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Form boundedness of the general second‐order differential Operator
Author(s) -
Maz′ya Vladimir G.,
Verbitsky Igor E.
Publication year - 2006
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20122
Subject(s) - mathematics , infinitesimal , differential operator , sobolev space , compact space , nabla symbol , order (exchange) , operator (biology) , laplace operator , pure mathematics , mathematical analysis , omega , chemistry , physics , finance , repressor , quantum mechanics , transcription factor , economics , gene , biochemistry
We give explicit necessary and sufficient conditions for the boundedness of the general second‐order differential operator$${\cal L} = \sum\limits_{i,\,j=1}^{n} a_{ij} \partial_{i} \partial_{j} + \sum\limits_{j=1}^{n} b_{j} \partial_{j} + c$$ with real‐ or complex‐valued distributional coefficients a ij , b j , and c , 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 Schrödinger operator $(i \nabla + \vec{a})^2 + q$ with magnetic vector potential $\vec{a} \in L^2_{{\rm loc}} (R^{n})^{n}$ and q ∈ D′ (ℝ n ). © 2005 Wiley Periodicals, Inc.