Bourgain's method for K-closedness in the semicommmutative setting

Abstract: In the early 1990s, J.Bourgain proved a general result $K$-closedness result in the context of classical harmonic analysis. In this paper, we extend Bourgain's method to the semicommutative setting, making use of the recent semicommutative Calderón-Zygmund decomposition introduced by L.Cadilhac, JM.Conde-Alonso and J.Parcet. As an application, we recover Pisier's result about $K$-closedness of noncommutative Hardy spaces on the torus, and we also establish new interpolation results for noncommutative Sobolev spaces on the torus.

• The paper extends Bourgain's K-closedness method to semicommutative settings by adapting Calderón-Zygmund decomposition for operator-valued Hardy spaces.
• It utilizes noncommutative projections and interpolation theory to establish quasi-complementation in Lₚ spaces with uniform constants.
• The findings yield new invariance and interpolation properties for noncommutative Hardy and Sobolev spaces, impacting quantum probability and operator theory.

Bourgain's Method for $K$-Closedness in the Semicommutative Setting

Introduction and Context

This paper presents a thorough extension of Bourgain's $K$-closedness method, originally rooted in harmonic analysis on commutative spaces, to the semicommutative—i.e., noncommutative but still tensorized with commutative measure spaces—setting. Bourgain's original work demonstrated that analytic Hardy spaces form an interpolation scale for the real method due to $K$-closedness of Hardy subcouples inside Lebesgue spaces. The present work generalizes this framework by leveraging the recently-developed semicommutative Calderón-Zygmund decompositions and applies the results to operator-valued Hardy and Sobolev spaces, primarily on the torus.

Technical Framework and Preliminaries

The starting point is the recognition that $K$-closedness (or quasi-complementation) of Hardy-type subcouples $(H_1, H_\infty)$ in their ambient Lebesgue couple $(L_1, L_\infty)$ is both necessary and sufficient for forming real interpolation scales. The classical result, due to Jones and Pisier, states that there is a universal constant $C$ such that for any $f$,

$K_t(f, H_1, H_\infty) \leq C K_t(f, L_1, L_\infty)$

where $K_t$ is the Peetre $K$0-functional.

Bourgain's commutative method utilized projections defined by singular convolution kernels, the Calderón-Zygmund decomposition, and interpolation theory to establish $K$1-closedness without recourse to complex variable techniques. The present work recasts this strategy in the context of semifinite von Neumann algebras and their associated $K$2 spaces, relying on the formalism of abstract Hardy spaces and operator-valued integration/interpolation theory.

Extension of Bourgain's Method

The main technical innovation is the adaptation of Bourgain's $K$3-closedness argument to the semicommutative framework via the noncommutative Calderón-Zygmund decomposition developed by Cadilhac, Conde-Alonso, and Parcet. Here, the tensor product $K$4 plays the role of operator-valued functions over a space of homogeneous type, equipped with a semifinite trace.

The principal theorem asserts that for a projection $K$5 defined by a singular kernel (with Calderón-Zygmund regularity and technical compatibility with the $K$6-structure), the subcouple $K$7 is quasi-complemented in $K$8 with uniform constants: $K$9 for all $K$0 in the algebraic sum $K$1 and all $K$2.

Further, the extension naturally covers projections acting as operator-valued Fourier multipliers, such as the Riesz and Leray projections on the (noncommutative) torus, since their kernels satisfy the required Calderón-Zygmund and Hörmander-type conditions in this setting. The technical apparatus includes identification and explicit construction of the required singular kernels.

Applications to Noncommutative Hardy and Sobolev Spaces

A major application is to the real and complex interpolation scales for operator-valued Hardy and Sobolev spaces on the torus. Specifically, the work recovers the Pisier-Xu interpolation theorem for operator-valued Hardy spaces without using complexification: $K$3 with equivalence of norms and constants depending only on $K$4, where $K$5 (2604.23864).

A second non-trivial application is to noncommutative Sobolev spaces. The paper proves an operator-valued analogue of the classical DeVore-Scherer result for Sobolev spaces: $K$6 with consequences for the entire interpolation scale: $K$7 and also for the complex interpolation scale: $K$8 with equivalence of norms and constants depending on the parameters only.

Technical Ingredients

The crux of the extension lies in two technical achievements:

1. Semicommutative Calderón-Zygmund Decomposition: The adaptation to the operator-valued (tensor) setting, where the key projection and decomposition steps (Cuculescu's construction, control of supports, norm estimates) are made compatible with noncommutative integration.
2. Interpolation Functoriality and Duality: Precise control of interpolated subspaces, duality via trace pairing, and rank/density/closure arguments ensure that the Hardy and Sobolev spaces retain their interpolation structure.

Notably, the technical apparatus accommodates a general class of spaces of homogeneous type, not only the torus, provided Ahlfors regularity and appropriate doubling/reverse-doubling conditions hold for the underlying measure metric spaces. The Hörmander and Lipschitz regularity conditions for kernels are verified explicitly for all operator-valued multipliers under consideration.

Implications and Directions

The results clarify longstanding technical obstructions in extending $K$9-closedness and interpolation for Hardy and Sobolev spaces to the operator-valued and noncommutative regime. These provide foundations not only for further developments in noncommutative harmonic analysis and quantum probability, but also for the more general study of abstract Hardy spaces in the context of $K$0- and von Neumann algebraic structures.

The methodology is robust and invites further study in several directions:

• Extending to other interpolation couples, e.g., operator-valued BMO and more general Calderón couples.
• Application to quantum groups, noncommutative geometry, and noncommutative martingale theory.
• Exploring the boundary cases and the open question of characterizing complex interpolation spaces for $K$1.

Conclusion

This work establishes a comprehensive extension of Bourgain's $K$2-closedness method to the semicommutative setting, relying on the semicommutative Calderón-Zygmund decomposition and advanced interpolation theory. It provides unified operator-valued proofs of $K$3-closedness for Hardy and Sobolev spaces, recovers classical results in the noncommutative domain, and yields new invariance and interpolation properties with broad ramifications for functional analysis and operator theory (2604.23864).