Boundedness of a class of bi-parameter square functions in the upper half-space

TL;DR

This paper establishes boundedness criteria for a class of bi-parameter square functions in the upper half-space using probabilistic averaging and Whitney cube techniques.

Contribution

It provides an efficient proof of boundedness for bi-parameter square functions employing modern probabilistic and geometric methods.

Findings

Boundedness criteria for bi-parameter square functions established.

Probabilistic averaging methods effectively used in the proof.

Control of double Whitney averages over good cubes demonstrated.

Abstract

We consider a class of bi-parameter kernels and related square functions in the upper half-space, and give an efficient proof of a boundedness criterion for them. The proof uses modern probabilistic averaging methods and is based on controlling double Whitney averages over good cubes.