Abstract Measure Grand Lebesgue Spaces and Applications

Abstract: We introduce abstract measure grand Lebesgue spaces endowed with ball basis structures and investigate their fundamental properties, together with the behavior of $\mathfrak{BO}$ operators. By exploiting sparse domination for $\mathfrak{BO}$ operators and norm inequalities for sparse operators, we derive norm estimates for this class of operators. As applications, we establish that $\mathfrak{BO}$ operators encompass the maximal operators, Calderón-Zygmund operators on homogeneous spaces and Carleson operators.

• The paper introduces abstract Grand Lebesgue spaces using ball basis structures to extend classical L^p-space theory.
• It develops bounded oscillation (BO) operators and demonstrates sharp sparse domination techniques along with explicit norm estimates.
• Key results include precise operator bounds for maximal, Calderón–Zygmund, and Carleson operators in broad, non-Euclidean settings.

Abstract Measure Grand Lebesgue Spaces and $\mathfrak{BO}$ Operators

Introduction and Motivation

This work systematically develops the theory of abstract Grand Lebesgue Spaces (GLS) $L^{p)}(X, \mu)$ over general measure spaces endowed with a ball basis structure, extending the foundational results from the Euclidean setting to more abstract frameworks. The study is motivated by advances in the analysis of nonlinear PDEs, the modern theory of sparse domination for singular integrals, and the desire to establish operator boundedness and structural properties in settings without a priori geometric regularity. The main contributions are the construction of these function spaces, the introduction and analysis of bounded oscillation ($\mathfrak{BO}$) operators, and sharp sparse domination and norm estimates for a broad class of singular and maximal operators.

Construction of Abstract Grand Lebesgue Spaces

Following the axiomatic framework of Karagulyan [Karagulyan, Trans. AMS, 2019], the paper introduces ball bases as collections of measurable sets satisfying finite measure, covering, approximation, and a doubling-type hull property. Within this structure, the abstract Grand Lebesgue space $L^{p)}(X, \mu)$ consists of functions for which

$\|f\|_{L^{p)}(X,\mu)} := \sup_{0 < \varepsilon \leq p-1} \left( \varepsilon \, \fint_X |f|^{p-\varepsilon} d\mu \right)^{1/(p-\varepsilon)} < \infty,$

with an associated small abstract Lebesgue (dual) space $L^{p)'}(X, \mu)$ defined by atomic decompositions. The constructed spaces are Banach function spaces with generalized Hölder inequalities and strict inclusions

$$L^p(X, \mu) \subsetneq L^{p)}(X, \mu) \subsetneq L^{p-\varepsilon}(X, \mu).$$

Key technical lemmas cover outer measure extension, measurability of rearrangement-invariant norms, and the duality structure following the foundational work of Fiorenza [Collect. Math., 2000].

Ball Basis Geometry and Sparse Domination

A careful analysis of the ball basis enables general covering and fragmentation results, critical for formulating sparse domination theorems. The authors provide generalizations of classical covering lemmas, demonstrate the Besicovitch covering property in this setting, and construct countable covering sequences with explicit doubling bounds.

The notion of sparse family—collections of balls with controlled overlap—is deployed for defining sparse operators. The corresponding sparse averaging operators $\mathcal{A}_{\mathcal{S}, r}$ inherit modular bounds pivotal for domination arguments.

Bounded Oscillation Operators ($\mathfrak{BO}$) and Main Theorems

A central focus is the class of $\mathfrak{BO}$ operators: sublinear operators satisfying two technical conditions. These guarantee, roughly, that localized truncations and the oscillation across balls and their hulls are dominated by averages of the input function, with constants $L^{p)}(X, \mu)$0 and $L^{p)}(X, \mu)$1 quantifying this control. The formal definition abstracts the behavior of classical Calderón-Zygmund and maximal operators to the setting of ball bases.

Three main theorems are established:

1. Sparse Domination: Every $L^{p)}(X, \mu)$2 operator $L^{p)}(X, \mu)$3 on $L^{p)}(X, \mu)$4 with a weak-$L^{p)}(X, \mu)$5 estimate can be pointwise-sparsely dominated in terms of sparse operators $L^{p)}(X, \mu)$6 modulo the operator's weak bounds and structural constants. This holds with a decomposition into two explicitly sparse families.
2. Norm Inequalities: The operator norm of $L^{p)}(X, \mu)$7 is dominated by a product of the characteristic constants, sparsity parameters, and the sparse operator norm—fully quantified via parameters of the ball basis and the sparse family.
3. Sharp Bounds under Besicovitch: If the ball basis satisfies the Besicovitch property, the operator norm estimate refines further, including explicit dependence on the Besicovitch constant and the operator structure.

These results extend the modern sparse domination paradigm, e.g., Lerner's $L^{p)}(X, \mu)$8 conjecture approach, to abstract measure and GLS settings, and demonstrate how the properties of the underlying basis and space geometry are reflected in the sharpness and structure of operator bounds.

Applications to Classical Operators

The developed framework is applied to demonstrate that:

• The abstract Hardy–Littlewood maximal operator $L^{p)}(X, \mu)$9 is a $\mathfrak{BO}$0 operator and admits sharp norm bounds on $\mathfrak{BO}$1 matching those derived in the abstract theorems.
• Calderón–Zygmund operators on homogeneous spaces and Carleson operators (given uniformly bounded families of $\mathfrak{BO}$2s) are subsumed in this framework. The expected weak and strong type bounds are obtained abstractly, and the proofs avoid reliance on Euclidean geometry.
• For each, both norm and weak-type inequalities are provided, with explicit dependence on sparsity, ball basis geometry, and Besicovitch constants.

Technical Advancements

Several key technical developments underpin the main results:

• Abstract extension of distribution function methods and maximal inequalities to measure spaces with merely a ball basis.
• Exhaustive treatment of covering arguments and hull operations to ensure the validity of the maximal and singular integral machinery.
• Unification of the sparse domination technique across diverse operator types, including oscillatory and maximal truncations, using only structure constants and covering parameters of the basis.

Notably, the results substantially generalize the boundedness and norm control previously known for GLS in Euclidean and homogeneous spaces to a broad class of measure spaces, providing a template for further extension to Orlicz, Morrey, and endpoint-type function spaces.

Implications and Future Directions

This framework decouples operator boundedness theory from explicit geometric structure. The results imply sharp control on singular integrals and maximal operators in nonclassic, possibly fractal or non-doubling settings, as soon as a ball basis satisfying mild geometric properties can be exhibited.

Possible directions opened by this work include:

• Extending the theory to multilinear, vector-valued, or time-frequency analysis contexts (especially considering recent advances in sparse bounds for such settings).
• Interfacing with abstract harmonic analysis, such as analytic capacity, quasimetric spaces, and general quasi-Banach function spaces.
• Developing optimal extrapolation results for PDE regularity or potential theory in spaces beyond the standard Lebesgue scale.

Conclusion

The paper achieves a significant generalization of sparse domination techniques and operator theory in Grand Lebesgue Spaces, providing precise, explicit norm bounds for Calderón–Zygmund, maximal, and Carleson operators on general measure spaces with minimal structural assumptions. The methodology is both robust and sharp, extending applicability to a diverse array of analytic settings and providing a foundation for further investigation into generalized function spaces and abstract operator theory.