$L^p$ regularity of homogeneous elliptic differential operators with constant coefficients on ${\mathbb{R}}^N$

Abstract

Let $A$ be a homogeneous elliptic differential operator of order $m$ on ${\mathbb{R}}^N$ with constant complex coefficients. A special case of the main result is as follows: suppose that $u\in L_{\mathrm{loc}}^1$ and that $Au\in L^p$ for some $1<p<\infty$. Then, all the partial derivatives of order $m$ of $u$ are in $L^p$ if and only if $|u|$ grows slower than $|x{|}^m$ at infinity, provided that growth is measured in an $L^1$-averaged sense over balls with increasing radii. The necessity provides an alternative answer to the pointwise growth question investigated with mixed success in the literature. Only very few special cases of the sufficiency are already known, even when $A=\Delta$.

The full result gives a similar necessary and sufficient growth condition for the derivatives of $u$ of any order $k\ge 0$ to be in $L^p$ when $Au$ satisfies a suitable (necessary) condition. This is generalized to exterior domains, which sometimes introduces mandatory restrictions on $N$ and $p,$ and to Douglis–Nirenberg elliptic systems whose entries are homogeneous operators with constant coefficients but possibly different orders, as the Stokes system.