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ON THE PARABOLIC REGULARITY, SOBOLEV EMBEDDINGS AND GLOBAL CARLEMAN ESTIMATES IN L(L) SPACES

In this paper we discuss some aspects related to regularity in parabolic problems with corollaries regarding anisotropic Sobolev embeddings. We use these results in the context of bootstrap arguments applied to global Carleman estimates for nonhomogeneous parabolic equations in Lqt (L p x) spaces, estimates which are fundamental in associated control and inverse problems. The arguments we use are characterizations of regularity in terms of domains of fractional powers of elliptic operators and then characterization of these domains as interpolation spaces and relations to Bessel potential and Sobolev-Slobodeckii spaces.