H^∞ functional calculus and maximal inequalities for semigroups of contractions on vector-valued L_p-spaces

2014/02/11 by Quanhua Xu, Xu, Quanhua
Mathematics · #42B25 #47A60. Secondary: 46B20 #Advanced Banach Space Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #Nonlinear Differential Equations Analysis #Primary: 47A35

paper · pdf · doi:10.48550/arxiv.1402.2344

openalex publication_date 2014/02/11 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let \T_t\_t>0 be a strongly continuous semigroup of positive contractions on L_p(X,μ) with 10\frac1t|∫₀^tT_s(f(⋅,ω))(x)ds|, (x,ω)∈ X×Ω.Then the following maximal ergodic inequality holds‖\mathcal M(f)‖_{Lp(X; E)}\lesssim ‖f‖_{Lp(X; E)}, f∈ L_p(X; E).If the semigroup \T_t\_tgt;0 is additionally assumed to be analytic, then \T_t\_tgt;0 extends to an analytic semigroup on L_p(X; E) and \mathcal M(f) in the above inequality can be replaced by the following sectorial maximal function\mathcal T_θ(f)(x, ω)=sup_|\rm arg(z)|0. Under the latter analyticity assumption and if E is a complex interpolation space between a Hilbert space and a UMD Banach space, then \T_t\_t>0 extends to an analytic semigroup on L_p(X; E) and its negative generator has a bounded H^∞(Σ_σ) calculus for some σ

Citations

• Some Maximal Inequalities

Related

• Strong q-variation inequalities for analytic semigroups
• Form Inequalities for Symmetric Contraction Semigroups
• Isometric dilations and H^∞ calculus for bounded analytic semigroups and Ritt operators
• Extremal properties of contraction semigroups on c_o
• A sharp equivalence between H^∞ functional calculus and square function estimates