Uniform Boundedness of Sequence of Operators Associated with the Walsh System and Their Pointwise Convergence

Abstract

Revisiting the main point of the almost everywhere convergence, it becomes clear that a weak (1,1)-type inequality must be established for the maximal operator corresponding to the sequence of operators. The better route to take in obtaining almost everywhere convergence is by using the uniform boundedness of the sequence of operator, instead of using the mentioned maximal type of inequality. In this paper it is proved that a sequence of operators, defined by matrix transforms of the Walsh–Fourier series, is convergent almost everywhere to the function $f\in L_{1}$ if they are uniformly bounded from the dyadic Hardy space $H_{1} \left( {\mathbb {I}}\right)$ to $L_{1}\left( \mathbb {I}\right)$. As a further matter, the characterization of the points are put forth where the sequence of the operators of the matrix transform is convergent.