We give an A_p type characterization for the pairs of weights (w,v) for which the maximal operator Mf(y) = sup 1/(b-a) ʃ_a^b |f(x)|dx, where the supremum is taken over all intervals [a,b] such that 0 ≤ a ≤ y ≤ b/ψ(b-a), is of weak type (p,p) with weights (w,v). Here ψ is a nonincreasing function such that ψ(0) = 1 and ψ(∞) = 0.
In this paper we study distribution and continuity properties of functions satisfying a vanishing mean oscillation property with a lag mapping on a space of homogeneous type.
The purpose of this paper is to prove that the higher order Riesz transform for Gaussian measure associated with the Ornstein-Uhlenbeck differential operator , x ∈ ℝ, need not be of weak type (1,1). A function in , where dγ is the Gaussian measure, is given such that the distribution function of the higher order Riesz transform decays more slowly than C/λ.
Nowak and Stempak (2006) proposed a unified approach to the theory of Riesz transforms and conjugacy in the setting of multi-dimensional orthogonal expansions, and proved their boundedness on L². Following them, we give easy to check sufficient conditions for their boundedness on , 1 < p < ∞. We also discuss the symmetrized version of these transforms.
Given a rotation invariant measure in , we define the maximal operator over circular sectors. We prove that it is of strong type for and we give necessary and sufficient conditions on the measure for the weak type inequality. Actually we work in a more general setting containing the above and other situations.
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