A solution to the equation $-\Delta u+u=f$ for $f\in L^2(\mathbb R^n)$ belongs in $H^2(\mathbb R^n)$. Is it possible to obtain a solution in $H^2\cap L^\infty(\mathbb R^n)$ if $f\in L^2\cap L^\infty(\mathbb R^n)$?
2026-03-28 14:05:38.1774706738
Boundedness of solutions for the Laplacian
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Yes, just notice that $u=(-\Delta +1)^{-1} f=G_2 * f$, where $\hat{G}_2(\xi)=(1+|\xi|^2)^{-1}$ is the Bessel potential and it follows from properties of $G_2$ (in particular that it is an $L^1$ function) that it maps $L^\infty$ to $L^\infty$ (see "Modern Fourier Analysis" by Grafakos for a proof of this fact).