On (C,1) summability for Vilenkin-like systems

G. Gát

Studia Mathematica (2001)

  • Volume: 144, Issue: 2, page 101-120
  • ISSN: 0039-3223

Abstract

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We give a common generalization of the Walsh system, Vilenkin system, the character system of the group of 2-adic (m-adic) integers, the product system of normalized coordinate functions for continuous irreducible unitary representations of the coordinate groups of noncommutative Vilenkin groups, the UDMD product systems (defined by F. Schipp) and some other systems. We prove that for integrable functions σₙf → f (n → ∞) a.e., where σₙf is the nth (C,1) mean of f. (For the character system of the group of m-adic integers, this proves a more than 20 years old conjecture of M. H. Taibleson [24, p. 114].) Define the maximal operator σ*f : = supₙ|σₙf|. We prove that σ* is of type (p,p) for all 1< p ≤ ∞ and of weak type (1,1). Moreover, | | σ * f | | ₁ ≤ c | | f | | H , where H is the Hardy space.

How to cite

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G. Gát. "On (C,1) summability for Vilenkin-like systems." Studia Mathematica 144.2 (2001): 101-120. <http://eudml.org/doc/284899>.

@article{G2001,
abstract = {We give a common generalization of the Walsh system, Vilenkin system, the character system of the group of 2-adic (m-adic) integers, the product system of normalized coordinate functions for continuous irreducible unitary representations of the coordinate groups of noncommutative Vilenkin groups, the UDMD product systems (defined by F. Schipp) and some other systems. We prove that for integrable functions σₙf → f (n → ∞) a.e., where σₙf is the nth (C,1) mean of f. (For the character system of the group of m-adic integers, this proves a more than 20 years old conjecture of M. H. Taibleson [24, p. 114].) Define the maximal operator σ*f : = supₙ|σₙf|. We prove that σ* is of type (p,p) for all 1< p ≤ ∞ and of weak type (1,1). Moreover, $||σ*f||₁ ≤ c||f||_\{H\}$, where H is the Hardy space.},
author = {G. Gát},
journal = {Studia Mathematica},
keywords = {Walsh system; -means; Vilenkin-like systems; group of 2-adic integers; UDMD systems},
language = {eng},
number = {2},
pages = {101-120},
title = {On (C,1) summability for Vilenkin-like systems},
url = {http://eudml.org/doc/284899},
volume = {144},
year = {2001},
}

TY - JOUR
AU - G. Gát
TI - On (C,1) summability for Vilenkin-like systems
JO - Studia Mathematica
PY - 2001
VL - 144
IS - 2
SP - 101
EP - 120
AB - We give a common generalization of the Walsh system, Vilenkin system, the character system of the group of 2-adic (m-adic) integers, the product system of normalized coordinate functions for continuous irreducible unitary representations of the coordinate groups of noncommutative Vilenkin groups, the UDMD product systems (defined by F. Schipp) and some other systems. We prove that for integrable functions σₙf → f (n → ∞) a.e., where σₙf is the nth (C,1) mean of f. (For the character system of the group of m-adic integers, this proves a more than 20 years old conjecture of M. H. Taibleson [24, p. 114].) Define the maximal operator σ*f : = supₙ|σₙf|. We prove that σ* is of type (p,p) for all 1< p ≤ ∞ and of weak type (1,1). Moreover, $||σ*f||₁ ≤ c||f||_{H}$, where H is the Hardy space.
LA - eng
KW - Walsh system; -means; Vilenkin-like systems; group of 2-adic integers; UDMD systems
UR - http://eudml.org/doc/284899
ER -

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