Multiple conjugate functions and multiplicative Lipschitz classes

Ferenc Móricz

Colloquium Mathematicae (2009)

  • Volume: 115, Issue: 1, page 21-32
  • ISSN: 0010-1354

Abstract

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We extend the classical theorems of I. I. Privalov and A. Zygmund from single to multiple conjugate functions in terms of the multiplicative modulus of continuity. A remarkable corollary is that if a function f belongs to the multiplicative Lipschitz class L i p ( α ₁ , . . . , α N ) for some 0 < α ₁ , . . . , α N < 1 and its marginal functions satisfy f ( · , x ₂ , . . . , x N ) ∈ L i p β ₁ , . . . , f ( x ₁ , . . . , x N - 1 , · ) ∈ L i p β N for some 0 < β ₁ , . . . , β N < 1 uniformly in the indicated variables x l , 1 ≤ l ≤ N, then f ̃ ( η ₁ , . . . , η N ) ∈ L i p ( α ₁ , . . . , α N ) for each choice of ( η ₁ , . . . , η N ) with η l = 0 or 1 for 1 ≤ l ≤ N.

How to cite

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Ferenc Móricz. "Multiple conjugate functions and multiplicative Lipschitz classes." Colloquium Mathematicae 115.1 (2009): 21-32. <http://eudml.org/doc/283958>.

@article{FerencMóricz2009,
abstract = {We extend the classical theorems of I. I. Privalov and A. Zygmund from single to multiple conjugate functions in terms of the multiplicative modulus of continuity. A remarkable corollary is that if a function f belongs to the multiplicative Lipschitz class $Lip(α₁,..., α_N)$ for some $0 < α₁,...,α_N < 1$ and its marginal functions satisfy $f(·,x₂,...,x_N) ∈ Lip β₁,...,f(x₁,...,x_\{N-1\},·) ∈ Lip β_N$ for some $0 < β₁,...,β_N < 1$ uniformly in the indicated variables $x_\{l\}$, 1 ≤ l ≤ N, then $f̃^\{(η₁, ..., η_N)\} ∈ Lip(α₁, ..., α_N)$ for each choice of $(η₁,...,η_N)$ with $η_\{l\} = 0$ or 1 for 1 ≤ l ≤ N.},
author = {Ferenc Móricz},
journal = {Colloquium Mathematicae},
keywords = {Fourier series; conjugate series; conjugate functions; multiplicative modulus of continuity; total modulus of continutiy; multiplicative Lipschitz classes ; ...; N)Lip (1; ...; N); marginal functions},
language = {eng},
number = {1},
pages = {21-32},
title = {Multiple conjugate functions and multiplicative Lipschitz classes},
url = {http://eudml.org/doc/283958},
volume = {115},
year = {2009},
}

TY - JOUR
AU - Ferenc Móricz
TI - Multiple conjugate functions and multiplicative Lipschitz classes
JO - Colloquium Mathematicae
PY - 2009
VL - 115
IS - 1
SP - 21
EP - 32
AB - We extend the classical theorems of I. I. Privalov and A. Zygmund from single to multiple conjugate functions in terms of the multiplicative modulus of continuity. A remarkable corollary is that if a function f belongs to the multiplicative Lipschitz class $Lip(α₁,..., α_N)$ for some $0 < α₁,...,α_N < 1$ and its marginal functions satisfy $f(·,x₂,...,x_N) ∈ Lip β₁,...,f(x₁,...,x_{N-1},·) ∈ Lip β_N$ for some $0 < β₁,...,β_N < 1$ uniformly in the indicated variables $x_{l}$, 1 ≤ l ≤ N, then $f̃^{(η₁, ..., η_N)} ∈ Lip(α₁, ..., α_N)$ for each choice of $(η₁,...,η_N)$ with $η_{l} = 0$ or 1 for 1 ≤ l ≤ N.
LA - eng
KW - Fourier series; conjugate series; conjugate functions; multiplicative modulus of continuity; total modulus of continutiy; multiplicative Lipschitz classes ; ...; N)Lip (1; ...; N); marginal functions
UR - http://eudml.org/doc/283958
ER -

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