Igor Verbitsky

Nonlinear equations and weighted norm inequalities

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 the solvability of the Dirichlet problem −∆u = v u + w, u ≥ 0 on Ω, u = 0 on ∂Ω, on a regular domain Ω in Rn in the “superlinear case” q > 1. The coefficients v, w are arbitrary positive measurable functions (or measures) on Ω. We also consider more general nonlinear differential and integral equations, and study the spaces of coefficients and solutions naturally associated with these problems, as well as the corresponding capacities. Our characterizations of the existence of positive solutions take into account the interplay between v, w, and the corresponding Green’s kernel. They are not only sufficient, but also necessary, and are established without any a priori regularity assumptions on v and w; we also obtain sharp two-sided estimates of solutions up to the boundary. Some of our results are new even if v ≡ 1 and Ω is a ball or half-space. The corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so-called 3G-inequality by an elementary “integration by parts” argument. This also gives a new unified proof for some classical inequalities including the Carleson measure theorem for Poisson integrals and trace inequalities for Riesz potentials and Green potentials.

The Schrödinger operator on the energy space: boundedness and compactness criteria

We characterize the class of measurable functions (or, more generally, realor complexvalued distributions) V such that the Schr5dinger operator H=-A§ maps the energy o space L~(R n) to its dual L~-I(Rn). Similar results are obtained for the inhomogeneous Sobolev space W~(Rn). In other words, we give a complete solution to the problem of the relative form-boundedness of the potential energy operator V with respect to the Laplacian --A, which is fundamental to quantum mechanics. Relative compactness criteria for the corresponding quadratic forms are established as well. We also give analogous boundedness and compactness criteria for Sobolev spaces on domains f t C R n under mild restrictions on 012. One of the main goals of the present paper is to give necessary and sufficient conditions for the classical inequality

Criteria of solvability for multidimensional Riccati equations

We study the solvability problem for the multidimensional Riccati equation −∇u=|∇u|q+ω, whereq>1 and ω is an arbitrary nonnegative function (or measure). We also discuss connections with the classical problem of the existence of positive solutions for the Schrödinger equation −Δu−ωu=0 with nonnegative potential ω. We establish explicit criteria for the existence of global solutions onRn in terms involving geometric (capacity) estimates or pointwise behavior of Riesz potentials, together with sharp pointwise estimates of solutions and their gradients. We also consider the corresponding nonlinear Dirichlet problem on a bounded domain, as well as more general equations of the type−Lu=f(x, u, ∇u)+ω where, andL is a uniformly elliptic operator.

Nonlinear potentials and trace inequalities

Characterizations of trace inequalities for Sobolev spaces of the type and their generalizations are considered; here ω is a Borel measure on a domain Ω ⊂ R n , and ∇ m is the gradient of order m. A survey of the known results is presented which reflects the pioneering work of V. Maz’ya on this problem. Recent developments involving nonlinear potentials are discussed. In particular, a simple proof of the Kerman-Sawyer theorem for q = p, and a new characterization of trace inequalities for 0 1, in the case Ω = R n are given.

WEIGHTED NORM INEQUALITIES FOR INTEGRAL OPERATORS

We consider a large class of positive integral operators acting on functions which are defined on a space of homogeneous type with a group structure. We show that any such operator has a discrete (dyadic) version which is always essentially equivalent in norm to the original operator. As an application, we study conditions of “testing type,” like those initially introduced by E. Sawyer in relation to the Hardy-Littlewood maximal function, which determine when a positive integral operator satisfies two-weight weak-type or strong-type (Lp, Lq) estimates. We show that in such a space it is possible to characterize these estimates by testing them only over “cubes”. We also study some pointwise conditions which are sufficient for strong-type estimates and have applications to solvability of certain nonlinear equations.

Infinitesimal form boundedness and Trudinger’s subordination for the Schrödinger operator

AbstractWe give explicit analytic criteria for two problems associated with the Schrödinger operator H=-Δ+Q on L2(ℝn) where Q∈D’(ℝn) is an arbitrary real- or complex-valued potential.First, we obtain necessary and sufficient conditions on Q so that the quadratic form

Wolff's Inequality for Radially Nonincreasing Kernels and Applications to Trace Inequalities

\langle{Q}\cdot,\ \cdot\rangle

Local Integral Estimates and Removable Singularities for Quasilinear and Hessian Equations with Nonlinear Source Terms

has zero relative bound with respect to the Laplacian. For Q∈L1loc(ℝn), this property can be expressed in the form of the integral inequality:

Local and Global Behaviour of Solutions to Nonlinear Equations with Natural Growth Terms

\left\vert\int_{\mathbb{R}^n} |u(x)|^2 Q(x) dx \right\vert\leq\epsilon\| \nabla u \|^2_{L^2(\mathbb{R}^n)} + C(\epsilon) \|u \|^2_{L^2(\mathbb{R}^n)}, \quad\forall u \in C^{\infty}_0(\mathbb{R}^n),