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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.

Pointwise convergence to the initial data for nonlocal dyadic diffusions

Marcelo ActisHugo Aimar — 2016

Czechoslovak Mathematical Journal

We solve the initial value problem for the diffusion induced by dyadic fractional derivative s in + . First we obtain the spectral analysis of the dyadic fractional derivative operator in terms of the Haar system, which unveils a structure for the underlying “heat kernel”. We show that this kernel admits an integrable and decreasing majorant that involves the dyadic distance. This allows us to provide an estimate of the maximal operator of the diffusion by the Hardy-Littlewood dyadic maximal operator....

Gradual doubling property of Hutchinson orbits

Hugo AimarMarilina CarenaBibiana Iaffei — 2015

Czechoslovak Mathematical Journal

The classical self-similar fractals can be obtained as fixed points of the iteration technique introduced by Hutchinson. The well known results of Mosco show that typically the limit fractal equipped with the invariant measure is a (normal) space of homogeneous type. But the doubling property along this iteration is generally not preserved even when the starting point, and of course the limit point, both have the doubling property. We prove that the elements of Hutchinson orbits possess the doubling...

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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