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<p>Consider a non-negative, self-adjoint, maximally subelliptic operator on a compact manifold. We show that the spectral multiplier is a singular integral operator under an appropriate Mihlin–Hörmander type condition. We establish the equivalence between non-isotropic Besov and Triebel–Lizorkin spaces adapted to the operator and those adapted to a Carnot–Carathéodory geometry on the manifold. We also give a Mihlin–Hörmander type condition for the boundedness of the spectral multiplier on non-isotropic <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3685_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> Sobolev spaces.</p>

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Spectral multipliers for maximally subelliptic operators

  • 期刊论文

  • 发表日期：

    2025年2月7日

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  • Lingxiao Zhang

摘要

Consider a non-negative, self-adjoint, maximally subelliptic operator on a compact manifold. We show that the spectral multiplier is a singular integral operator under an appropriate Mihlin–Hörmander type condition. We establish the equivalence between non-isotropic Besov and Triebel–Lizorkin spaces adapted to the operator and those adapted to a Carnot–Carathéodory geometry on the manifold. We also give a Mihlin–Hörmander type condition for the boundedness of the spectral multiplier on non-isotropic \(L^p\) L p Sobolev spaces.