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No abstract uploaded!

We show that the$L$^{$p$}norms, 0<$p$<∞, of the nontangenital maximal function and area integral of solutions and normalized adjoint solutions to second order nondivergence form elliptic equations, are comparable when integrated on the boundary of a Lipschitz domain with respect to measures, which are respectively$A$_{∞}with respect to the corresponding harmonic measure or normalized harmonic measure.

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We give a sufficient condition on a Lévy measure$μ$which ensures that the generator$L$of the corresponding pure jump Lévy process is (locally) hypoelliptic, i.e., $\mathop {\mathrm {sing\,supp}}u\subseteq \mathop {\mathrm {sing\,supp}}Lu$ for all admissible$u$. In particular, we assume that $\mu|_{\mathbb {R}^{d}\setminus \{0\}}\in C^{\infty}(\mathbb {R}^{d}\setminus \{0\})$ . We also show that this condition is necessary provided that $\mathop {\mathrm {supp}}\mu$ is compact.