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The Kato problem for operators with weighted ellipticity. (English) Zbl 1326.47051

In this paper, the authors consider the Kato problem, which can be stated basically as: given a complex-valued \(n\times n\) elliptic matrix \({\mathbf A}\), the square root \(\mathcal{L}^{1/2}\) of the operator \(\mathcal{L}=-\operatorname{div} \mathbf {A}\nabla\) has domain \(H^1\) and satisfies \[ \|\mathcal{L}^{1/2}f\|_{L^2} \approx \|\nabla f\|_{L^2}. \]
They prove that, if \(w\) is a Muckemhoupt \(A_2\) weight, then the Kato problem is solved for the operators \(\mathcal{L}_w=-w^{-1}\operatorname{div}{\mathbf {A}}_w\nabla\), where \({\mathbf A}_w\) is a complex-valued \(n\times n\) matrix such that \(w^{-1}{\mathbf A}_w\) is complex elliptic, in the weighted space \(L^2(w)\).

MSC:

47F05 General theory of partial differential operators
35J15 Second-order elliptic equations
35D30 Weak solutions to PDEs
47D06 One-parameter semigroups and linear evolution equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
35B45 A priori estimates in context of PDEs
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References:

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