# Weak type (1,1) multipliers on LCA groups

Studia Mathematica (1997)

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

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topRaposo, 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

top- [ABB] N. Asmar, E. Berkson and J. Bourgain, Restrictions from ${\mathbb{R}}^{n}$ to ${\mathbb{Z}}^{n}$ of weak type (1, 1) multipliers, Studia Math. 108 (1994), 291-299. Zbl0827.42008
- [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. Zbl0855.43002
- [GR] J. García Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud. 46, North-Holland, 1985.
- [Ru] W. Rudin, Fourier Analysis on Groups, Wiley, 1990. Zbl0698.43001
- [Sk] S. B. Stechkin, On the best lacunary systems of functions, Izv. Akad. Nauk SSSR 25 (1961), 357-366 (in Russian).
- [Sz] S. J. Szarek, On the best constants in the Khinchin inequality, Studia Math. 58 (1976), 197-208. Zbl0424.42014

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