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Classical Herz spaces (in the Euclidean setting \(\mathbb {R}^{n-1}\) ) are defined in the spirit of the Littlewood-Paley theory, in which one localizes functions on concentric dyadic annuli, measures their size using \(L^p\) norms, and assembles these numerical pieces all together using a “weighted” \(\ell ^q\) norm. In this same philosophy, we introduce a new class of Herz space, which we refer to as a Geometric Herz space. This new brand of Herz space generalizes its classical counterpart in that now, the functions under consideration are localized on progressively more distant “tubular” regions mimicking the shape of a given fixed closed set in \(\mathbb {R}^{n-1}\) of (Hausdorff) dimension \(d\in [0,n-1)\) . Additionally, we show that these spaces are stable under the real method of interpolation. This permits us to obtain basic boundedness and embedding results which open the door for the treatment of boundary value problems. In particular, we show how Poisson kernels can be effectively employed to ultimately prove well-posedness results for the Dirichlet Problem with boundary data prescribed in such Geometric Herz spaces.

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Geometric Herz Spaces and Poisson Kernels in the Upper Half-Space

  • 会议论文

  • 发表日期：

    2025年

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  • J. Cruz-Lugo,
  • M. Laurel,
  • M. Mitrea,
  • P. Takemura

摘要

Classical Herz spaces (in the Euclidean setting \(\mathbb {R}^{n-1}\) ) are defined in the spirit of the Littlewood-Paley theory, in which one localizes functions on concentric dyadic annuli, measures their size using \(L^p\) norms, and assembles these numerical pieces all together using a “weighted” \(\ell ^q\) norm. In this same philosophy, we introduce a new class of Herz space, which we refer to as a Geometric Herz space. This new brand of Herz space generalizes its classical counterpart in that now, the functions under consideration are localized on progressively more distant “tubular” regions mimicking the shape of a given fixed closed set in \(\mathbb {R}^{n-1}\) of (Hausdorff) dimension \(d\in [0,n-1)\) . Additionally, we show that these spaces are stable under the real method of interpolation. This permits us to obtain basic boundedness and embedding results which open the door for the treatment of boundary value problems. In particular, we show how Poisson kernels can be effectively employed to ultimately prove well-posedness results for the Dirichlet Problem with boundary data prescribed in such Geometric Herz spaces.