1. Introduction We are going to investigate the equation ut(t, x ... - ICMS

4result can be proved for equations with variable continuous coefficients. However, inthe main application, which we have in mind, to stochastic partial differential equationsfrom filtering theory, the regularity of a ij in time is hard to control. Therefore, wealways deal only with measurable coefficients and, in Sec. 6 after some additional work,we prove our main result for equations in R × R d +in full generality.The arguments of Sec. 6 are based on Lemma 2.5, which also allows us to give adifferent proof of the result in R × R d by using the Marcinkiewicz interpolation theoremrather than the Calderón-Zygmund theorem.The proofs of part of the results are based on a general theorem saying, roughlyspeaking that, whatever estimate is true for the heat equation, it is also true for parabolicequations with coefficients depending only on time. This theorem is equally applicableto Sobolev and Hölder spaces.Finally, it is worth noting that we also give results for the initial value problems. Togive the reader a flavor of our results in R×R d + we state a particular case of Theorem 5.2.