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.

The geometric descriptions of the (essential) spectra of Toeplitz operators with piecewise continuous symbols are among the most beautiful results about Toeplitz operators on Hardy spaces Hp with 1<p<∞. In the Hardy space H1, the essential spectra of Toeplitz operators are known for continuous symbols and symbols in the Douglas algebra C+H∞. It is natural to ask whether the theory for piecewise continuous symbols can also be extended to H1. We answer this question in the negative and show in particular that the Toeplitz operator is never bounded on H1 if its symbol has a jump discontinuity.

We consider the discrete Laplacian $\Delta$ on the cubic lattice $\mathbb{Z}^d$, and deduce estimates on the group $e^{i t \Delta}$ and the resolvent $(\Delta-z)^{-1}$, weighted by $\ell^q (\mathbb{Z}^d)$-weights for suitable $q \geqslant 2$. We apply the obtained results to discrete Schrödinger operators in dimension $d \geqslant 3$ with potentials from $\ell^p (\mathbb{Z}^d)$ with suitable $p \geqslant1$.

We characterize the indecomposable injective objects in the category of finitely presented representations of an interval finite quiver.

A generalization of a result of Wermer concerning the existence of polynomial hulls without analytic discs is presented. As a consequence it is shown that there exists a Cantor set X in C3 whose polynomial hull is strictly larger than X but contains no analytic discs.