Approximation via partial Hausdorff integrals on $H^1(\mathbb{R})$

TL;DR

The paper addresses approximating a function $f$ in the Hardy–Sobolev space $H^1(\mathbb{R})$ using partial Hausdorff integrals. It extends a known $L^p$ approximation result to $H^1(\mathbb{R})$ by employing $K$-functional theory and the theory of homogeneous $H^1$ multipliers, and proves uniform $H^1$-boundedness of the partial Hausdorff integrals $F_\epsilon$ along with their convergence to $f$ at a rate governed by the $K$-functional $K_{\sigma}(f,\epsilon)_{H^1}$. Under additional structural assumptions on the generating functions, the authors show that $F_\epsilon$ approximate $f$ in $H^1$ and identify the associated multiplier as homogeneous in $H^1$. The paper also presents four explicit examples of partial Hausdorff integrals, illustrating the broad applicability of the method to various Hausdorff-type operators.

Abstract

We obtain the result of approximating \( f \) in the \( H^1(\mathbb{R}) \) norm using partial Hausdorff integrals. Specifically, by leveraging the homogeneous multiplier theory of \( H^1(\mathbb{R}) \) and the \( K \) functional theory, one result from Pinos and Liflyand [CMB,~2021,~64,~no.3] is extended from \( L^p(\mathbb{R}) \) ( \( 1 \leq p \leq \infty \)) to \( H^1(\mathbb{R}) \). As applications, four examples of partial Hausdorff integrals are also given.

Theorems & Definitions (18)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Proposition 2.6
• Definition 2.7
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3