Hi,


it is easy to prove the $W^{2,2}(\mathcal S^2)$ regularity for the laplace on the (2 dimensional-) standard sphere $\mathcal S^2:=\lbrace x \in\mathbb R^3: \vert x\vert=1 \rbrace\hookrightarrow\mathbb R^3$ by partial integration, getting
 $$\Vert f\Vert\_{W^{2,2}(\mathcal S^2)} \le C ( \Vert \triangle f\Vert\_{L^2(\mathcal S^2)} + \Vert f\Vert\_{L^2(\mathcal S^2)}).$$



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> **But**: How do you prove an equivalent $W^{2,p}(\mathcal S^2)$ regularity for $p>2$ in a more or less direct matter? The only interessting part is to prove a
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> > 
> > ### $L^p$-inequality for $D^2f$:
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> > 
> > For $p>2$ there exists $1\le q<\infty$ such that
> >  $$\Vert \nabla(\nabla f) \Vert\_{L^p(\mathcal S^2)} \le C\_{p,q}( \Vert \triangle f\Vert\_{L^p(\mathcal S^2)} + \Vert f\Vert\_{W^{1,q}(\mathcal S^2)}).$$
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> As the rest follows by the sobolev-inequalites and the $W^{2,2}$-case.
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It's essential for me to get a direct proof (in particular no contradiction), because I have to adapt it in an approximated sphere on an asymptotic flat Riemannian Manifold.




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In fact it would be enough for me to prove
 $$\Vert \nabla(\nabla f) \Vert\_{L^p(\mathcal S^2)} \le C\_{p,q,r}( \Vert \triangle f\Vert\_{L^r(\mathcal S^2)} + \Vert f\Vert\_{W^{1,q}(\mathcal S^2)})$$
for some $1\le q<\infty$ and some $1\le r\le\infty$, because of my special situation, but I do not believe that this is true if the inequality above is not.


Thx for some hints...


Elgrimm




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**Edit**: I mean the sphere $\mathcal S^2:=\lbrace x\in\mathbb R^3 : \vert x \vert=1\rbrace$ as *riemannian submanifold* of $\mathbb R^3$ and the corresponding laplace-beltrami-operator $\triangle$ on $\mathcal S^2$ **not** the ball as substet of $\mathbb R^3$.




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**Edit**:
For me it would even be sufficient to prove a



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> > 
> > ### $L^\infty$-inequality for $\nabla f$:
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> > 
> > There exists $1\le p<\infty$ and $1\le r\le\infty$ such that
> >  $$ \Vert \nabla f\Vert\_{L^\infty(\mathcal S^2)} \le C(\Vert f\Vert\_{L^p(\mathcal S^2)} + \Vert\triangle f\Vert\_{L^r(\mathcal S^2)}).$$
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Which could be concluded from the results above with the sobolev-inequalities.




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**Edit** (sorry for the mass of edits): By looking at $\nabla f$ instead of $f$ it's quite obvious, that it would be a even stronger result to prove the



> 
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> > 
> > ### Weak inequality:
> > 
> > 
> > For $p>2$ and $\frac1p+\frac1q=1$ it's true that
> >  $$ \Vert f\Vert\_{W^{1,p}(\mathcal S^2)} \le C(\Vert f\Vert\_{L^p(\mathcal S^2)} + \Vert \triangle f\Vert\_{W^{-1,q}(\mathcal S^2)}), $$
> >  where $W^{-1,q}(\mathcal S^2)$ is the dual of $W^{1,q}(\mathcal S^2)$, in particular
> >  $$\Vert \triangle f\Vert\_{W^{-1,q}(\mathcal S^2)}:=\sup\_{\Vert\varphi\Vert\_{W^{1,q}(\mathcal S^2)}=1}\left\vert\int\_{\mathcal S^2}\nabla f\cdot\nabla\varphi\right\vert.$$
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$W^{2,p}$ or $W^{1,q}$ regularity for the laplace on a euclidean sphere
### Sobolev-Norms:


Let $(M,g)$ be a two-dimensional compact remannian-manifold without boundary $r>0$, define for $f\in\mathcal C^\infty(M)$ the sobolev-inequalites
$$ \Vert f\Vert\_{W^{k,p}(M)} := \sum\_{l=0}^k r^l\cdot\left\Vert \vert\nabla^l f\vert\_g\right\Vert\_{L^p(M)}\qquad\forall f\in\mathcal C^1(M),\ k\in\mathbb N\_{\ge0}, $$
where $\nabla$ is the levi-civita connection. $W^ {k,p}(M)$ is as usual the completion of $\mathcal C^\infty(M)$ for this norm, and as usual we identifiy elements of $W^{k,p}$-with almost every-where defined functions.




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### $\triangle$-$W^{1,\infty}$-, -$W^{2,2}$-, -$L^\infty$-regularities:


Let $M$ be a two-dimensional compact manifold without boundary and $c,r\in\mathbb R\_{>0}$, such that
$$ \vert M\vert \le cr^2, \quad \Vert \text{Ric}(M) \Vert\_{L^\infty(M)} \le \frac c{r^2},\quad \Vert f\Vert\_{L^2(M)} \le \frac cr\Vert f\Vert\_{W^{1,1}(M)} \quad\forall f\in W^{1,1}(M) $$
and $c>\lambda\_1$, where $\lambda\_1$ is the smallest positiv eigenvalue of $-\triangle$.
There is a constant $C=C(c)$ such that
$$ \Vert f\Vert\_{W^{2,2}(M)} \le C(r^2\Vert \triangle f\Vert\_{L^2(M)}+\Vert f\Vert\_{L^2(M)}) \qquad\forall f\in W^{2,2}(M), \tag{1} $$
$$ \Vert f\Vert\_{L^\infty(M)} \le C(r\Vert \triangle f\Vert\_{L^2(M)}+r^{-1}\Vert f\Vert\_{L^2(M)}) \qquad\forall f\in W^{2,2}(M), \tag{2} $$
$$ \Vert f\Vert\_{W^{1,\infty}(M)} \le Cr^{-\frac2p}(r^2\Vert \triangle f\Vert\_{L^p(M)}+\Vert f\Vert\_{L^p(M)}) \qquad\forall f\in W^{2,p}(M),\ p>2, \tag{3} $$
if $r>C$.




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Proof [as Deane Yang suggested by a Moser-iteration]:
Be setting $g:=\vert f\vert^{\frac 12}$, we get the usual sobolev-inequalites
$$\Vert f\Vert\_{L^q(M)} \le \frac cr\Vert f\Vert\_{W^{1,p}(M)} \quad\forall f\in W^{1,p}(M),\ 1\le p< 2: \ q=\frac{2p}{2-p}. \tag 4$$
By an argument as as in [1] we conclude the additional sobolev-inequality
$$\Vert f\Vert\_{L^\infty(M)} \le cr^{-\frac2p}\Vert f\Vert\_{W^{1,p}(M)} \qquad\forall f\in W^{1,p}(M), 2< p. \tag 5 $$
Using that $div\nabla X=g^{ij}Ric(X,e\_i)e\_j+\nabla div X$ for any frame $\lbrace e\_1,e\_2\rbrace$ and any smooth vector field $X$, we conclude $(1)$ by defining $X=\nabla f$ and partial integration and therefore we can deduce $(2)$ with $(5)$.


Let $f\in W^{2,p}(M)$ be such that $\int f=0$, in particular $\Vert f\Vert\_{L^2(M)} \le c \Vert\triangle f\Vert\_{L^2(M)}$, and assume there exists a polnom $P$ and a constant $C$ such that for $q=\frac{2p}{p-2}$
$$ \Vert \vert\nabla f\vert^{2s}\Vert\_{L^p(M)} \le P(s)\left(\left(\frac Csr^{\frac2p}\Vert\triangle f\Vert\_{L^q(M)}\right)^{2r}+r^{-\frac2p}\Vert\vert\nabla f\vert^r\Vert\_{L^p(M)}\right). \tag6 $$
Setting $a\_i:=(r^{2^i-\frac2p}\Vert\vert\nabla f\vert^{2^i}\Vert\_{L^p(M)})^{2^{-i}}$, we conclude by basic analysis
$$ r\Vert \nabla f\Vert\_{L^\infty(M)} \le C\left(r^{1-\frac2p}\Vert\triangle f\Vert\_{L^p(M)}+r^{1+\frac2p}\Vert\nabla f\Vert\_{L^{\frac{2p}{2-p}}(M)}\right). $$
This inequality obviously also holds for $f+d$ ($d\in\mathbb R)$. So we only have to prove $(6)$ for a polnyom and a constant not depending on $f\in\mathcal C^\infty(M)$ with $\int f\ d\mu=0$. By basic analysis and some partial integrations we see
$$ \int \vert\nabla(\vert\nabla f\vert^s)\vert^2\ d\mu \le -s\int\vert\nabla f\vert^{2s-2}g(\nabla f,\triangle\nabla f). $$
We conclude by hölder-inequalities and partial integration
$$ \int \vert\nabla(\vert\nabla f\vert^s)\vert^2\ d\mu \le Csr^{\frac2{ps}}\Vert \vert\nabla f\vert^{2s}\Vert\_{L^p(M)}^{\frac{s-1}s}\underbrace{\left(\begin{aligned}\Vert\text{Ric}\Vert\_{L^\infty(M)}\cdot\Vert\nabla f\Vert\_{L^{\frac{2p}{p-1}}(M)}^2+\Vert\triangle f\Vert\_{L^{\frac{2p}{p-1}}(M)}^2 \\\\ + \Vert\triangle f\Vert\_{L^2(M)}\cdot\Vert\triangle f\Vert\_{L^{\frac{2p}{p-2}}(M)} \\\\ + \Vert\text{ric}\Vert\_{L^\infty(M)}^{\frac12}\Vert\nabla f\Vert\_{L^2(M)}\cdot\Vert\triangle f\Vert\_{L^{\frac{2p}{p-2}}(M)}\end{aligned}\right)}\_{=: c\_f}. $$
By some Yang- and Hölderinequalities and using $\Vert f\Vert\_{L^2(M)}\le c\Vert \triangle f\Vert\_{L^2(M)}$ we see for $q:=\frac{2p}{2-p}$ and the inequality for $\text{ric}$
$$ c\_f \le C\_pr^{\frac2p}\Vert\triangle f\Vert\_{L^q(M)}^2. $$
By the Sobolev- and Young-inequalities we therefore can conclude $(6)$ for $P(s)=s^6$ using the inequality for $\text{ric}$.




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[1]: Stampacchia, Guido: Régularisation des solutions de problèmes aux limites
elliptiques à données discontinues. In: Proceedings of the International Sympo-
sium on Linear Spaces. Hebrew University of Jerusalem : Jerusalem Acad. Pr.,
July 1960 (A publication of the Israel Academy of Sciences and Humanities),
S. 399–408


