TY - JOUR

T1 - On the maximal operators of weighted Marcinkiewicz type means of two-dimensional Walsh-Fourier series

AU - Nagy, Károly

AU - Salim, Mohamed

N1 - Publisher Copyright:
© 2021 Walter de Gruyter GmbH, Berlin/Boston 2022.

PY - 2022/2/1

Y1 - 2022/2/1

N2 - Goginava proved that the maximal operator σ α, ∗ σα,∗ (0 < α < 1 0<α<1) of two-dimensional Marcinkiewicz type (C, α) (C,α) means is bounded from the two-dimensional dyadic martingale Hardy space H p (G 2) Hp(G2 to the space L p (G 2) Lp(G2) for p > 2 2 + α p>22+α. Moreover, he showed that assumption p > 2 2 + α p>22+α is essential for the boundedness of the maximal operator σ α, ∗ σα,∗. It was shown that at the point p 0 = 2 2 + α p0=22+α the maximal operator σ α, ∗ σα,∗ is bounded from the dyadic Hardy space H 2/(2 + α) (G 2) H2/(2+α)(G2) to the space weak-L 2/(2 + α) (G 2) L2/(2+α)(G2)}. The main aim of this paper is to investigate the behaviour of the maximal operators of weighted Marcinkiewicz type σ α, ∗ σα,∗}} means (0 < α < 1 0<α<1) in the endpoint case p 0 = 2 2 + α p0=22+α. In particular, the optimal condition on the weights is given which provides the boundedness from H 2/(2 + α) (G 2) H2/(2+α)(G2) to L 2/(2 + α) (G 2) L2/(2+α)(G2). Furthermore, a strong summation theorem is stated for functions in the dyadic martingale Hardy space H 2/(2 + α) (G 2) H2/(2+α)(G2).

AB - Goginava proved that the maximal operator σ α, ∗ σα,∗ (0 < α < 1 0<α<1) of two-dimensional Marcinkiewicz type (C, α) (C,α) means is bounded from the two-dimensional dyadic martingale Hardy space H p (G 2) Hp(G2 to the space L p (G 2) Lp(G2) for p > 2 2 + α p>22+α. Moreover, he showed that assumption p > 2 2 + α p>22+α is essential for the boundedness of the maximal operator σ α, ∗ σα,∗. It was shown that at the point p 0 = 2 2 + α p0=22+α the maximal operator σ α, ∗ σα,∗ is bounded from the dyadic Hardy space H 2/(2 + α) (G 2) H2/(2+α)(G2) to the space weak-L 2/(2 + α) (G 2) L2/(2+α)(G2)}. The main aim of this paper is to investigate the behaviour of the maximal operators of weighted Marcinkiewicz type σ α, ∗ σα,∗}} means (0 < α < 1 0<α<1) in the endpoint case p 0 = 2 2 + α p0=22+α. In particular, the optimal condition on the weights is given which provides the boundedness from H 2/(2 + α) (G 2) H2/(2+α)(G2) to L 2/(2 + α) (G 2) L2/(2+α)(G2). Furthermore, a strong summation theorem is stated for functions in the dyadic martingale Hardy space H 2/(2 + α) (G 2) H2/(2+α)(G2).

KW - Cesàro mean

KW - Hardy space

KW - Marcinkiewicz mean

KW - Walsh-Paley system

KW - bounded operator

KW - maximal operator

KW - strong summation

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U2 - 10.1515/gmj-2021-2109

DO - 10.1515/gmj-2021-2109

M3 - Article

AN - SCOPUS:85117394540

SN - 1572-9176

VL - 29

SP - 71

EP - 82

JO - Georgian Mathematical Journal

JF - Georgian Mathematical Journal

IS - 1

ER -