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 - 1072-947X
VL - 29
SP - 71
EP - 82
JO - Georgian Mathematical Journal
JF - Georgian Mathematical Journal
IS - 1
ER -