We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for theâ€¦ (More)

HyperkÃ¤hler embeddings and holomorphic symplectic geometry I. 0. Introduction. In this paper we are studying complex analytic subvarieties of a given KÃ¤hler manifold which is endowed with aâ€¦ (More)

We study trace inequalities of the type T k f L q (dÂµ) â‰¤ C f L p (dÏƒ) , f âˆˆ L p (dÏƒ), in the " upper triangle case " 1 â‰¤ q < p for integral operators T k with positive kernels, where dÏƒ and dÂµ areâ€¦ (More)

We give explicit analytic criteria for two problems associated with the SchrÃ¶dinger operator H = âˆ’âˆ† + Q on L 2 (R n) where Q âˆˆ D â€² (R n) is an arbitrary real-or complex-valued potential. First, weâ€¦ (More)

Background Joint work with Fedor Nazarov and Igor Verbitsky (preprint) Start with general functional analysis theorem: Background Joint work with Fedor Nazarov and Igor Verbitsky (preprint) Startâ€¦ (More)

A weighted norm inequality of Muckenhoupt-Wheeden type is obtained for gradients of solutions to a class of quasilinear equations with measure data on Reifenberg flat domains. This essentially leadsâ€¦ (More)

Let M be a complete non-compact Riemannian manifold and let Ïƒ be a Radon measure on M . We study the problem of existence or non-existence of positive solutions to a semilinear elliptic inequaliy âˆ’âˆ†uâ€¦ (More)

In this paper we give necessary and sufficient conditions for the existence of solutions to quasilinear equations of Lane-Emden type with measure data on a Carnot group G of arbitrary step. Theâ€¦ (More)

Let M be a complete non-compact Riemannian manifold and let Ïƒ be a Radon measure on M . We study the problem of existence or non-existence of positive solutions to a semilinear elliptic inequaliy âˆ’Î”uâ€¦ (More)