Let be a sequence in the upper half plane. If and ifhas solution in the class of Poisson integrals of functions for any sequence , then we show that is an interpolating sequence for . If , has solution in the class of Poisson integrals of BMO functions whenever , then is again an interpolating sequence for . A somewhat more general theorem is also proved and a counterexample for the case is described.
We study the high-dimensional Hausdorff operators on the Morrey space and on the Campanato space. We establish their sharp boundedness on these spaces. Particularly, our results solve an open question posted by E. Liflyand (2013).
In this short note we present new integral formulas for the Hessian determinant. We use them for new definitions of Hessian under minimal regularity assumptions. The Hessian becomes a continuous linear functional on a Sobolev space.
We derive various integral representation formulas for a function minus a polynomial in terms of vector field gradients of the function of appropriately high order. Our results hold in the general setting of metric spaces, including those associated with Carnot-Carathéodory vector fields, under the assumption that a suitable to Poincaré inequality holds. Of particular interest are the representation formulas in Euclidean space and stratified groups, where polynomials exist and to Poincaré...
We study spaces of analytic functions generated by homogeneous polynomials from the dual space to the symmetric Hilbertian tensor product of a Hilbert space. In particular, we introduce an analogue of the classical Hardy space H² on the Hilbert unit ball and investigate spectral decomposition of unitary operators on this space. Also we prove a Wiener-type theorem for an algebra of analytic functions on the Hilbert unit ball.
We introduce an alternative proof of the existence of certain Ck barrier maps, with polynomial explosion of the derivatives, on weakly pseudoconvex domains in Cn. Barriers of this sort have been constructed very recently by J. Michel and M.-C. Shaw, and have various applications. In our paper, the adaptation of Hörmander's L2 techniques to suitable vector-valued functions allows us to give a very simple approach of the problem and to improve some aspects of the result of Michel and Shaw, regarding...
Let D be a bounded strict pseudoconvex non-smooth domain in Cn. In this paper we prove that the estimates in Lp and Lipschitz classes for the solutions of the ∂-equation with Lp-data in regular strictly pseudoconvex domains (see [2]) are also valid for D. We also give estimates of the same type for the ∂b in the regular part of the boundary of these domains.
We prove that every Sobolev function defined on a metric space coincides with a Hölder continuous function outside a set of small Hausdorff content or capacity. Moreover, the Hölder continuous function can be chosen so that it approximates the given function in the Sobolev norm. This is a generalization of a result of Malý [Ma1] to the Sobolev spaces on metric spaces [H1].