  Let $A$ be a homogeneous elliptic differential operator of order $m$ on $%
\Bbb{R}^{N}$ with constant complex coefficients. A partial version of the main
result is as follows: Suppose that $u\in L_{loc}^{1}$ 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 a few
special cases of the sufficiency are already known, mostly when $A=\Delta .$
  The full result gives a similar necessary and sufficient growth condition for
the derivatives of $u$ of any order $k\geq 0$ to be in $L^{p}$ when $Au$
satisfies a suitable (necessary) condition. This is generalized to exterior
domains under mandatory restrictions on $N$ and $p$ and to Douglis-Nirenberg
elliptic systems whose entries are homogeneous operators with constant
coefficients and possibly different orders.
