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Quantitative estimates for bounded holomorphic semigroups

Tuomas Hytönen, Stefanos Lappas

Abstract

In this paper we revisit the theory of one-parameter semigroups of linear operators on Banach spaces in order to prove quantitative bounds for bounded holomorphic semigroups. Subsequently, relying on these bounds we obtain new quantitative versions of two recent results of Xu related to the vector-valued Littlewood--Paley--Stein theory for symmetric diffusion semigroups.

Quantitative estimates for bounded holomorphic semigroups

Abstract

In this paper we revisit the theory of one-parameter semigroups of linear operators on Banach spaces in order to prove quantitative bounds for bounded holomorphic semigroups. Subsequently, relying on these bounds we obtain new quantitative versions of two recent results of Xu related to the vector-valued Littlewood--Paley--Stein theory for symmetric diffusion semigroups.
Paper Structure (6 sections, 22 theorems, 144 equations)

This paper contains 6 sections, 22 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical semigroup theory made quantitative
  4. Quantitative analyticity of diffusion semigroups in uniformly convex spaces
  5. Littlewood--Paley--Stein inequalities: First approach
  6. Littlewood--Paley--Stein inequalities: Second approach

Key Result

Theorem 2.5

(Hille--Yosida Generation Theorem) Let $A$ be a linear operator on $X$ and let $M\ge 1, \omega\in\mathbb{R}$. The following conditions are equivalent:

Theorems & Definitions (53)

  • Definition 2.1: Strongly Continuous Semigroup
  • Remark 2.2
  • Definition 2.3: Infinitesimal Generator
  • Definition 2.4
  • Theorem 2.5: EN, Theorem II.3.8 or P, Theorem I.5.3 and Remark I.5.4
  • Definition 2.6: Open/Closed Sector
  • Definition 2.7: Holomorphic Semigroup
  • Theorem 2.8: EN, IV.1.2
  • Lemma 3.1: EN, Theorem 4.6; P, Chapter 2, Theorem 5.2; F, Theorem 1.1.23
  • proof
  • ...and 43 more