### A new proof of Harnack's inequality for elliptic partial differential equations in divergence form.

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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 $L:={d}^{2}/d{x}^{2}-2xd/dx$, x ∈ ℝ, need not be of weak type (1,1). A function in ${L}^{1}\left(d\gamma \right)$, 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 ${L}^{p}$, 1 < p < ∞. We also discuss the symmetrized version of these transforms.

Given a rotation invariant measure in ${\mathbb{R}}^{n}$, 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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