<p>In this paper, we characterize the weighted infinitesimal boundedness: for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varvec{0}\varvec{<}\varvec{\alpha }\varvec{<}\varvec{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mn mathvariant="bold">0</mn> </mrow> <mrow> <mo mathvariant="bold"><</mo> </mrow> <mrow> <mi mathvariant="bold-italic">α</mi> </mrow> <mrow> <mo mathvariant="bold"><</mo> </mrow> <mrow> <mi mathvariant="bold-italic">n</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varvec{1}\varvec{<}\varvec{p}\varvec{<}\varvec{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mn mathvariant="bold">1</mn> </mrow> <mrow> <mo mathvariant="bold"><</mo> </mrow> <mrow> <mi mathvariant="bold-italic">p</mi> </mrow> <mrow> <mo mathvariant="bold"><</mo> </mrow> <mrow> <mi mathvariant="bold-italic">∞</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <Equation ID="Equ1"> <EquationSource Format="TEX">$\begin{aligned} \Vert \varvec{V}\varvec{\phi }\Vert _{\varvec{L}^{\varvec{p}}\varvec{(w)}}^{\varvec{p}}\varvec{\le }\varvec{\epsilon }\Vert \varvec{(-\Delta )}^{\frac{\varvec{\alpha }}{\varvec{2}}}\varvec{\phi }\Vert _{\varvec{L}^{\varvec{p}}\varvec{(w)}}^{\varvec{p}}\varvec{+}\varvec{C}\varvec{(\epsilon )}\Vert \varvec{\phi }\Vert _{\varvec{L}^{\varvec{p}}\varvec{(w)}}^{\varvec{p}}. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msubsup> <mrow> <mo stretchy="false">‖</mo> <mrow> <mi mathvariant="bold-italic">V</mi> </mrow> <mrow> <mi mathvariant="bold-italic">ϕ</mi> </mrow> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mrow> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow> <mi mathvariant="bold-italic">p</mi> </mrow> </msup> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">w</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </mrow> <mrow> <mi mathvariant="bold-italic">p</mi> </mrow> </msubsup> <mrow> <mrow> <mo mathvariant="bold">≤</mo> </mrow> <mrow> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mo stretchy="false">‖</mo> </mrow> <msup> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mo mathvariant="bold">-</mo> <mi mathvariant="bold">Δ</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> <mfrac> <mrow> <mi mathvariant="bold-italic">α</mi> </mrow> <mrow> <mn mathvariant="bold">2</mn> </mrow> </mfrac> </msup> <msubsup> <mrow> <mrow> <mi mathvariant="bold-italic">ϕ</mi> </mrow> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mrow> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow> <mi mathvariant="bold-italic">p</mi> </mrow> </msup> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">w</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </mrow> <mrow> <mi mathvariant="bold-italic">p</mi> </mrow> </msubsup> <mrow> <mo mathvariant="bold">+</mo> </mrow> <mrow> <mi mathvariant="bold-italic">C</mi> </mrow> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">ϵ</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> <msubsup> <mrow> <mo stretchy="false">‖</mo> <mrow> <mi mathvariant="bold-italic">ϕ</mi> </mrow> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mrow> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow> <mi mathvariant="bold-italic">p</mi> </mrow> </msup> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">w</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </mrow> <mrow> <mi mathvariant="bold-italic">p</mi> </mrow> </msubsup> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>In particular, we extend the classical result due to Maz’ya and Verbitsky by using Carleson condition, localization estimates and capacity theory.</p>

Characterization of Weighted Infinitesimal Boundedness of Schrödinger Operator

期刊论文
发表日期：

2025年12月8日

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Yanhan Chen

摘要

In this paper, we characterize the weighted infinitesimal boundedness: for $\varvec{0}\varvec{<}\varvec{\alpha }\varvec{<}\varvec{n}$ $0 < α < n$ and $\varvec{1}\varvec{<}\varvec{p}\varvec{<}\varvec{\infty }$ $1 < p < \infty$ , $\begin{aligned} \Vert \varvec{V}\varvec{\phi }\Vert _{\varvec{L}^{\varvec{p}}\varvec{(w)}}^{\varvec{p}}\varvec{\le }\varvec{\epsilon }\Vert \varvec{(-\Delta )}^{\frac{\varvec{\alpha }}{\varvec{2}}}\varvec{\phi }\Vert _{\varvec{L}^{\varvec{p}}\varvec{(w)}}^{\varvec{p}}\varvec{+}\varvec{C}\varvec{(\epsilon )}\Vert \varvec{\phi }\Vert _{\varvec{L}^{\varvec{p}}\varvec{(w)}}^{\varvec{p}}. \end{aligned}$ $\begin{matrix} {‖ V ϕ ‖}_{L^{p} (w)}^{p} \leq ϵ ‖ {(- Δ)}^{\frac{α}{2}} {ϕ ‖}_{L^{p} (w)}^{p} + C (ϵ) {‖ ϕ ‖}_{L^{p} (w)}^{p} . \end{matrix}$ In particular, we extend the classical result due to Maz’ya and Verbitsky by using Carleson condition, localization estimates and capacity theory.