Function spaces with respect to the Gaussian measure

Abstract

One of the main goals of functional spaces is to interpret and quantify the smoothness of functions. In this chapter, we discuss the analogs of classical functional spaces with respect to the Gaussian measure. We see that almost all classical spaces with respect to the Lebesgue measure have an analog for the Gaussian measure; nevertheless, we see that in some cases, for instance, Hardy spaces, the analogs to classical spaces are still incomplete and/or imperfect. On the other hand, most of the time, even if the spaces look similar, most of the proofs are different, mainly because the Gaussian measure is not invariant by translation, which implies the need for completely different techniques.