2015 | OriginalPaper | Buchkapitel
A Quantitative Coulhon–Lamberton Theorem
verfasst von : Tuomas P. Hytönen
Erschienen in: Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics
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Let X be a Banach space and $$p\;\in\;(1,\infty)$$ . We denote the Lp -norms of several operators on X-valued functions as follows: the norm of martingale transforms (i.e., the UMD constant) by $$\beta_{p,X}$$ , the norm of the Hilbert transform by $$\hbar_{p,X}$$ , and the norm of the maximal regularity operator for the Poisson semigroup by $$\mathfrak{m}_{p,X}$$ . Qualitatively, all three are known to be finite or infinite simultaneously. We prove the quantitative relation $$\frac{1}{2}\mathrm{max}(\beta_{p,X},\hbar_{p,X})\leq\mathfrak{m}_{p,X}\leq \beta_{p,X}\;+\;\hbar_{p,X}$$ .