arXiv subjects

I. E. Verbitsky

Publications and source records attributed to I. E. Verbitsky.

6 recordsLinked to original sources

arXiv2019

Accretivity and form boundedness of second order differential operators

Let $\mathcal{L}$ be the general second order differential operator with complex-valued distributional coefficients $A=(a_{jk})_{j, k=1}^n$, $\vec{b}=(b_{j})_{j=1}^n$, and $c$ in an open set $Ω\subseteq \mathbb{R}^n$ ($n \ge 1$), with principal part either in the divergence form, $\mathcal{L} u= {\rm div} \, (A \nabla u) + \vec{b} \cdot\nabla u + c \, u$, or non-divergence form, $ \mathcal L u= \sum_{j, \, k=1}^n \, a_{jk} \, \partial_j \partial_k u + \vec{b} \cdot\nabla u + c \, u $. We give a survey of the results by the authors which characterize the following two properties of $\mathcal{L}$: (1) $-\mathcal{L}$ is accretive, i.e., ${\rm Re} \, \langle -\mathcal L u, \, u\rangle \ge 0$; (2) $\mathcal L$ is form bounded, i.e., $\vert \langle \mathcal L u, u \rangle \vert \le C \, \Vert \nabla u \Vert_{L^2(Ω)}^2$, for all complex-valued $u \in C^\infty_0(Ω)$.

math.AP↗

arXiv2018

Accretivity of the general second order linear differential operator

For the general second order linear differential operator $$\mathcal L_0 = \sum_{j, \, k=1}^n \, a_{jk} \, \partial_j \partial_k + \sum_{j=1}^n \, b_{j} \, \partial_j + c$$ with complex-valued distributional coefficients $a_{jk}$, $b_{j}$, and $c$ in an open set $Ω\subseteq \mathbf{R}^n$ ($n \ge 1$), we present conditions which ensure that $-\mathcal L_0$ is accretive, i.e., ${\rm Re} \, \langle -\mathcal L_0 ϕ, ϕ\rangle \ge 0$ for all $ϕ\in C^\infty_0(Ω).$

math.AP↗

arXiv2004

Form boundedness of the general second order differential operator

We give explicit necessary and sufficient conditions for the boundedness of the general second order differential operator L with real- or complex-valued distributional coefficients acting from the Sobolev space W^{1,2}(R^n) to its dual W^{-1,2}(R^n). This enables us to obtain analytic criteria for the fundamental notions of relative form boundedness, compactness, and infinitesimal form boundedness of L with respect to the Laplacian on L^2(R^n). In particular, we establish a complete characterization of the form boundedness of the Schroedinger operator (i \nabla + a)^2 + q with magnetic vector potential a \in L^2_{loc} and q \in D'(\R^n).

math.AP↗

arXiv2004

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

We give explicit analytic criteria for two problems associated with the Schrödinger operator $H = -Δ+ Q$ on $L^2(\R^n)$ where $Q\in D'(\R^n)$ is an arbitrary real- or complex-valued potential. First, we obtain necessary and sufficient conditions on $Q$ so that the quadratic form $ $ has zero relative bound with respect to the Laplacian. For $Q\in L^1_{\rm loc}(\R^n)$, this property can be expressed in the form of the integral inequality: $$ | \int_{\R^n} |u(x)|^2 Q(x) dx | \leq ε||\nabla u||^2_{L^2(\R^n)} + C(ε) ||u||^2_{L^2(\R^n)}, \quad \forall u \in C^\infty_0(\R^n), $$ for an arbitrarily small $ε>0$ and some $C(ε)> 0$. Secondly, we characterize Trudinger's subordination property where $C(ε)$ in the above inequality is subject to the condition $C(ε) \le c {ε^{-β}}$ ($β>0$) as $ε\to +0$. Such quadratic form inequalities can be understood entirely in the framework of Morrey--Campanato spaces, using mean oscillations of $\nabla (1-Δ)^{-1} Q$ and $(1-Δ)^{-1} Q$ on balls or cubes. As a consequence, we characterize the class of those $Q$ which satisfy a multiplicative quadratic from inequality of Nash's type.

math.FA↗

arXiv2003

Nonlinear potentials and two weight trace inequalities for general dyadic and radial kernels

We study trace inequalities of the type $$ \| T_k f\|_{L^q(dμ)}\leq C \|f\|_{L^p(dσ)}, \qquad f \in L^p(dσ), $$ in the ``upper triangle case'' $1 \leq q<p$ for integral operators $T_k$ with positive kernels, where $dσ$ and $dμ$ are positive Borel measures on $\R^n$. Our main tool is a generalization of Th. Wolff's inequality which gives two-sided estimates of the energy ${\mathcal E}_{k, σ} [μ]=\int_{\R^n} (T_k [μ])^{p'} d σ$ through the $L^1(dμ)$-norm of an appropriate nonlinear potential $W_{k, σ}[μ]$ associated with the kernel $k$ and measures $dμ$, $d σ$. We initially work with a dyadic integral operator with kernel $K_{\mathcal D}(x, y) = \sum_{Q\in{\mathcal D}} K(Q) χ_Q(x) χ_Q(y)$, where $\mathcal D=\{Q\}$ is the family of all dyadic cubes in $\R^n$, and $K: {\mathcal D}\to \R^+$. The corresponding continuous versions of Wolff's inequality and trace inequalities are derived from their dyadic counterparts.

math.FA↗

arXiv2003

The form boundedness criterion for the relativistic Schrödinger operator

We establish necessary and sufficient conditions for the boundedness of the relativistic Schrödinger operator $\mathcal{H} = \sqrt{-Δ} + Q$ from the Sobolev space $W^{1/2}_2 (\R^n)$ to its dual $W^{-1/2}_2 (\R^n)$, for an arbitrary real- or complex-valued potential $Q$ on $\R^n$. %Analogous results for %$\mathcal{H}_m = \sqrt{-Δ+ m^2} - m + Q$, as well as %the corresponding compactness criteria are obtained. In other words, we give a complete solution to the problem of the domination of the potential energy by the kinetic energy in the relativistic case characterized by the inequality $$ | \int_{\R^n} |u(x)|^2 Q(x) dx | \leq \text{const} ||u||^2_{W_2^{1/2}}, \quad u \in C^\infty_0(\R^n), $$ where the ``indefinite weight'' $Q$ is a locally integrable function (or, more generally, a distribution) on $\R^n$. Along with necessary and sufficient results, we also present new broad classes of admissible potentials $Q$ in the scale of Morrey spaces of negative order, and discuss their relationship to well-known $L_p$ and Fefferman-Phong conditions.

math-ph↗