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Some remarks on K-closedness of the couples of real Hardy spaces

Ioann Vasilyev

Abstract

In this paper K closedness is proved in the case of the couple of real Hardy spaces in the corresponding couple of Lebesgue spaces. This means roughly that any measurable decomposition of an analytic function gives rise to an "analytic" decomposition with summands of roughly the same size. The proof uses Bourgain's method, the atomic decomposition for Hardy spaces and the subharmonic property of the gradient of a system of conjugate harmonic functions.

Some remarks on K-closedness of the couples of real Hardy spaces

Abstract

In this paper K closedness is proved in the case of the couple of real Hardy spaces in the corresponding couple of Lebesgue spaces. This means roughly that any measurable decomposition of an analytic function gives rise to an "analytic" decomposition with summands of roughly the same size. The proof uses Bourgain's method, the atomic decomposition for Hardy spaces and the subharmonic property of the gradient of a system of conjugate harmonic functions.
Paper Structure (2 sections, 3 theorems, 69 equations)

This paper contains 2 sections, 3 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. The main theorem

Key Result

Theorem 1

The couple of real Hardy spaces $(H^{p_1}(\mathbb R ^n),H^{p_2}(\mathbb R ^n))$ is $K-$closed in the couple of corresponding Lebesgue spaces $(L^{p_1}(\mathbb R ^n),L^{p_2}(\mathbb R ^n))$ for $\frac{n-1}{n} < p_1 < 1< p_2.$

Theorems & Definitions (9)

  • Definition 1
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Remark 2