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Related Experiment Videos

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
PubMed
Summary
This summary is machine-generated.

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.

Keywords:
Elliptic operatorSharp weighted estimatesSparse boundsSquare functions

Related Experiment Videos

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.