On the maximal operators of weighted Marcinkiewicz type means of two-dimensional Walsh–Fourier series

Abstract

Abstract Goginava proved that the maximal operator σ α , * {\sigma^{\alpha,*}} ( 0 < α < 1 {0<\alpha<1} ) of two-dimensional Marcinkiewicz type ( C , α ) {(C,\alpha)} means is bounded from the two-dimensional dyadic martingale Hardy space H p ⁢ ( G 2 ) {H_{p}(G^{2})} to the space L p ⁢ ( G 2 ) {L^{p}(G^{2})} for p > 2 2 + α {p>\frac{2}{2+\alpha}} . Moreover, he showed that assumption p > 2 2 + α {p>\frac{2}{2+\alpha}} is essential for the boundedness of the maximal operator σ α , * {\sigma^{\alpha,*}} . It was shown that at the point p 0 = 2 2 + α {p_{0}=\frac{2}{2+\alpha}} the maximal operator σ α , * {\sigma^{\alpha,*}} is bounded from the dyadic Hardy space H 2 / ( 2 + α ) ⁢ ( G 2 ) {H_{2/(2+\alpha)}(G^{2})} to the space weak- L 2 / ( 2 + α ) ⁢ ( G 2 ) {L^{2/(2+\alpha)}(G^{2})} . The main aim of this paper is to investigate the behaviour of the maximal operators of weighted Marcinkiewicz type σ α , * {\sigma^{\alpha,*}} means ( 0 < α < 1 {0<\alpha<1} ) in the endpoint case p 0 = 2 2 + α {p_{0}=\frac{2}{2+\alpha}} . In particular, the optimal condition on the weights is given which provides the boundedness from H 2 / ( 2 + α ) ⁢ ( G 2 ) {H_{2/(2+\alpha)}(G^{2})} to L 2 / ( 2 + α ) ⁢ ( G 2 ) {L^{2/(2+\alpha)}(G^{2})} . Furthermore, a strong summation theorem is stated for functions in the dyadic martingale Hardy space H 2 / ( 2 + α ) ⁢ ( G 2 ) {H_{2/(2+\alpha)}(G^{2})} .