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.
The CMO-Dirichlet problem for the Schrödinger equation in the upper half-space and characterizations of CMO
Square functions with general measures
Boundedness of non-homogeneous square functions and $L^q$ type testing conditions with $q \in (1,2)$
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