My question is rather simple. Given $f$ a $L^2(\mathbb{R}^2)$ function with zero mean but supported in the whole plane, is there a bound of the form $$ \|\nabla(\Delta)^{-1} f\|_{L^2(\mathbb{R}^2)}\leq C\|f\|_{L^2(\mathbb{R}^2)}?. $$ Any comment or reference is very welcome!
2026-03-29 07:39:17.1774769957
Bounds on solutions of elliptic pde's in the whole plane
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