[Paper Review] Gelfand and Kolmogorov numbers of embeddings in weighted function spaces

Shun Zhang, Fang Gensun|arXiv (Cornell University)|Feb 3, 2011

Advanced Harmonic Analysis Research22 references3 citations

TL;DR

This paper establishes sharp estimates for Gelfand and Kolmogorov numbers of compact embeddings between weighted Besov and Triebel-Lizorkin spaces with polynomial weights, extending to the quasi-Banach setting in the non-limiting case. The analysis provides precise asymptotic behavior of these entropy-type numbers, offering a complete characterization in terms of the underlying function space parameters.

ABSTRACT

In this paper we study the Gelfand and Kolmogorov numbers of compact embeddings between weighted function spaces of Besov and Triebel-Lizorkin type with polynomial weights. The sharp estimates are determined in the non-limiting case where the quasi-Banach setting is included.

Motivation & Objective

• To determine sharp asymptotic estimates for Gelfand and Kolmogorov numbers of embeddings between weighted function spaces.
• To extend existing results on entropy numbers to the quasi-Banach setting, particularly for non-limiting embeddings.
• To analyze the role of polynomial weights in shaping the asymptotic behavior of these entropy-type numbers.
• To unify and generalize previous findings in the theory of function space embeddings using spectral and entropy-type invariants.

Proposed method

• The analysis employs techniques from functional analysis and interpolation theory to study compact embeddings between weighted Besov and Triebel-Lizorkin spaces.
• The authors use the duality between Gelfand and Kolmogorov numbers to derive estimates via symmetry and duality arguments in the context of quasi-Banach spaces.
• Key estimates are derived using the entropy function method and asymptotic analysis of the entropy numbers in relation to the smoothness and integrability parameters of the function spaces.
• The non-limiting case is treated by applying precise asymptotic expansions and comparison techniques based on the behavior of the weight functions.
• The method relies on the structure of polynomial weights to control the growth of the embedding norms and to derive sharp bounds.
• The framework incorporates interpolation methods and Lorentz space embeddings to relate the entropy numbers to the underlying function space scales.

Experimental results

Research questions

• RQ1 What are the sharp asymptotic estimates for Gelfand and Kolmogorov numbers of embeddings between weighted Besov and Triebel-Lizorkin spaces with polynomial weights?
• RQ2 How do these entropy-type numbers behave in the quasi-Banach setting, particularly when the Lebesgue exponent is less than one?
• RQ3 To what extent do polynomial weights influence the asymptotic decay rate of the Gelfand and Kolmogorov numbers?
• RQ4 Can the duality between Gelfand and Kolmogorov numbers be exploited to derive sharp estimates in the non-limiting case?
• RQ5 How do the smoothness and integrability parameters of the function spaces affect the asymptotic behavior of these numbers?

Key findings

• Sharp asymptotic estimates are established for both Gelfand and Kolmogorov numbers in the non-limiting case, including the quasi-Banach regime.
• The estimates are shown to depend critically on the smoothness, integrability, and weight parameters of the function spaces involved.
• The behavior of the numbers is fully characterized in terms of the underlying function space parameters, with precise decay rates derived.
• The duality between Gelfand and Kolmogorov numbers is used to derive symmetric estimates, confirming consistency across the duality framework.
• The results extend previous findings in the limiting case to the non-limiting case, providing a complete picture for polynomial weights.
• The analysis confirms that the asymptotic behavior is governed by the interplay between the smoothness and the weight growth, with polynomial weights leading to explicit and sharp bounds.