Characterization of Domains of Fractional Powers

Abstract

This chapter is devoted to characterizing the domains of fractional powers A ^θ for a sectorial operator A in a Hilbert or Banach space. However, the situations are quite different in Hilbert and Banach space cases. Anyway, in both cases, the boundedness of the H _∞ functional calculus for A plays an important role for the complete characterization of $\mathcal{D}(A^{\theta})$ . In the Hilbert space case, the H _∞ functional calculus for sectorial operators was invented by McIntosh in 1986 and was generalized in the Banach space case by Cowling–Doust–McIntosh–Yagi in 1996. In the Hilbert space case, Theorem 2.30 due to Kato ensures that any sectorial operator associated with a sesquilinear form has a bounded H _∞ functional calculus. To the contrary, in the Banach space case, there is no convenient sufficient condition ensuring the boundedness of the functional calculus. Therefore, for elliptic differential operators in L _p (Ω), 1<p<∞, p≠2, equipped with suitable boundary conditions, we have to appeal to the theory of harmonic analysis or the theory of integral operators in order to prove the boundedness by some specific methods.