In this paper we decompose D into diadic annuli {An : n ∈ N} and consider the class Sp,q of Toeplitz operators Tϕ for which the sequence of Schatten norms Tϕ n p n∈N belongs to q , where ϕn = ϕχA n . We study the boundedness and compactness of the operators in Sp,q and we describe the operators Tϕ, ϕ ≥ 0 in these spaces in terms of weighted Herz norms of the averaging operator of the symbols ϕ. (2000). Primary 47B35, Secondary 46E30. Mathematics Subject Classification

We study the interplay between the geometry of Hardy spaces and functional analytic properties of singular integral operators (SIO's), such as the Riesz transforms as well as Cauchy-Clifford and harmonic double-layer operator, on the one hand and, on the other hand, the regularity and geometric properties of domains of locally finite perimeter. Among other things, we give several characterizations of Euclidean balls, their complements, and half-spaces, in terms of the aforementioned SIO's.

We obtain harmonic extensions to the upper half-space of distributions in the weighted spaces w n+1 D L 1 , which according to [1] are the optimal spaces of tempered distributions S -convolvable with the classical euclidean version of the Poisson kernel . We also characterize the class of harmonic functions in the upper half-space with boundary * Partially supported by PAPIIT-IN105801. values in w n+1 D L 1 , generalizing in this way a classical result in the theory of Hardy spaces. Some facts concerning harmonic extensions of distributions in D L p , 1 < p ≤ ∞, are also approached in this paper, as well as natural relations among these spaces and the weighted spaces w n+1 D L 1 . We can also obtain n-harmonic extensions of appropriate weighted integrable distributions associated to a natural product domain version of the Poisson kernel.

The purpose of this article is to study the Hilbert space W 2 consisting of all solutions of the Helmholtz equation u + u = 0 in R 2 that are the image under the Fourier transform of L 2 densities in the unit circle. We characterize this space as a close subspace of the Hilbert space H 2 of all functions belonging to L 2 (|x| −3 dx) jointly with their angular and radial derivatives, in the complement of the unit disk in R 2 .