Inverse Laplace operator $\Delta^{-1}$ and Sobolev spaces

Question

I'm looking for some regularity results for the inverse Laplace operator. More precisely - we're set in $\mathbb{R}^3$ and we are looking at the operator $$ \Delta^{-1}f = \frac{x}{|x|^3} \ast f$$ I'd very much appreciate a reference or a name of a result that would let me conclude in what Sobolev space $W^{k,p}$ the $\Delta^{-1}$ is if I know that $f \in L^q$ for some $q$. In particular I'm interested in the case $f \in L^1$ and I'd like to deduce that $\Delta^{-1}f \in W^{1,p}(B)$ for $p < 3/2$ and with $B$ being a ball of some radius. Does it hold? If so how can I estimate the $W^{1,p}$ norm for $\Delta^{-1}f$?

Answer 1

Look up elliptic regularity: http://en.wikipedia.org/wiki/Elliptic_operator#Elliptic_regularity_theorem and Sobolev embeddings: http://en.wikipedia.org/wiki/Sobolev_inequality ...

If $\Delta u\in L^1(\mathbb{R}^3)$ then $u\in W^{2,1}(\mathbb{R}^3)$, and this combined with the Sobolev embedding gives that $u\in W^{1,{3/2}}(\mathbb{R}^3)$. If you are just considering a ball $B$ then since the $L^p(B)$ spaces are decreasing you get $u\in W^{1,p}(B)$ for all $p<3/2.$

Answer 2

We can also estimate directly with the integral kernel of $(-\Delta)^{-1}$. Write $$Tf(x) = \int_{\mathbf{R}^3} \frac{f(y)}{|x-y|} \, dy.$$ Then $$|\partial_x Tf(x)| \le \int_{\mathbf{R}^3} \frac{|f(y)|}{|x-y|^2} \, dy.$$ For a fixed compact set $B$, H\"{o}lder and Fubini-Tonelli imply that $$ | \langle Tf, g \rangle| \lesssim \int_B\int_{\mathbf{R}^3} \frac{|g(x)| |f(y)|}{ |x-y|} \, dy \, dx \lesssim \|f\|_{L^1} \|g\|_{L^{\frac{3}{2}+}(B)}$$ $$| \langle \partial_x Tf, g \rangle| \lesssim \int_{B} \int_{\mathbf{R}^3} \frac{|g(x)| |f(y)|}{|x-y|^2} \, dy \, dx \lesssim \|f\|_{L^1} \|g\|_{L^{3+}(B)}$$

Thus by duality we see that $\|Tf\|_{L^{3-\delta}(B)} + \| \partial_x Tf\|_{L^{3/2 - \delta}(B)} \lesssim_{\delta} \|f\|_{L^1}$.