Sharp $L^p$--$L^q$ bounds for the strong circular maximal operator

Abstract

We study $L^p$--$L^q$ boundedness of the strong circular maximal operator $\M$, the maximal function over axis-parallel ellipses centered at the origin whose two semiaxes range independently over $[1,2]$. We prove that $\M$ is bounded from $L^p$ to $L^q$ whenever $(1/p,1/q)$ lies in the interior of the triangle with vertices $(0,0)$, $(2/7,1/7)$, and $(1/3,1/3)$, the two-parameter analogue of the triangle with vertices $(0,0)$, $(2/5,1/5)$, and $(1/2,1/2)$ obtained by Schlag and Schlag--Sogge for the circular maximal function. The range is sharp up to the boundary; in particular, weak type $(p,q)$ bounds fail for every $p\le 3$. The proof is based on a bilinear estimate for frequencies in separated angular sectors, uniform as the sectors approach the coordinate axes. We also prove the analogous sharp result for axis dilations of a translated circle $c+{\mathbb{S}}^1$ with fixed $c\ne 0$, where the vertex $(2/7,1/7)$ is replaced by $(1/4,1/8)$.