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

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Let X be a complete positive σ–finite measure space and {At}t≥0 be a symmetric diffusion semigroup of contraction operators on Lp(X). We prove that for 1<p<∞, the domain of the infinitesimal generator of the semigroup is precisely the space ∫01Ash ds:h∈Lp(X). We also establish that for 1<p<∞, the function spaces 2n-1∫02-(n-1)Ash ds|h∈Lp(X) are equal for every n∈ℤ.