As one can imagine, Sobolev fractional regularity of duality solutions can be deduced if we have some information on the exact behavior of the fundamental solution for this operator. For the sake of exposition we consider the case of anisotropic α-stable process and we readapt the result in [16] in order to get sharp local Sobolev regularity for the solution of (2.6).

Let Ω be a bounded open set of ${\mathbb{R}}^{n}$. Consider

$\{\begin{array}{cccc}\hfill {L}^{\alpha}u& =\mu \hfill & & \hfill \text{in}\mathrm{\Omega},\\ \hfill u& =0\hfill & & \hfill \text{in}{\mathbb{R}}^{n}\backslash \mathrm{\Omega},\end{array}$(3.1)

where μ is a Radon bounded measure on Ω.
The operator ${L}^{\alpha}$ is given by

${L}^{\alpha}u(x)=\mathrm{PV}{\int}_{{\mathbb{R}}^{n}}\left(u(x)-u(x+y)\right)\frac{a(y/|y|)}{{|y|}^{n+\alpha}}\mathit{d}y,$

where $0<\alpha <2$ and $a:{S}^{n-1}\to \mathbb{R}$ is a non-negative and symmetric function that satisfies the uniform ellipticity condition

$\lambda \le a(\theta )\le \mathrm{\Lambda}\mathit{\hspace{1em}}\text{for all}\theta \in {S}^{n-1}$

with $0<\lambda \le \mathrm{\Lambda}$.

These operators are infinitesimal generators of a very special class of Lévy processes: the so-called α-stable processes. With respect to the previous section here we use the usual convention $\alpha =2s$.

It is proved in [29] that the potential kernel *K* associated to ${L}^{\alpha}$ (i.e., the fundamental solution of the operator) satisfies

$\frac{{c}_{1}}{{|y|}^{n-\alpha}}\le K(y)\le \frac{{c}_{2}}{{|y|}^{n-\alpha}}$

for suitable positive constants ${c}_{1}\le {c}_{2}$, and

$K(y)={|y|}^{\alpha -n}K\left(\frac{y}{|y|}\right).$

Moreover, by [7, Theorem 1], we have that *K* belongs to the Hölder space ${C}_{\mathrm{loc}}^{2\alpha -\u03f5}({\mathbb{R}}^{n}\setminus \{0\})$ for all $\u03f5>0$.

Thus, it follows that *K* is a fractional kernel of order α and regularity $2\alpha -\u03f5$ in the sense of [15, Definition 4.1]. In particular, we have the following
result (see [15, Theorem 5.2]).

*The Riesz potential associated to ${L}^{\alpha}$,*

${I}^{\alpha}(f)(x)={\int}_{{\mathbb{R}}^{n}}f(y)K(x-y)\mathit{d}y,$(3.2)

*maps ${L}^{p}\mathit{}\mathrm{(}{\mathbb{R}}^{n}\mathrm{)}$ boundedly into ${C}^{\alpha \mathrm{-}\frac{n}{p}}\mathit{}\mathrm{(}{\mathbb{R}}^{n}\mathrm{)}$ whenever $p\mathrm{>}n\mathrm{/}\alpha $. That is,*

${\parallel {I}^{\alpha}(f)\parallel}_{{C}^{\alpha -\frac{n}{p}}({\mathbb{R}}^{n})}\le C{\parallel f\parallel}_{{L}^{p}({\mathbb{R}}^{n})},$

*with **C* depending only on *p*, *n*, λ, Λ and α.

Due to Theorem 3.1 and in particular to the integral representation (3.2) it is straightforward to reproduce the argument in [16] in order to show the existence of a unique duality solution $\stackrel{~}{u}$ for problem

${L}^{\alpha}u=\mu \mathit{\hspace{1em}}\text{in}{\mathbb{R}}^{n},$

whenever μ is set to be zero outside Ω. Moreover, $\stackrel{~}{u}\in {W}_{\mathrm{loc}}^{1-(2-\alpha )/q,q}({\mathbb{R}}^{n})$
for any $q<\frac{n+2-\alpha}{n+1-\alpha}$.

For simplicity, in what follows we also assume $\alpha >1$, i.e., $s>\frac{1}{2}$
(see [16, p. 2] and Remark 3.4 below for further comments on this restriction),
and μ to be non-negative.

We have the following local result.

*The duality solution **u* of problem (3.1) belongs to ${W}_{\mathrm{loc}}^{\mathrm{1}\mathrm{-}\mathrm{(}\mathrm{2}\mathrm{-}\alpha \mathrm{)}\mathrm{/}q\mathrm{,}q}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$
for any $q\mathrm{<}\frac{n\mathrm{+}\mathrm{2}\mathrm{-}\alpha}{n\mathrm{+}\mathrm{1}\mathrm{-}\alpha}$.

#### Proof.

Let $\stackrel{~}{u}$ be the duality solution of

${L}^{\alpha}\stackrel{~}{u}=\mu \mathit{\hspace{1em}}\text{in}{\mathbb{R}}^{n},$

where, as we said, μ is extended by zero outside Ω.
Let us consider now $v:=\stackrel{~}{u}-u$.
This function is a non-negative function in all of ${\mathbb{R}}^{n}$, and it satisfies ${(-\mathrm{\Delta})}^{s}v=0$ in Ω.
Moreover, $v=\stackrel{~}{u}$ outside Ω, and in particular $\stackrel{~}{u}\in {L}^{\mathrm{\infty}}({\mathbb{R}}^{n}\backslash \mathrm{\Omega})$. This implies that *v* is locally smooth inside Ω (see, e.g., [26, 8]). Therefore *v* also satisfies that $v\in {W}_{\mathrm{loc}}^{1-(2-\alpha )/q,q}(\mathrm{\Omega})$
for any $q<\frac{n+2-\alpha}{n+1-\alpha}$. The proof is complete since $u=\stackrel{~}{u}-v$.
∎

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