CMA Research Report

MRR02-004

On square functions associated to sectorial operators

Abstract:  We give new results on square functions |x|_F = |(\int₀^\infty |F(tA)x|² d/ dt)^1/2|_p associated to a sectorial operator A on L^p for 1<p<\infty. Under the assumption that A is actually R-sectorial, we prove equivalences of the form K^-1 |x|_G \leq |x|_F\leq K|x|_G for suitable functions F, G. We also show that A has a bounded H^\infty functional calculus with respect to | |_F. Then we apply our results to the study of conditions under which we have an estimate |(∫₀^∞ |Ce^-tA (x)|² dt)^1/2|_q ≤M |x|_p, when -A generates a bounded semigroup e^-tA on L^p and C: D(A)-> L^q is a linear mapping.

AMS Classification:  47A60
Date:  22 February 2002

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