Estimates of three classical summations on the spaces $\dot F_p^{\alpha ,q} (\mathbb{R}^n )(0 < p \leqslant 1)$

Abstract

We obtain the boundedness on \(\dot F_p^{\alpha ,q} (\mathbb{R}^n )(0 < p \leqslant 1)\) for the Poisson summation and Gauss summation. Their maximal operators are proved to be bounded from \(\dot F_p^{\alpha ,q} (\mathbb{R}^n )(0 < p \leqslant 1)\) to \(L^\infty (\mathbb{R}^n )\). For the maximal operator of the Bochner-Riesz summation, we prove that it is bounded from \(\dot F_p^{\alpha ,q} (\mathbb{R}^n )(0 < p \leqslant 1)\) to \(L^{\tfrac{{pn}} {{n - p\alpha }},\infty } (\mathbb{R}^n ) \).