The following question comes from Brian Hall's "Quantum Theory for Mathematicians".
Let $D(\Delta)\subset L^2(\mathbb{R}^n)$ denote the domain of the Laplacian, which is given by $$D(\Delta)=\{\psi\in L^2(\mathbb{R}^n):|\textbf{k}|^2|\hat\psi(\textbf{k})|\in L^2(\mathbb{R}^n)\}$$ and assume that $n\le3$. Show that each $\psi\in D(\Delta)$ is continuous and that there exists constants $c_1$ and $c_2$ such that $$|\psi(\textbf{x})|\le c_1\|\psi\|+c_2\big\||\textbf{k}|^{9/5}|\hat\psi(\textbf{k})|\big\|,$$ for all $\psi\in D(\Delta)$.
I was able to show that $\psi$ is continuous by showing that $\hat\psi\in L^1(\mathbb{R})$ here. But my attempts at a similar procedure for bounding $|f(\textbf{x})|$ failed. Does anyone know how to get this inequality?
Hint: In fact you can show $$||\hat\psi||_1\le c_1\|\psi\|_2+c_2\big\||\textbf{k}|^{9/5}|\hat\psi(\textbf{k})|\big\|_2.$$ Apply Cauchy-Schwarz one way for $\int_{|k|\le1}|\hat\psi(k)|\,dk$ and in a slightly different way for $|k|>1$. ($|\psi|=1\cdot|\psi|=|k|^{-\alpha}(|k|^\alpha|\hat\psi(k)|)$.)