<p>In this paper, we introduce the class <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_626_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {E}_{m,F}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">E</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>F</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and solve complex <i>m</i>-Hessian equations in the class <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_626_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {E}_{m,F}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">E</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>F</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. After that, we study subextension in the class <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_626_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {E}_{m,F}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">E</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>F</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with the weighted Hessian measure of subextension unchanged. This is an extensive version of the result in [<CitationRef CitationID="CR24">24</CitationRef>] in the case when the smaller domain is relatively compact inside the large one.</p>

Weighted energy class of m-subharmonic functions

期刊论文
发表日期：

2025年2月14日

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Hoang Thieu Anh,
Nguyen Van Phu,
Nguyen Quang Dieu

摘要

In this paper, we introduce the class $\mathcal {E}_{m,F}(\Omega )$ $E_{m, F} (Ω)$ and solve complex m-Hessian equations in the class $\mathcal {E}_{m,F}(\Omega )$ $E_{m, F} (Ω)$ . After that, we study subextension in the class $\mathcal {E}_{m,F}(\Omega )$ $E_{m, F} (Ω)$ with the weighted Hessian measure of subextension unchanged. This is an extensive version of the result in [24] in the case when the smaller domain is relatively compact inside the large one.