Eigenvalue estimates and asymptotics for weighted pseudodifferential operators with singular measures in the critical case

  • Grigori Rozenblum

    Chalmers University of Technology, Göteborg, Sweden; St. Petersburg State University; Russian Federation
  • Eugene Shargorodsky

    King's College London, UK; Technische Universität Dresden, Germany
Eigenvalue estimates and asymptotics for weighted pseudodifferential operators with singular measures in the critical case cover
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Abstract

We consider self-adjoint operators of the form TP,A=APA\mathbf{T}_{P,\mathfrak{A}}=\mathfrak{A}^* P \mathfrak{A} in a domain ΩRN\Omega\subset\mathbb{R}^\mathbf{N}, where A\mathfrak{A} is an order l=N2-l=-\frac{\mathbf{N}}{2} pseudodifferential operator in Ω\Omega and PP is a signed Borel measure with compact support in Ω\Omega. Measure PP may contain singular component. For a wide class of measures we establish eigenvalue estimates for operator TP,A\mathbf{T}_{P,\mathfrak{A}}. In case of measure PP being absolutely continuous with respect to the Hausdorff measure on a Lipschitz surface of an arbitrary dimension, we find the eigenvalue asymptotics. The order of eigenvalue estimates and asymptotics does not depend on dimensional characteristics of the measure, in particular, on the dimension of the surface supporting the measure.