Weak type (1,1) multipliers on LCA groups

José Raposo

Studia Mathematica (1997)

  • Volume: 122, Issue: 2, page 123-130
  • ISSN: 0039-3223

Abstract

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In [ABB] Asmar, Berkson and Bourgain prove that for a sequence ϕ j j = 1 ∞ of weak type (1, 1) multipliers in ℝ n and a function k ∈ L 1 ( ℝ n ) the weak type (1,1) constant of the maximal operator associated with k ⁎ ϕ j j is controlled by that of the maximal operator associated with ϕ j j . In [ABG] this theorem is extended to LCA groups with an extra hypothesis: the multipliers must be continuous. In this paper we prove a more general version of this last result without assuming the continuity of the multipliers. The proof arises after simplifying the one in [ABB] which becomes then extensible to LCA groups.

How to cite

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Raposo, José. "Weak type (1,1) multipliers on LCA groups." Studia Mathematica 122.2 (1997): 123-130. <http://eudml.org/doc/216364>.

@article{Raposo1997,
abstract = {In [ABB] Asmar, Berkson and Bourgain prove that for a sequence $\{ϕ_j\}^∞_\{j=1\} $ of weak type (1, 1) multipliers in $ℝ^n$ and a function $k ∈ L^1(ℝ^n)$ the weak type (1,1) constant of the maximal operator associated with $\{k⁎ϕ_j\}_j$ is controlled by that of the maximal operator associated with $\{ϕ_j\}_j$. In [ABG] this theorem is extended to LCA groups with an extra hypothesis: the multipliers must be continuous. In this paper we prove a more general version of this last result without assuming the continuity of the multipliers. The proof arises after simplifying the one in [ABB] which becomes then extensible to LCA groups.},
author = {Raposo, José},
journal = {Studia Mathematica},
keywords = {weak type multipliers; maximal operators; vector inequalities; locally compact abelian group; operator; weak type (1,1) multiplier},
language = {eng},
number = {2},
pages = {123-130},
title = {Weak type (1,1) multipliers on LCA groups},
url = {http://eudml.org/doc/216364},
volume = {122},
year = {1997},
}

TY - JOUR
AU - Raposo, José
TI - Weak type (1,1) multipliers on LCA groups
JO - Studia Mathematica
PY - 1997
VL - 122
IS - 2
SP - 123
EP - 130
AB - In [ABB] Asmar, Berkson and Bourgain prove that for a sequence ${ϕ_j}^∞_{j=1} $ of weak type (1, 1) multipliers in $ℝ^n$ and a function $k ∈ L^1(ℝ^n)$ the weak type (1,1) constant of the maximal operator associated with ${k⁎ϕ_j}_j$ is controlled by that of the maximal operator associated with ${ϕ_j}_j$. In [ABG] this theorem is extended to LCA groups with an extra hypothesis: the multipliers must be continuous. In this paper we prove a more general version of this last result without assuming the continuity of the multipliers. The proof arises after simplifying the one in [ABB] which becomes then extensible to LCA groups.
LA - eng
KW - weak type multipliers; maximal operators; vector inequalities; locally compact abelian group; operator; weak type (1,1) multiplier
UR - http://eudml.org/doc/216364
ER -

References

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  1. [ABB] N. Asmar, E. Berkson and J. Bourgain, Restrictions from ℝ n to ℤ n of weak type (1, 1) multipliers, Studia Math. 108 (1994), 291-299. 
  2. [ABG] N. Asmar, E. Berkson and T. A. Gillespie, Convolution estimates and generalized de Leuw Theorems for multipliers of weak type (1, 1), Canad. J. Math. 47 (1995), 225-245. 
  3. [GR] J. García Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud. 46, North-Holland, 1985. 
  4. [Ru] W. Rudin, Fourier Analysis on Groups, Wiley, 1990. 
  5. [Sk] S. B. Stechkin, On the best lacunary systems of functions, Izv. Akad. Nauk SSSR 25 (1961), 357-366 (in Russian). 
  6. [Sz] S. J. Szarek, On the best constants in the Khinchin inequality, Studia Math. 58 (1976), 197-208. 

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