<p>In this paper, we establish weak type estimates on the Lorentz space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10224_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">$L^{r_0,1}({\mathbb R}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <msub> <mi>r</mi> <mn>0</mn> </msub> <mo>,</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for spectral multipliers and Bochner-Riesz means related to the Schrödinger operator with inverse-square potential <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10224_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="133" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {L_a}=-\Delta +a|x|^{-2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">L</mi> <mi mathvariant="script">a</mi> </msub> <mo>=</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo>+</mo> <mi>a</mi> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>. Such kind of endpoint estimates are peculiar for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10224_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {L_a}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">L</mi> <mi mathvariant="script">a</mi> </msub> </math></EquationSource> </InlineEquation>. The proofs are based on some rearrangement inequalities.</p>

Endpoint Estimates for Two Operators Related to Schrödinger Operators with Inverse-square Potential

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发表日期：

2025年5月13日

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Xudong Lai

摘要

In this paper, we establish weak type estimates on the Lorentz space $L^{r_0,1}({\mathbb R}^d)$ $L^{r_{0}, 1} (R^{d})$ for spectral multipliers and Bochner-Riesz means related to the Schrödinger operator with inverse-square potential $\mathcal {L_a}=-\Delta +a|x|^{-2}$ $L_{a} = - Δ + a {| x |}^{- 2}$ . Such kind of endpoint estimates are peculiar for $\mathcal {L_a}$ $L_{a}$ . The proofs are based on some rearrangement inequalities.