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

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 $$0\le \delta <\delta (d,p):=d(\frac{1}{2}-\frac{1}{p})-\frac{1}{2},\frac{2d}{d-1}<p<\mathrm{\infty },$$ 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 ${\mathbb{R}}^d\times (0,\mathrm{\infty })$. 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 ${\mathcal{C}}_{p,\delta }\mathrm{\ne }{L}^p({\mathbb{R}}^d)$, 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 $t\to \mathrm{\infty }$. A key ingredient in our argument is a multiparameter variant of the Bochner–Riesz means.