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<p>We prove a converse of Fatou type result for certain eigenfunctions of the Laplace–Beltrami operator on harmonic <i>NA</i> groups relating sectorial convergence and admissible convergence of Poisson type integrals of complex (signed) measures. This result improves and extends several results of this kind proved earlier in the context of the classical upper half space <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_836_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb R_+^{n+1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">R</mi> <mo>+</mo> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> </math></EquationSource> </InlineEquation>. Similar results are also obtained in the degenerate case of the real hyperbolic spaces.</p>

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Admissible and sectorial convergence of generalized Poisson integrals on harmonic \(\varvec{N\!A}\) groups

  • 期刊论文

  • 发表日期：

    2025年9月22日

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  • Utsav Dewan

摘要

We prove a converse of Fatou type result for certain eigenfunctions of the Laplace–Beltrami operator on harmonic NA groups relating sectorial convergence and admissible convergence of Poisson type integrals of complex (signed) measures. This result improves and extends several results of this kind proved earlier in the context of the classical upper half space \(\mathbb R_+^{n+1}\) R + n + 1 . Similar results are also obtained in the degenerate case of the real hyperbolic spaces.