Situation of Nirenberg-Sobolev embedding

Question

Suppose $f\in L^2(\mathbb{R}^2)$ with compact support and $\frac{\partial ^{(k)}} {\partial x^{K}}f, \frac{\partial ^{(k)}} {\partial y^{K}}f\in L^2(\mathbb{R}) \; \forall k\in\mathbb{N}$. Can we conclude that $f\in C^\infty$. It seems that it should follow from Nirenberg-Sobolev embedding.

Answer 1

Since you are not assuming integrability of mixed partials, this is not so much a Sobolev embedding situation, but more of elliptic regularity result. Luckily everything is in $L^2$, which leads to conclusion quickly. Consider the Fourier transform $\hat f(\eta,\xi)$ where $(\eta,\xi)\in\mathbb R^2$. Your assumptions imply $$|\eta|^k \hat f \in L^2, \quad |\xi|^k \hat f \in L^2,\quad k=0,1,2,\dots$$ Given a polynomial $P(\eta,\xi)$, let $Q(\eta,\xi)= P(\eta,\xi)(1+ \eta ^2+\xi^2)$. Since $Q$ is dominated by $1+|\eta|^k +|\xi|^k$ when $k$ is large enough (consider what happens in the unit disk, and use homogeneity outside of the disk) it follows that
$Q\hat f \in L^2$. By Cauchy-Schwarz, $$ \int_{\mathbb R^2} |P(\eta,\xi)\hat f(\eta,\xi)| \le \|Q\hat f \|_{L^2} \|(1+ \eta ^2+\xi^2)^{-1}\|_{L^2} <\infty$$ The function $P(\eta,\xi)\hat f(\eta,\xi)$ is (up to some constants) the Fourier transform of a differential operator with constant coefficient applied to $f$: such as a partial derivative of some order, possibly mixed. Having an integrable Fourier transform, this derivative is continuous. $\quad\Box$