  In $R^d$, define a maximal function in the directions $v\in
\directions\subset\{x \mid \abs x=1\}$ by $$ M^\directions
f(x)=\sup_{v\in\directions} \sup_{\zve} \int_{-\ze}^\ze \abs{f(x-vy)} dy. $$
For a function $f$ on $\ZR^d$, let $S_\zw f$ denote the Fourier restriction of
$f$ to a region $\zw$. We are especially interested taking \zw to be a sector
of $R^d$ with base points at the origin. A sector is a product of the interval
$(0,\infty)$ with respect to a choice of (non orthogonal) basis. What is most
important is that the basis is a subset of $\directions$. Consider a collection
$\zW$ of pairwise disjoint sectors $\zw$ as above. Assume that $M^\directions $
maps $L^p$ into $L^p$, for some $1<p<\zI $. Then we have the following
Littlewood--Paley inequality $$ \NORm \Bigl[\sum_{\zw\in\zW}\abs{S_\zw
f}^2\Bigr]^{1/2}.q.\lesssim{}\norm f.q., \qquad 2\le q<2 \frac p{p-1}. $$ The
one dimensional analogue of this inequality is due to Rubio de Francia.
  The conclusion when the set of vectors is a fixed basis is known, is due to
Journ\'e. Our method of proof relies on a phase plane analysis. We introduce a
notion of Carleson measures adapted to $\directions$, and demonstrate a John
Nirenberg inequality for these measures. The John Nirenberg inequality, and an
obvious $L^2$ estimate will prove the Theorem.
