The multiplier problem for the polygon

Abstract

Let us denote by $P$ a polygon of $N$ sides in $\mathbf{R}^2$ and consider the operator defined by $\widehat{Tf}(\xi) = \chi_P(\xi)\hat{f}(\xi)$, where $\chi_P$ is the characteristic function of $P$ and $\hat{f}$ denotes the Fourier transform of $f$; it has been known for a long time that $T$ is a bounded operator on $L^p(\mathbf{R}^2)$, $1\lt p\lt \infty$. On the other hand $\widehat{S_0f}(\xi) = \xi_B(\xi)\hat{f}(\xi)$, where $B$ is the unit ball, is only bounded on $L^2(\mathbf{R}^2)$, while the Fourier multiplier method defined by $\widehat{S_af}(\xi) = \mathcal{m}_\alpha(\xi)\hat{f}(\xi)$ (where $\mathcal{m}_\alpha(\xi) = (1-|\xi|^2)^\alpha$ if $|\xi| < 1$ and $\mathcal{m}_\alpha(\xi) = 0$ otherwise), $1/2 > \alpha >0$, is bounded on $L^p(\mathbf{R}^2)$ for every $p$ between $4/(3+2\alpha)$ and $4/(1-2\alpha)$. The polygon problem (which appears in the work of C. Fefferman as a natural step toward the understanding of the behavior of the Bochner-Riesz multipliers $S_\alpha$) asks for sharp estimates for the norm of the operator $T$. Associated to the operator $T$ there is a maximal function $M$ defined by
$Mf(x) = \sup_{x\in R} \frac{1}{|R|} \int_R |f(y)|dy,$
where the “sup” is taken over all the rectangles in $\mathbf{R}^2$ having sides parallel to one of the sides of the polygon. The boundedness properties of $M$ and $T$ are, in some sense, equivalent. Part I of this paper is devoted to the maximal function and we prove the following theorem.

THEOREM 1. For every $p\geqq 2$, there exist constants $a(p)$, $C_p$, independent of $N$, such that $\Vert Mf\Vert_p\leqq C_p(\log N)^{a(p)}\Vert f\Vert _p$, for every $f\in L^p(\mathbf{R}^2)$.

Part II is devoted to the multiplier operator, and we prove:

THEOREM 2. For every $p$, $4/3\leqq p\leqq 4$, there exist constants $C_p$, $a(p)$, independent of $N$, such that $\Vert Tf\Vert _p\leqq C_p(\log N)^{a(p)}\Vert f\Vert_p$, for every $f \in L^p(\mathbf{R}^2)$.

These results are sharp and explain the different behavior of $S_0$ and $S_\alpha$, $\alpha>0$. For the sake of simplicity, we will present the proof of the theorems for regular polygons, but this restriction can be relaxed.

Antonio Córdoba