TY - JOUR
T1 - Restricted summability of the multi-dimensional Cesàro means of Walsh–Kaczmarz–Fourier series
AU - Nagy, Károly
AU - Salim, Mohamed
N1 - Publisher Copyright:
© 2019 University of Debrecen, Institute of Mathematics. All rights reserved.
PY - 2019
Y1 - 2019
N2 - The properties of the maximal operator of the (C, α)-means (α = (α1, . . ., αd)) of the multi-dimensional Walsh–Kaczmarz–Fourier series are discussed, where the set of indices is inside a cone-like set. We prove that the maximal operator is bounded from dyadic Hardy space Hp γ to Lebesgue space Lp for p0 < p (p0 = max{1/(1 + αk): k = 1, . . ., d}) and is of weak type (1, 1). As a corollary, we get a theorem of Simon on the a.e. convergence of cone-restricted two-dimensional Fejér means of integrable functions. In the endpoint case p = p0, we show that the maximal operator σL κ,α, is not bounded from the dyadic Hardy space Hp γ 0 to the Lebesgue space Lp0.
AB - The properties of the maximal operator of the (C, α)-means (α = (α1, . . ., αd)) of the multi-dimensional Walsh–Kaczmarz–Fourier series are discussed, where the set of indices is inside a cone-like set. We prove that the maximal operator is bounded from dyadic Hardy space Hp γ to Lebesgue space Lp for p0 < p (p0 = max{1/(1 + αk): k = 1, . . ., d}) and is of weak type (1, 1). As a corollary, we get a theorem of Simon on the a.e. convergence of cone-restricted two-dimensional Fejér means of integrable functions. In the endpoint case p = p0, we show that the maximal operator σL κ,α, is not bounded from the dyadic Hardy space Hp γ 0 to the Lebesgue space Lp0.
KW - A.e. convergence
KW - Cesàro means
KW - Maximal operator
KW - Multi-dimensional system
KW - Restricted summability
KW - Walsh–Kaczmarz system
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U2 - 10.5486/PMD.2019.8344
DO - 10.5486/PMD.2019.8344
M3 - Article
AN - SCOPUS:85071052751
SN - 0033-3883
VL - 94
SP - 381
EP - 394
JO - Publicationes Mathematicae Debrecen
JF - Publicationes Mathematicae Debrecen
IS - 3-4
ER -