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A Functorial Approach to Multi-Space Interpolation with Function Parameters

Thomas Lamby, Samuel Nicolay

TL;DR

This work develops a functorial, function-parameterized extension of real interpolation to multi-space settings, enabling explicit construction of intermediate spaces for $(A_0,\dots,A_n)$ and stability under operations such as powers and convex combinations. Central to the approach are Boyd functions, which govern the indexing and scaling via indices $\underline{b}(\phi)$ and $\overline{b}(\phi)$, and enable a unified treatment of limiting cases. The authors define multi-space $K$- and $J$-type interpolants $K_p^{\phi_1,\dots,\phi_n}(\mathbf{A})$ and $J_p^{\phi_1,\dots,\phi_n}(\mathbf{A})$, establish their fundamental embeddings, and prove (i) non-equivalence in general for $n>1$, (ii) conditions $(\mathcal{H}_2)$ and $(\mathcal{H}_1)$ under which exact or bounded interpolation holds, and (iii) a power theorem linking scaled spaces $\mathbf{A}^{(q)}$. Applications include the interpolation of multiple generalized Sobolev spaces to generalized Besov spaces, and three-space reiteration results that extend the Stein-Weiss framework to block-Lorentz spaces, with potential impact on multi-parameter interpolation in Sobolev/Lorentz-type scales.

Abstract

We introduce an extension of interpolation theory to more than two spaces by employing a functional parameter, while retaining a fully functorial and systematic framework. This approach allows for the construction of generalized intermediate spaces and ensures stability under natural operations such as powers and convex combinations. As a significant application, we demonstrate that the interpolation of multiple generalized Sobolev spaces yields a generalized Besov space. Our framework provides explicit tools for handling multi-parameter interpolation, highlighting both its theoretical robustness and practical relevance.

A Functorial Approach to Multi-Space Interpolation with Function Parameters

TL;DR

This work develops a functorial, function-parameterized extension of real interpolation to multi-space settings, enabling explicit construction of intermediate spaces for and stability under operations such as powers and convex combinations. Central to the approach are Boyd functions, which govern the indexing and scaling via indices and , and enable a unified treatment of limiting cases. The authors define multi-space - and -type interpolants and , establish their fundamental embeddings, and prove (i) non-equivalence in general for , (ii) conditions and under which exact or bounded interpolation holds, and (iii) a power theorem linking scaled spaces . Applications include the interpolation of multiple generalized Sobolev spaces to generalized Besov spaces, and three-space reiteration results that extend the Stein-Weiss framework to block-Lorentz spaces, with potential impact on multi-parameter interpolation in Sobolev/Lorentz-type scales.

Abstract

We introduce an extension of interpolation theory to more than two spaces by employing a functional parameter, while retaining a fully functorial and systematic framework. This approach allows for the construction of generalized intermediate spaces and ensures stability under natural operations such as powers and convex combinations. As a significant application, we demonstrate that the interpolation of multiple generalized Sobolev spaces yields a generalized Besov space. Our framework provides explicit tools for handling multi-parameter interpolation, highlighting both its theoretical robustness and practical relevance.
Paper Structure (9 sections, 30 theorems, 154 equations, 1 figure)

This paper contains 9 sections, 30 theorems, 154 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Boyd functions
  3. Interpolation
  4. On the equivalence theorem
  5. Some properties
  6. Power theorem
  7. Stability
  8. Interpolation of generalized Sobolev spaces
  9. Extension of the Stein-Weiss Theorem

Key Result

Proposition 3.13

Let $p\in [1,\infty]$ and $\phi_1,\dotsc,\phi_n\in \mathcal{B}$ satisfying $(\mathcal{H}_1)$; the functor $K_p^{\phi_1,\dotsc,\phi_n}$ is an exact interpolation functor of type $f$, where Moreover,

Figures (1)

  • Figure 1: Illustration inspired of Spa:74 of Theorem \ref{['thm pour fig1']} with $m=3$ and $n=2$.

Theorems & Definitions (83)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Definition 3.7
  • ...and 73 more