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Quadratic Sparse Domination and Weighted Estimates for Non-integral Square Functions
Julian Bailey1, Gianmarco Brocchi1, Maria Carmen Reguera1
1School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT UK.
Journal of Geometric Analysis
|June 1, 2026
Summary
This study establishes a quadratic sparse domination result for non-integral square functions, enabling optimal norm estimates in weighted L^p spaces. The findings apply to elliptic operators and the Laplace-Beltrami operator.
Area of Science:
- Harmonic Analysis
- Functional Analysis
- Partial Differential Equations
Background:
- Square functions are fundamental objects in harmonic analysis, often associated with differential operators.
- Sparse domination techniques provide powerful tools for analyzing function spaces and operators.
- Weighted norm inequalities are crucial for understanding the behavior of functions in various contexts.
Purpose of the Study:
- To prove a quadratic sparse domination result for general non-integral square functions.
- To extend these results to square functions associated with divergence form elliptic operators and the Laplace-Beltrami operator.
- To derive optimal norm estimates in weighted L^p spaces using the established sparse domination.
Main Methods:
- Development of a quadratic sparse domination estimate for general non-integral square functions.
- Application of the sparse domination to specific classes of square functions (elliptic operators, Laplace-Beltrami operator).
- Utilizing the sparse domination to obtain weighted norm estimates in L^p(w) spaces.
Main Results:
- A quadratic sparse domination result is proven for general non-integral square functions S.
- The result holds for p0 in [1, 2) and q0 in (2, infinity], providing estimates in terms of a sparse collection of cubes.
- Optimal norm estimates are derived in the weighted space L^p(w).
Conclusions:
- The established quadratic sparse domination is a significant advancement for analyzing square functions.
- The findings have broad applicability, covering important classes of differential operators.
- The derived optimal norm estimates in weighted spaces offer valuable insights into function space theory.
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