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Weighted weak type inequalities for certain maximal functions

Hugo AimarLiliana Forzani — 1991

Studia Mathematica

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.

The higher order Riesz transform for Gaussian measure need not be of weak type (1,1)

Liliana ForzaniRoberto Scotto — 1998

Studia Mathematica

The purpose of this paper is to prove that the higher order Riesz transform for Gaussian measure associated with the Ornstein-Uhlenbeck differential operator L : = d 2 / d x 2 - 2 x d / d x , x ∈ ℝ, need not be of weak type (1,1). A function in L 1 ( d γ ) , 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.

L p boundedness of Riesz transforms for orthogonal polynomials in a general context

Liliana ForzaniEmanuela SassoRoberto Scotto — 2015

Studia Mathematica

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 L p , 1 < p < ∞. We also discuss the symmetrized version of these transforms.

On maximal functions over circular sectors with rotation invariant measures

Hugo A. AimarLiliana ForzaniVirginia Naibo — 2001

Commentationes Mathematicae Universitatis Carolinae

Given a rotation invariant measure in n , we define the maximal operator over circular sectors. We prove that it is of strong type ( p , p ) for p > 1 and we give necessary and sufficient conditions on the measure for the weak type ( 1 , 1 ) inequality. Actually we work in a more general setting containing the above and other situations.

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