Suppose$μ$is a positive measure on $\mathbb{R}^{2}$ given by$μ$=$ν$×$λ$, where$ν$and$λ$are Radon measures on $\mathcal{S}^{1}$ and $\mathbb{R}^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .1ex\hbox {$\scriptscriptstyle +$}} {\scriptscriptstyle +}}}$ , respectively, which do not vanish on any open interval. We prove that if for either$ν$or$λ$there exists a set of positive measure$A$in its domain for which the upper and lower$s$-densities, 0<$s$≤1, are positive and finite for every$x$∈$A$then the uncentered Hardy–Littlewood maximal operator$M$_{$μ$}is weak-type (1,1) if and only if$ν$is doubling and$λ$is doubling away from the origin. This generalizes results of Vargas concerning rotation-invariant measures on $\mathbb{R}^{n}$ when$n$=2.

We study differentiability properties of a potential of the type K⋆μ, where μ is a finite Radon measure in RN and the kernel K satisfies |∇jK(x)|≤C|x|−(N−1+j),j=0,1,2. We introduce a notion of differentiability in the capacity sense, where capacity is classical capacity in the de la Vallée Poussin sense associated with the kernel |x|−(N−1). We require that the first order remainder at a point is small when measured by means of a normalized weak capacity “norm” in balls of small radii centered at the point. This implies weak LN/(N−1) differentiability and thus Lp differentiability in the Calderón–Zygmund sense for 1≤p<N/(N−1). We show that K⋆μ is a.e. differentiable in the capacity sense, thus strengthening a recent result by Ambrosio, Ponce and Rodiac. We also present an alternative proof of a quantitative theorem of the authors just mentioned, giving pointwise Lipschitz estimates for K⋆μ. As an application, we study level sets of newtonian potentials of finite Radon measures.

No abstract uploaded!

We give an explicit formula for the logarithmic potential of the asymptotic zero-counting measure of the sequence {(dn/dzn)(R(z)expT(z))}∞n=1. Here, R(z) is a rational function with at least two poles, all of which are distinct, and T(z) is a polynomial. This is an extension of a recent measure-theoretic refinement of Pólya’s Shire theorem for rational functions.

We show that sparse and Carleson coefficients are equivalent for every countable collection of Borel sets and hence, in particular, for dyadic rectangles, the case relevant to the theory of bi-parameter singular integrals. The key observation is that a dual refomulation by I. E. Verbitsky for Carleson coefficients over dyadic cubes holds also for Carleson coefficients over general sets.