Elliptic functions, area integrals and the exponential square class on $B_{1}(0) \subseteq {\Bbb R}^{n},n>2$

Caroline Sweezy¹

¹ Department of Mathematical Sciences New Mexico State University Las Cruces, NM 88003, U.S.A.

Studia Mathematica, Tome 164 (2004) no. 1, pp. 1-28

Résumé

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.

Détail

DOI : 10.4064/sm164-1-1

Keywords: strictly elliptic operators unit ball mathbb whose coefficients have difference function satisfies carleson type condition shown pointwise comparison concerning lusin area integrals valid result prove infty n n lies exponential square class whenever operator infty implies n exponential square class here lusin area integral exponential square theorem first proved thomas wolff harmonic functions upper half space proved constant coefficient operator solutions giving family operators methods proof include martingales stopping time arguments