A note on the strong summability of the Riesz means of multiple Fourier series

Abstract:

Let $S_t^\alpha (x,f)$ be the Riesz means of order $\alpha$ of an integrable function $f(x)$ on $N$-dimensional torus ${T^N}(N \geqslant 2)$, that is, \[ S_t^\alpha (x,f) = \sum \limits _{{{\left | m \right |}^2} < t} {{{\left ( {1 - \frac {{{{\left | m \right |}^2}}}{t}} \right )}^\alpha }\hat f(m){e^{2\pi imx}}.} \] E. M. Stein has shown that if $1 < p \leqslant 2$ and $\alpha > {\alpha _p}$ where \[ {\alpha _p} = \frac {{N - 1}}{2}\left ( {\frac {2}{p} - 1} \right ) - \frac {1}{{p’}} = \frac {{N - 1}}{2} - \frac {N}{{p’}},\] then for any function $f(x) \in {L^p}({T^N})S_t^\alpha (x,f)$ is strong summable to $f(x)$, that is, \[ \lim \limits _{T \to \infty } \frac {1}{T}\int _0^T {{{\left | {S_t^\alpha (x,f) - f(x)} \right |}^2}dt = 0} \] for almost every $x$. In this paper we shall show that if $1 \leqslant p \leqslant 2$ and $- 1 < \alpha < {\alpha _p}$, then there exists a function $f(x) \in {L^p}({T^N})$ such that \[ \frac {1}{T} \int _0^T |S_t^\alpha (x,f)|^2 dt - \Omega (T^{\alpha _p - \alpha } \log ^{-2\tau } T) \quad \text {as $T \to \infty $} \] for every $x$ and every $\tau > 1/p$, in particular, \[ \overline {\lim }\limits _{T \to \infty } \frac {1}{T} \int _0^T | S_t^\alpha (x,f) |^2 dt = \infty \] for every $x$, where we can take for $f(x),{f_{\sigma \tau }}(x)$ such that ${\hat f_{\sigma \tau }}(m) = 1/{\left | m \right |^\sigma }{\log ^\tau }\left | m \right |,\left | m \right | > 1$.