using the map @xmath518 , @xmath519 , we will often identify smooth functions on @xmath47 with the corresponding right @xmath19-invariant function on @xmath44 . [ lemma : suppfkak ] let @xmath521 and suppose that @xmath282 is @xmath19-bi - invariant . then @xmath522 where @xmath523 denotes the restriction of @xmath4 to @xmath513 . moreover , @xmath524 is a @xmath357-invariant compact subset of @xmath513 . denote by @xmath525 the ( commutative ) algebra of @xmath44-invariant differential operators on @xmath47 . we identify @xmath407 can be identified with the submanifold @xmath526 . then , for every @xmath527 , there is a unique @xmath357-invariant differential operator , @xmath528 on @xmath407 , called the _ radial part of @xmath2 _ , so that for all @xmath529 one has @xmath530 see @xcite , p. 259 . define a differential operator on @xmath52 , also denoted by @xmath531 , by @xmath532 in this way , we can consider @xmath528 as a ( singular ) @xmath357-invariant differential operator on the euclidean space @xmath52 . to simplify our notation , we shall adopt the identification @xmath533 using the exponential map . then @xmath534 . we then write @xmath535 instead of @xmath523 , and the above mentioned decomposition of a @xmath19-bi - invariant subset of @xmath44 will be written as @xmath536 instead of @xmath537 . with these identifications , the operator @xmath528 turns out to be a hypergeometric differential operator associated with the triple @xmath538 as considered in subsection [ ex : hyper ] . when @xmath539 is @xmath19-invariant , we can moreover extend ( [ eq : radialpart ] ) by @xmath357-invariance to obtain @xmath540    [ lemma : almostdconvex ] let @xmath47 be a riemannian symmetric space of the noncompact type . let @xmath527 and let @xmath515 be a compact , convex and @xmath357-invariant subset of @xmath52 . set @xmath541 . then for all @xmath521 , we have @xmath542    we need to prove that @xmath543 implies @xmath544 . the first step is , as in @xcite , to reduce to the case in which the support of @xmath4 is left-@xmath19-invariant . the function @xmath4 can in fact be expanded as a sum of @xmath19-finite functions . since @xmath545 is left-@xmath19-invariant , we will obtain @xmath544 if the support of each @xmath19-finite summand is contained in @xmath545 . we can therefore assume that @xmath4 , and hence @xmath9 , are @xmath19-finite . as the support of a @xmath19-finite function is left @xmath19-invariant , we obtain from lemma [ lemma : suppfkak ] that @xmath546 and @xmath547 where @xmath548 and @xmath549 are @xmath357-invariant and compact . since @xmath543 , we then conclude that @xmath550 . since @xmath528 is a hypergeometric differential operator , corollary [ cor : mainhypw ] with @xmath551 yields that @xmath552 . this proves the required inclusion . [ thm : dconvex ] let @xmath47 and @xmath527 be as in lemma [ lemma : almostdconvex ] . then @xmath47 is @xmath2-convex , that is for every compact subset @xmath245 of @xmath47 there is a compact set @xmath553 so that for every @xmath554 the inclusion @xmath555 implies @xmath556 . here @xmath24 denotes the formal transpose of @xmath2 . choose @xmath557 so that @xmath558 the set @xmath559 is @xmath19-bi - invariant , hence @xmath560 for some @xmath357-invariant compact subset @xmath515 of @xmath561 . since @xmath562 we can apply lemma [ lemma : suppfkak ] to it . so @xmath563 . we can then select @xmath564 . we conclude this section by a short discussion of the application of the theorem of our support theorem in harmonic analysis corresponding the the @xmath48-hypergeometric transform , c.f . because of the technical nature of this application , a detailed exposition would require a certain amount of notation and of background information . instead , we prefer here just to outline the main ideas involved . we refer the interested reader to @xcite for further information . as already remarked in the introduction , the support theorem plays a role in the study of the _ paley - wiener space_. this is the set of images under a suitable generalization @xmath565 of the fourier transform , of the compactly supported smooth functions . the paley - wiener space generally consists of entire or meromorphic functions with exponential growth and possibly satisfying additional symmetry conditions . the size the support of the original function is linked to the exponential growth of its fourier transform . the thrust of paley - wiener type theorems is usually to prove that , if a function @xmath453 in the paley - wiener space has a given exponential growth , then the support of the associated `` wave packet '' @xmath566 ( obtained by formal application to @xmath453 of the inverse fourier transform @xmath567 of @xmath565 ) is compact and has the correct size . in the classical situation of the fourier transform on @xmath1 , this is proven by a suitable `` shift '' of contour of integration by means of cauchy s theorem . the shift is allowed because @xmath566 is given by integration of an entire function of exponential type and rapidly decreasing . a suitable generalization of this argument was applied also to the spherical fourier transform on riemannian symmetric spaces . see @xcite , ch . iv ,  7.2 , or @xcite ,  6.6 . for pseudo - riemannian symmetric spaces the situation is more complicated . here the wave packages are integrals of meromorphic functions and the required shift of integration would generally require addition of certain `` residues '' . for some kinds of fourier transforms , like the spherical fourier - laplace transform on noncompactly causal symmetric spaces with even multiplicities @xcite , the singularities of the integrand are cancelled by multiplication by a certain polynomial function @xmath429 . hence , for every function @xmath453 in the paley - wiener space the wave - packet @xmath568 is given by integration of an entire function . furthermore , there is an invariant differential operator @xmath2 so that @xmath569 . the possibility of shifting the contour of integration in @xmath568 allows us then to determine the size of the support of @xmath570 . the generalization of the support theorem of lions and titchmarsch presented in this papers allows us finally to recover the size of the support of @xmath571 . a concrete application of this procedure using theorem [ thm : mainfunction ] can be found in @xcite in the context of the @xmath48-hypergeometric transform . the latter transform , stated in a setting of transforms associated with root systems , is a common generalization of opdam s hypergeometrical transform ( hence of harish - chandra s spherical transform on riemannian symmetric spaces of noncompact type ) and of the fourier - laplace transform on noncompactly causal symmetric spaces @xcite . a similar application would also be needed for determining the paley - wiener space for the transform associated with the @xmath48-bessel functions of @xcite . g.  j. heckman , . in : proceedings of the special session on hypergeometric functions on domains of positivity , jack polynomials and applications at the ams meeting in tampa , fa , march 2223 ( contemp . * 138 * ( 1992 ) ) , am . soc . , providence , ri .
let @xmath0 be a finite coxeter group acting linearly on @xmath1 . in this article we study support properties of @xmath0-invariant partial differential operator @xmath2 on @xmath1 with real analytic coefficients . 
 our assumption is that the principal symbol of @xmath2 has a special form , related to the root system corresponding to @xmath0 . in particular the zeros of the principal symbol 
 are supposed to be located on hyperplanes fixed by reflections in @xmath0 . we show that @xmath3 holds for all compactly supported smooth functions @xmath4 so that @xmath5 is @xmath0-invariant . 
 the main tools in the proof are holmgren s uniqueness theorem and some elementary convex geometry . 
 several examples and applications linked to the theory of special functions associated with root systems are presented .