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Abstract <p>We consider a “superposition operator" obtained through the continuous superposition of operators of mixed fractional order, modulated by a signed finite Borel measure defined over the set [0,&#xa0;1]. The relevance of this operator is rooted in the fact that it incorporates special and significant cases of interest, like the mixed operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(-\Delta + (-\Delta )^s\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo>+</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> <mi>s</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, the (possibly) infinite sum of fractional Laplacians and allows to consider operators carrying a “wrong sign". We first outline weak and strong maximum principles for this type of operators. Then, we complete the spectral analysis for the related Dirichlet eigenvalue problem started in&#xa0;[<CitationRef CitationID="CR10">10</CitationRef>].</p> Graphic Abstract <p>Portrait of Sandro by Luigi Serafini.<InlineMediaObject> <ImageObject Color="Color" FileRef="MediaObjects/44007_2026_204_Figa_HTML.gif" Format="GIF" Height="336" Rendition="HTML" Resolution="120" Type="Halftone" Width="462" /> </InlineMediaObject></p>

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Maximum principles and spectral analysis for the superposition of operators of fractional order

  • 期刊论文

  • 发表日期：

    2026年5月6日

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  • Serena Dipierro,
  • Edoardo Proietti Lippi,
  • Caterina Sportelli,
  • Enrico Valdinoci

摘要

Abstract

We consider a “superposition operator" obtained through the continuous superposition of operators of mixed fractional order, modulated by a signed finite Borel measure defined over the set [0, 1]. The relevance of this operator is rooted in the fact that it incorporates special and significant cases of interest, like the mixed operator \(-\Delta + (-\Delta )^s\) - Δ + ( - Δ ) s , the (possibly) infinite sum of fractional Laplacians and allows to consider operators carrying a “wrong sign". We first outline weak and strong maximum principles for this type of operators. Then, we complete the spectral analysis for the related Dirichlet eigenvalue problem started in [10].

Graphic Abstract

Portrait of Sandro by Luigi Serafini.