We consider the space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textrm{D}^\alpha _n(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mtext>D</mtext> <mi>n</mi> <mi>α</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> consisting of functions u(x), harmonic in a bounded domain <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^{d+1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> with smooth boundary <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>, satisfying <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$|\nabla ^n u(x)|^2 \rho (x)^{\alpha }\in L_1(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mi>n</mi> </msup> <msup> <mrow> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mi>ρ</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>α</mi> </msup> <mo>∈</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha >-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\rho (x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is the distance from x to the boundary. For a Borel measure <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> and a weight function V, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>-measurable, we study the operator defined by means of the quadratic form <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\pmb {\mu }_V[u]=\int V(x) |\gamma u(x)|^2 \mu (dx)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mi mathvariant="bold-italic">μ</mi> </mrow> <mi>V</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>u</mi> <mo stretchy="false">]</mo> </mrow> <mo>=</mo> <mo>∫</mo> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">|</mo> <mi>γ</mi> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> is, properly defined, operator of restriction of functions <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$u\in \textrm{D}^\alpha _n(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>∈</mo> <msubsup> <mtext>D</mtext> <mi>n</mi> <mi>α</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>. Main interest is directed to the case of a singular measure <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> possessing some Ahlfors regularity properties. For such operators, we establish two-sided estimates for singular values, and, under some geometrical conditions, Weyl type eigenvalue asymptotics.

Boundary Toeplitz type operators in weighted harmonic Sobolev spaces

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2025年8月18日

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Grigori Rozenblum

摘要

We consider the space $\textrm{D}^\alpha _n(\Omega )$ $D_{n}^{α} (Ω)$ consisting of functions u(x), harmonic in a bounded domain $\Omega \subset \mathbb {R}^{d+1}$ $Ω \subset R^{d + 1}$ with smooth boundary $\Gamma $ $Γ$ , satisfying $|\nabla ^n u(x)|^2 \rho (x)^{\alpha }\in L_1(\Omega )$ $| \nabla^{n} {u (x) |}^{2} ρ {(x)}^{α} \in L_{1} (Ω)$ , $\alpha >-1$ $α > - 1$ , where $\rho (x)$ $ρ (x)$ is the distance from x to the boundary. For a Borel measure $\mu $ $μ$ on $\Gamma $ $Γ$ and a weight function V, $\mu $ $μ$ -measurable, we study the operator defined by means of the quadratic form $\pmb {\mu }_V[u]=\int V(x) |\gamma u(x)|^2 \mu (dx)$ $μ_{V} [u] = \int V (x) {| γ u (x) |}^{2} μ (d x)$ , where $\gamma $ $γ$ is, properly defined, operator of restriction of functions $u\in \textrm{D}^\alpha _n(\Omega )$ $u \in D_{n}^{α} (Ω)$ to $\Gamma $ $Γ$ . Main interest is directed to the case of a singular measure $\mu $ $μ$ possessing some Ahlfors regularity properties. For such operators, we establish two-sided estimates for singular values, and, under some geometrical conditions, Weyl type eigenvalue asymptotics.