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$R$-boundedness of Poisson operators

Robert Denk, Nick Lindemulder, Jörg Seiler

TL;DR

This work develops a comprehensive framework for the $R$-boundedness of parameter-dependent Poisson operators on the half-space, connecting symbol-kernel structures $S^d_P$ and weak variants to maximal $L^q$-regularity for boundary-value problems with dynamic boundary conditions. By establishing $R$-boundedness across Besov, Triebel–Lizorkin, weak $L^p$–$L^q$, Sobolev, and weighted/edge-degenerate spaces, the authors enable robust maximal-regularity results for PDEs with boundary dynamics, including scalar diffusion, Cahn–Hilliard-type dynamics, and Kolmogorov–Petrovskii–Pisconov models. The analysis hinges on parameter-dependent multiplier techniques (Mikhlin), transmission properties, and carefully designed Poisson symbol classes, together with interpolation stability to transfer $R$-bounds across scales. The results have broad applicability to bulk–surface problems and PDEs with dynamic or nonlocal boundary interactions, providing a solid foundation for well-posedness and regularity in time-dependent settings. Overall, the paper advances the PDE boundary-value theory by linking $R$-bounded families of Poisson operators to maximal regularity in diverse functional frameworks.

Abstract

We investigate the $R$-boundedness of parameter-dependent families of Poisson operators on the half-space $\mathbb R^n_+$ in various scales of function spaces. Applications concern maximal $L_q$-regularity for boundary value problems with dynamic boundary conditions.

$R$-boundedness of Poisson operators

TL;DR

This work develops a comprehensive framework for the -boundedness of parameter-dependent Poisson operators on the half-space, connecting symbol-kernel structures and weak variants to maximal -regularity for boundary-value problems with dynamic boundary conditions. By establishing -boundedness across Besov, Triebel–Lizorkin, weak –, Sobolev, and weighted/edge-degenerate spaces, the authors enable robust maximal-regularity results for PDEs with boundary dynamics, including scalar diffusion, Cahn–Hilliard-type dynamics, and Kolmogorov–Petrovskii–Pisconov models. The analysis hinges on parameter-dependent multiplier techniques (Mikhlin), transmission properties, and carefully designed Poisson symbol classes, together with interpolation stability to transfer -bounds across scales. The results have broad applicability to bulk–surface problems and PDEs with dynamic or nonlocal boundary interactions, providing a solid foundation for well-posedness and regularity in time-dependent settings. Overall, the paper advances the PDE boundary-value theory by linking -bounded families of Poisson operators to maximal regularity in diverse functional frameworks.

Abstract

We investigate the -boundedness of parameter-dependent families of Poisson operators on the half-space in various scales of function spaces. Applications concern maximal -regularity for boundary value problems with dynamic boundary conditions.

Paper Structure

This paper contains 19 sections, 31 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Some background on R-boundedness
  3. Poisson operators
  4. An example
  5. Parameter-dependent symbol-kernels
  6. Characterizations of Poisson operators
  7. A first result on $R$-boundedness of Poisson operators
  8. $R$-boundedness of parameter-dependent Poisson operators
  9. $R$-boundedness in Besov spaces
  10. $R$-boundedness in (weak) $L^p$-$L^q$-spaces
  11. $R$-boundedness in classical Sobolev spaces
  12. $R$-boundedness in other function spaces
  13. Spaces with totally characteristic normal derivative
  14. Edge degenerate Sobolev spaces
  15. Weighted spaces
  16. ...and 4 more sections

Key Result

Theorem 2.2

Let $X$ be a Banach space and let $A\subset\mathscr{C}^\infty(\mathbb{R}^{n}\setminus\{0\},\mathscr{L}(X))$ be such that is an $R$-bounded subset of $\mathscr{L}(X)$. If $X$ is a UMD space and has Pisier's property $(\alpha)$ then $\{a(D)\mid a\in A\}$ is an $R$-bounded subset of $\mathscr{L}(H^s_p(\mathbb{R}^n;X))$, $\mathscr{L}(B^s_{pq}(\mathbb{R}^n;X))$, and $\mathscr{L}(F^s_{pq}(\mathbb{R}^n;

Theorems & Definitions (65)

  • Definition 2.1
  • Theorem 2.2: Mikhlin's multiplier theorem
  • Corollary 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Lemma 2.8
  • proof
  • Corollary 2.9
  • ...and 55 more