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 \[ 0\le \delta < \delta(d,p):=d\Big(\frac12-\frac1p\Big)-\frac12, \qquad \frac{2d}{d-1}<p<\infty, \] 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 $\mathcal C_{p,\delta}\neq L^p(\mathbb R^d)$, 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.