Elliptic functions, area integrals and the exponential square class on $B_{1}(0) \subseteq {\Bbb R}^{n},n>2$
For two strictly elliptic operators L0 and L1 on the unit ball in R, 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 Su1 ∈ L∞(Sn−1) then u1|Sn−1 = f lies in the exponential square class whenever L0 is an operator so that L0u0 = 0 and Su0 ∈ L∞ implies u0|Sn−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 B1(0) for constant coefficient operator solutions, thus giving a family of operators for L0. Methods of proof include martingales and stopping time arguments. The subject treated here is the exponential square class for elliptic operator solutions in the unit ball in Rn. The main result is given in Theorem 4, where it is proved that if solutions for one operator L0u0 = 0 in B1(0) satisfy the exponential square result (i.e. if the Lusin area integral Su0 is in L∞(∂B1(0)) then u0|∂B1(0) = f lies in the exponential square class) then solutions u1 to L1u1 = 0 in B1(0) will also satisfy the exponential square theorem if the coefficients of L1 satisfy a Carleson-type condition with vanishing trace with respect to the coefficients of L0. The main ingredient in the proof is to establish a comparison of area integrals by methods similar to those of Fefferman, Kenig and Pipher [FKP] in their proof of Dahlberg’s preservation of Bq for elliptic measures. In order to ensure that the result is not vacuous, the exponential square theorem is proved on B1(0) for harmonic functions. The martingale argument of Chang, Wilson and Wolff [CWW] can be adapted to the geometry on B1(0). This result is established in part I in detail, using spherical coordinates. To establish the formula that served in place of the Calderón reproducing formula, and to establish the martingale proof, it seemed better to use spherical coordinates. A similar formula shows that solutions to constant coefficient equations also satisfy the exponential square result. 2000 Mathematics Subject Classification: 35J25, 42B25.