# On the maximal function for rotation invariant measures in ${\mathbb{R}}^{n}$

Studia Mathematica (1994)

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

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topVargas, 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

top- [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
- [S] P. Sjögren, A remark on the maximal function for measures in ${\mathbb{R}}^{n}$, Amer. J. Math. 105 (1983), 1231-1233.

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