TY - CHAP
T1 - Eigenvalue estimates and asymptotics for weighted pseudodifferential operators with singular measures in the critical case
AU - Shargorodsky, Eugene
AU - Rozenblum, Grigori
PY - 2021/6
Y1 - 2021/6
N2 - We consider self-adjoint operators of the form $ \mathbf{T}_{P,\mathfrak{A}}=\mathfrak{A}^* P \mathfrak{A}$ in a domain $\Omega\subset \mathbb{R}^\mathbf{N},$ where $\mathfrak{A}$ is an order $-l=-\mathbf{N}/2$ pseudodifferential operator in $\Om$ and $P$ is a signed Borel measure with compact support in $\Omega$. Measure $P$ may contain singular component. For a wide class of measures we establish eigenvalue estimates for operator $\mathbf{T}_{P,\mathfrak{A}}.$ In case of measure $P$ 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.
AB - We consider self-adjoint operators of the form $ \mathbf{T}_{P,\mathfrak{A}}=\mathfrak{A}^* P \mathfrak{A}$ in a domain $\Omega\subset \mathbb{R}^\mathbf{N},$ where $\mathfrak{A}$ is an order $-l=-\mathbf{N}/2$ pseudodifferential operator in $\Om$ and $P$ is a signed Borel measure with compact support in $\Omega$. Measure $P$ may contain singular component. For a wide class of measures we establish eigenvalue estimates for operator $\mathbf{T}_{P,\mathfrak{A}}.$ In case of measure $P$ 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.
M3 - Chapter
SP - 331
EP - 354
BT - Partial Differential Equations, Spectral Theory, and Mathematical Physics: The Ari Laptev Anniversary Volume
PB - European Mathematical Society
ER -