Strong-type endpoint obstructions for bilinear spherical maximal operators

Abstract

For the bilinear spherical averages in dimension $d\ge 2$, we show that the lacunary maximal operator is not bounded from $L^1({\mathbb{R}}^d)\times L^q({\mathbb{R}}^d)$ to $L^r({\mathbb{R}}^d)$ whenever $1\le q\le \infty$ and $1/r=1+1/q$. The obstruction is realized by fixed nonnegative inputs; a second construction proves quantitative failure on smooth compactly supported functions and finitely many dyadic scales. The same conclusions hold for the full maximal operator. We give a sharp Lorentz-index criterion in the range $q>d/(d-1)$ and extend the negative result to all multilinear spherical maximal operators with an $L^1$ input. We indicate the precise strong-type boundary statements to which these conclusions apply, without drawing conclusions about separate weak-type or interior estimates.

arXiv comment7 pages