Rubio de Francia Littlewood Paley Inequalities and Directional Maximal Functions

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.