On the maximal function for rotation invariant measures in ℝ n

Ana Vargas

Studia Mathematica (1994)

  • Volume: 110, Issue: 1, page 9-17
  • ISSN: 0039-3223

Abstract

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Given a positive measure μ in ℝ n , there is a natural variant of the noncentered Hardy-Littlewood maximal operator M μ f ( x ) = s u p x ∈ B 1 / μ ( B ) ʃ B | f | d μ , where the supremum is taken over all balls containing the point x. In this paper we restrict our attention to rotation invariant, strictly positive measures μ in ℝ n . We give some necessary and sufficient conditions for M μ to be bounded from L 1 ( d μ ) to L 1 , ∞ ( d μ ) .

How to cite

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Vargas, Ana. "On the maximal function for rotation invariant measures in $ℝ^{n}$." Studia Mathematica 110.1 (1994): 9-17. <http://eudml.org/doc/216102>.

@article{Vargas1994,
abstract = {Given a positive measure μ in $ℝ^n$, there is a natural variant of the noncentered Hardy-Littlewood maximal operator $M_\{μ\}f(x) = sup_\{x ∈ B\} 1/μ(B) ʃ_\{B\} |f|dμ$, where the supremum is taken over all balls containing the point x. In this paper we restrict our attention to rotation invariant, strictly positive measures μ in $ℝ^n$. We give some necessary and sufficient conditions for $M_μ$ to be bounded from $L^\{1\}(dμ)$ to $L^\{1,∞\}(dμ)$.},
author = {Vargas, Ana},
journal = {Studia Mathematica},
keywords = {maximal operators; weak type estimates; rotation invariant measures; Hardy-Littlewood maximal operator},
language = {eng},
number = {1},
pages = {9-17},
title = {On the maximal function for rotation invariant measures in $ℝ^\{n\}$},
url = {http://eudml.org/doc/216102},
volume = {110},
year = {1994},
}

TY - JOUR
AU - Vargas, Ana
TI - On the maximal function for rotation invariant measures in $ℝ^{n}$
JO - Studia Mathematica
PY - 1994
VL - 110
IS - 1
SP - 9
EP - 17
AB - Given a positive measure μ in $ℝ^n$, there is a natural variant of the noncentered Hardy-Littlewood maximal operator $M_{μ}f(x) = sup_{x ∈ B} 1/μ(B) ʃ_{B} |f|dμ$, where the supremum is taken over all balls containing the point x. In this paper we restrict our attention to rotation invariant, strictly positive measures μ in $ℝ^n$. We give some necessary and sufficient conditions for $M_μ$ to be bounded from $L^{1}(dμ)$ to $L^{1,∞}(dμ)$.
LA - eng
KW - maximal operators; weak type estimates; rotation invariant measures; Hardy-Littlewood maximal operator
UR - http://eudml.org/doc/216102
ER -

References

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  1. [M-S] B. Muckenhoupt and E. M. Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965), 17-92. Zbl0139.29002
  2. [S] P. Sjögren, A remark on the maximal function for measures in ℝ n , Amer. J. Math. 105 (1983), 1231-1233. 

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