Riesz means of Fourier series and integrals: Strong summability at the critical index

We consider spherical Riesz means of multiple Fourier series and some generalizations. While almost everywhere convergence of Riesz means at the critical index $(d-1)\mathrm{/}2$ may fail for functions in the Hardy space $h^1(\mathbb T^d)$, we prove sharp positive results for strong summability almost everywhere. For functions in $L^p({\mathbb{T}}^d)$, $1<p<2$, we consider Riesz means at the critical index $d(1\mathrm{/}p-1\mathrm{/}2)-1\mathrm{/}2$ and prove an almost sharp theorem on strong summability. The results follow via transference from corresponding results for Fourier integrals. We include an endpoint bound on maximal operators associated with generalized Riesz means on Hardy spaces $H^p(\mathbb R^d)$for$0<p<1$.