Generalized Sobolev-Orlicz spaces based on the Riesz fractional gradient as interpolation and potential spaces

Abstract: In this work we establish that the recently introduced fractional Sobolev spaces based on the Riesz fractional gradient of Musielak-Orlicz functions by one of the authors, coincide with the space of Bessel potentials of functions on such generalized Orlicz setting. Moreover, we identify them as complex interpolation spaces, and exploiting the well known properties for interpolation of operators we obtain several structural properties for those spaces.

• The paper demonstrates that the fractional Sobolev-Orlicz space defined by the Riesz fractional gradient is equivalent to the Bessel-Orlicz potential space, unifying classical and Musielak-Orlicz settings.
• It characterizes these spaces as complex interpolation spaces between generalized L_A and first-order Orlicz-Sobolev spaces using advanced harmonic analysis and multiplier techniques.
• The study presents embedding, duality, and compactness results that enhance understanding of nonlocal PDEs and variational problems with nonstandard growth.

Generalized Sobolev-Orlicz Spaces via the Riesz Fractional Gradient: Interpolation and Potential Space Characterization

Introduction

This work rigorously investigates the structure and functional-analytic properties of generalized fractional Sobolev-Orlicz spaces associated with the Riesz fractional gradient. The central theme is the identification of these spaces both as Bessel potential spaces and as complex interpolation spaces within the Musielak-Orlicz setting. The authors employ advanced machinery from harmonic analysis, interpolation theory, and nonlocal calculus of variations to demonstrate equivalences and structural properties relevant to analysis and applications in PDEs, especially in nonlocal and long-range interaction models.

Framework and Principal Definitions

The study focuses on Sobolev-type spaces constructed via the Riesz fractional gradient $V^s$ (following Shieh and Spector) in the generalized Musielak-Orlicz framework. These spaces generalize both the classical Orlicz spaces and the fractional Sobolev spaces. The setup considers generalized Young functions $A \in \mathscr{F}(\Omega)$ and defines the associated modular and Luxemburg norm, giving rise to spaces $L_A(\Omega)$, $W^{1,A}(\Omega)$, and their fractional analogs.

The Riesz fractional gradient $V^s u$ for $u \in C_c^\infty(\mathbb{R}^n)$ is defined through a nonlocal integral operator involving singular kernels and admits Fourier representation, facilitating harmonic analysis arguments. The associated generalized space, denoted $H^{s,A}(\mathbb{R}^n)$, is defined via the Bessel potential $A_s$ as those temperate distributions $u$ such that $A_{-s} u \in L_A(\mathbb{R}^n)$.

Main Results

Equivalence of Fractional Orlicz-Sobolev and Bessel-Orlicz Potential Spaces

A cornerstone of the paper is the theorem establishing that the fractional Orlicz-Sobolev space defined by the Riesz fractional gradient (the space $A \in \mathscr{F}(\Omega)$0) coincides with the Bessel-Orlicz potential space $A \in \mathscr{F}(\Omega)$1, with equivalence of norms. The proof leverages the Hörmander-Mikhlin multiplier theorem adapted to the Musielak-Orlicz context and boundedness properties of Calderón-Zygmund operators on these generalized spaces. This equivalence generalizes classical results in the Lebesgue/exponent case to the Musielak-Orlicz setting.

Complex Interpolation Characterization

The authors show that $A \in \mathscr{F}(\Omega)$2 is precisely the complex interpolation space of exponent $A \in \mathscr{F}(\Omega)$3 between the base $A \in \mathscr{F}(\Omega)$4 space and the first-order generalized Orlicz-Sobolev space:

$A \in \mathscr{F}(\Omega)$5

The argument relies on technical multiplier estimates and the functorial properties of complex interpolation in the Banach setting, exploiting density, reflexivity, and UMD properties of the spaces under the standing assumptions on the Young function $A \in \mathscr{F}(\Omega)$6.

Embedding and Compactness Properties

Fractional Sobolev and Rellich-Kondrachov type embeddings are established in this generalized setting. If $A \in \mathscr{F}(\Omega)$7 satisfies suitable growth and doubling properties, and for $A \in \mathscr{F}(\Omega)$8, the space $A \in \mathscr{F}(\Omega)$9 embeds continuously and compactly into further Musielak-Orlicz spaces $L_A(\Omega)$0, with precise relationships between $L_A(\Omega)$1 and $L_A(\Omega)$2 detailed via inverse function scaling (e.g., $L_A(\Omega)$3).

Duality and Gagliardo-Nirenberg Inequalities

The dual spaces are identified via dual interpolation and scaling. Gagliardo-Nirenberg type inequalities are derived for the fractional gradient in these spaces, providing fine interpolation between various norms. The methodology relies on operator retraction arguments, precise multiplier norm bounds, and reiteration theorems.

Implications, Scope, and Limitations

This framework robustly generalizes classical potential space theory and interpolation results to settings crucial for analysis of nonlocal PDEs with nonstandard growth—prevalent in models for heterogeneous materials, nonlocal elasticity, and continuum mechanics.

The equivalences and embeddings established yield immediate consequences for the regularity theory of nonlocal PDEs and variational integrals in the Musielak-Orlicz setting, providing a rigorous foundation relevant for the development of compactness, existence, and approximation results.

A significant constraint is the requirement that the underlying Young function is non-weighted (respects condition (A0)), leaving analysis of weighted fractional Orlicz-Sobolev spaces for future investigation. Moreover, some advanced interpolation methods (e.g., the $L_A(\Omega)$4-method and its variants) are primarily developed for classical Orlicz spaces and cannot be systematically extended to the full Musielak setting without further technical advances.

Future Directions

Potential extensions include:

• Full development of the theory for weighted Musielak-Orlicz spaces.
• Further identification of the precise class of Triebel-Lizorkin and Besov-type spaces within this framework.
• Applications to fine properties of minimizers and solutions to variational and PDE problems, especially in peridynamics, nonlocal mechanics, and image processing models.
• Extension of the interpolation and potential space identities to stochastic-function spaces and anisotropic function spaces.

Conclusion

This work rigorously demonstrates that fractional Sobolev-Orlicz spaces based on the Riesz fractional gradient in the generalized Musielak-Orlicz setting can be fully characterized as Bessel potential spaces and, equivalently, as complex interpolation spaces between $L_A(\Omega)$5 and $L_A(\Omega)$6. The analysis establishes structural properties—embedding, interpolation, duality, and compactness—that extend and unify classical results, thereby providing substantial technical tools for the mathematical study of nonlocal and variable-exponent PDEs and their applications (2606.17770).