<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_701_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="150" /> </InlineMediaObject> <EquationSource Format="TEX">$\{T_{\alpha ,t}^{L}\}_{t>0}~(\alpha \in (0,1))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">{</mo> <msubsup> <mi>T</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>t</mi> </mrow> <mi>L</mi> </msubsup> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </msub> <mspace width="3.33333pt" /> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the fractional heat semigroup related to the Schrödinger operator <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_701_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="99" /> </InlineMediaObject> <EquationSource Format="TEX">$L:=-\Delta +V$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>:</mo> <mo>=</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo>+</mo> <mi>V</mi> </mrow> </math></EquationSource> </InlineEquation>, where <i>V</i> is non-negative potential belonging to the reverse Hölder class. In this paper, we establish the boundedness properties of the variation operator associated with the fractional heat semigroup <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_701_Article_IEq3.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$\{T_{\alpha ,t}^{L}\}_{t>0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msubsup> <mi>T</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>t</mi> </mrow> <mi>L</mi> </msubsup> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> on weighted Lebesgue spaces, weighted Morrey spaces and weighted BMO type spaces, respectively.</p>

Weighted boundedness of variation operators for fractional heat semigroups related to the Schrödinger operator

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发表日期：

2025年5月26日

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Zhiyong Wang,
Nan Zhao,
Pengtao Li,
Yu Liu

摘要

Let $\{T_{\alpha ,t}^{L}\}_{t>0}~(\alpha \in (0,1))$ ${T_{α, t}^{L}}_{t > 0} (α \in (0, 1))$ be the fractional heat semigroup related to the Schrödinger operator $L:=-\Delta +V$ $L : = - Δ + V$ , where V is non-negative potential belonging to the reverse Hölder class. In this paper, we establish the boundedness properties of the variation operator associated with the fractional heat semigroup $\{T_{\alpha ,t}^{L}\}_{t>0}$ ${T_{α, t}^{L}}_{t > 0}$ on weighted Lebesgue spaces, weighted Morrey spaces and weighted BMO type spaces, respectively.