Displaying similar documents to “On maximal functions over circular sectors with rotation invariant measures”

On the maximal function for rotation invariant measures in ℝ n

Ana Vargas (1994)

Studia Mathematica

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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 ) .

A new characterization of RBMO ( μ ) by John-Strömberg sharp maximal functions

Guoen Hu, Dachun Yang, Dongyong Yang (2009)

Czechoslovak Mathematical Journal

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Let μ be a nonnegative Radon measure on ℝ d which only satisfies μ ( B ( x , r ) ) ≤ C 0 r n for all x ∈ ℝ d , r > 0 , with some fixed constants C 0 > 0 and n ∈ ( 0 , d ] . In this paper, a new characterization for the space RBMO ( μ ) of Tolsa in terms of the John-Strömberg sharp maximal function is established.

Weak-star continuous homomorphisms and a decomposition of orthogonal measures

B. J. Cole, Theodore W. Gamelin (1985)

Annales de l'institut Fourier

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We consider the set S ( μ ) of complex-valued homomorphisms of a uniform algebra A which are weak-star continuous with respect to a fixed measure μ . The μ -parts of S ( μ ) are defined, and a decomposition theorem for measures in A ⊥ ∩ L 1 ( μ ) is obtained, in which constituent summands are mutually absolutely continuous with respect to representing measures. The set S ( μ ) is studied for T -invariant algebras on compact subsets of the complex plane and also for the infinite polydisc algebra.

Pointwise estimates for densities of stable semigroups of measures

Paweł Głowacki, Waldemar Hebisch (1993)

Studia Mathematica

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Let μ t be a symmetric α-stable semigroup of probability measures on a homogeneous group N, where 0 < α < 2. Assume that μ t are absolutely continuous with respect to Haar measure and denote by h t the corresponding densities. We show that the estimate h t ( x ) ≤ t Ω ( x / | x | ) | x | - n - α , x≠0, holds true with some integrable function Ω on the unit sphere Σ if and only if the density of the Lévy measure of the semigroup belongs locally to the Zygmund class LlogL(N╲e). The problem turns out to be related to the properties...

Metastability in the Furstenberg-Zimmer tower

Jeremy Avigad, Henry Towsner (2010)

Fundamenta Mathematicae

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According to the Furstenberg-Zimmer structure theorem, every measure-preserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measure-preserving systems to provide a perspicuous proof of Szemerédi’s theorem. Beleznay and Foreman showed that, in general, the transfinite construction of the maximal distal factor of a separable measure-preserving system can extend arbitrarily far into the countable...

Mean values and associated measures of δ -subharmonic functions

Neil A. Watson (2002)

Mathematica Bohemica

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Let u be a δ -subharmonic function with associated measure μ , and let v be a superharmonic function with associated measure ν , on an open set E . For any closed ball B ( x , r ) , of centre x and radius r , contained in E , let ℳ ( u , x , r ) denote the mean value of u over the surface of the ball. We prove that the upper and lower limits as s , t → 0 with 0 < s < t of the quotient ( ℳ ( u , x , s ) - ℳ ( u , x , t ) ) / ( ℳ ( v , x , s ) - ℳ ( v , x , t ) ) , lie between the upper and lower limits as r → 0 + of the quotient μ ( B ( x , r ) ) / ν ( B ( x , r ) ) . This enables us to use some well-known measure-theoretic results to prove new variants...

On the weak L 1 space and singular measures

Robert Kaufman (1982)

Annales de l'institut Fourier

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We study the class of singular measures whose Fourier partial sums converge to 0 in the metric of the weak L 1 space; symmetric sets of constant ratio occur in an unexpected way.