Elliptic functions, area integrals and the exponential square class on $B_{1}(0) \subseteq {\Bbb R}^{n},n>2$
Volume 164 / 2004
Abstract
For two strictly elliptic operators $L_{0}$ and $L_{1}$ on the unit ball in ${\mathbb R}^{n}$, 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 $L_1u_1 = 0$ in $B_1(0)$ and $Su_{1}\in L^{\infty }(S^{n-1})$ then $u_{1}|_{S^{n-1}}=f$ lies in the exponential square class whenever $L_{0}$ is an operator so that $L_0u_0 = 0$ and $Su_{0}\in L^{\infty }$ implies $u_{0}|_{S^{n-1}}$ 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 $B_{1}(0) $ for constant coefficient operator solutions, thus giving a family of operators for $L_{0}$. Methods of proof include martingales and stopping time arguments.
