On almost everywhere convergence of Bochner--Riesz means below the critical index

Abstract

In this paper, we study the almost everywhere convergence problem for the Bochner--Riesz means $S_t^{\delta }f$ for $f\in L^p({\mathbb{R}}^d)$ in the subcritical range \[ 0\le δ< δ(d,p):=d\Big(\frac12-\frac1p\Big)-\frac12, \qquad \frac{2d}{d-1}<p<\infty, \] where $d\ge 2$. In this regime, the operator need not be well defined for fixed $t>0$, even as a tempered distribution. Nevertheless, the family $\{S_t^{\delta }f{\}}_{t>0}$ can still be interpreted as a distribution on ${\mathbb{R}}^d\times (0,\infty )$. We introduce an admissible class ${\mathcal{C}}_{p,\delta }\subset L^p({\mathbb{R}}^d)$ on which this distribution has sufficient regularity in the $t$ variable to formulate the almost everywhere convergence problem. We establish three results concerning this class. First, we show that ${\mathcal{C}}_{p,\delta }\ne L^p({\mathbb{R}}^d)$, and thus the almost everywhere convergence problem cannot be formulated for all $L^p$ functions below the critical index. Second, we show that this admissible class is nevertheless large in the sense that, for every $f\in L^p({\mathbb{R}}^d)$, the pullback $f_V=f(V^{-1}\cdot )$ is admissible for Haar-a.e. volume-preserving upper triangular matrix $V$ with positive diagonal entries. Finally, we construct an $f\in {\mathcal{C}}_{p,\delta }$ for which $S_t^{\delta }f$ fails to converge almost everywhere as $t\to \infty$. A key ingredient in our argument is a multiparameter variant of the Bochner--Riesz means.

30 pages. Major revision; substantially expanded and reorganized, with a new title