AN EXPLICIT CHARACTERIZATION OF THE DOMAIN OF THE INFINITESIMAL GENERATOR OF A SYMMETRIC DIFFUSION SEMIGROUP ON LP OF A COMPLETE POSITIVE SIGMA-FINITE MEASURE SPACE

Abstract

Let $X$ be a complete positive $\sigma$–finite measure space and $\{A_t{\}}_{t\ge 0}$ be a symmetric diffusion semigroup of contraction operators on $L_p\left(X\right)$. We prove that for , the domain of the infinitesimal generator of the semigroup is precisely the space $\left\{{\int }_0^1{A}_{s}h ds:h\in L_p\left(X\right)\right\}$. We also establish that for , the function spaces

$$\left\{2^{n-1}{\int }_0^{2-(n-1)}{A}_{s}h ds|h\in L_p(X)\right\}$$

are equal for every $n\in \mathrm{\mathbb{Z}}$.