For two strictly elliptic operators L₀ and L₁ on the unit ball in ℝⁿ, whose coefficients have a difference function that satisfies a Carleson-type condition, it is shown that a pointwise comparison concerning Lusin area integrals is valid. This result is used to prove that if L₁u₁ = 0 in B₁(0) and then lies in the exponential square class whenever L₀ is an operator so that L₀u₀ = 0 and implies is in the exponential square class; here S is the Lusin area integral. The exponential square theorem, first proved by Thomas Wolff for harmonic functions in the upper half-space, is proved on B₁(0) for constant coefficient operator solutions, thus giving a family of operators for L₀. Methods of proof include martingales and stopping time arguments.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm164-1-1,
author = {Caroline Sweezy},
title = {Elliptic functions, area integrals and the exponential square class on B1(0) [?] Rn, n > 2},
journal = {Studia Mathematica},
volume = {162},
year = {2004},
pages = {1-28},
zbl = {1112.35058},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm164-1-1}
}
Caroline Sweezy. Elliptic functions, area integrals and the exponential square class on B₁(0) ⊆ ℝⁿ, n > 2. Studia Mathematica, Tome 162 (2004) pp. 1-28. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm164-1-1/