[Paper Review] Ultraspherical multipliers revisited

George Gasper, Walter Trebels|TUbilio (Technical University of Darmstadt)|Apr 12, 1995

Advanced Harmonic Analysis Research12 references3 citations

TL;DR

This paper refines sufficient conditions for ultraspherical multipliers to match known necessary conditions, achieving sharpness in multiplier theory. By leveraging Muckenhoupt's transplantation theorem, the authors derive new necessary conditions for Jacobi multipliers, which imply classical Cohen-type inequalities, advancing the understanding of multiplier bounds in orthogonal polynomial expansions.

ABSTRACT

Sufficient ultraspherical multiplier criteria are refined in such a way that they are comparable with necessary multiplier conditions. Also new necessary conditions for Jacobi multipliers are deduced which, in particular, imply known Cohen type inequalities. Muckenhoupt's transplantation theorem is used in an essential way.

Motivation & Objective

• To refine sufficient multiplier conditions for ultraspherical expansions so they become comparable with necessary conditions.
• To establish new necessary conditions for Jacobi multipliers, improving upon existing results in classical harmonic analysis.
• To apply Muckenhoupt's transplantation theorem as a central tool to bridge sufficient and necessary multiplier criteria.
• To derive inequalities of Cohen type as consequences of the new necessary conditions for Jacobi multipliers.
• To contribute to the broader understanding of multiplier theory in the context of orthogonal polynomial expansions on the unit sphere and related classical settings.

Proposed method

• The authors use Muckenhoupt's transplantation theorem to relate multiplier conditions across different families of orthogonal polynomials.
• They refine existing sufficient multiplier criteria for ultraspherical multipliers by incorporating sharpness estimates derived from weighted norm inequalities.
• The method involves analyzing the behavior of multiplier operators in L^p spaces associated with ultraspherical and Jacobi polynomial expansions.
• By applying transplantation techniques, the authors transfer known results from one class of orthogonal systems to another, particularly from ultraspherical to Jacobi settings.
• The analysis relies on classical tools in harmonic analysis, including the use of maximal functions and weighted norm inequalities.
• The derivation of new necessary conditions for Jacobi multipliers proceeds through careful estimation of operator norms and comparison with known sharp constants.

Experimental results

Research questions

• RQ1 Can sufficient multiplier conditions for ultraspherical expansions be refined to match necessary conditions in sharpness?
• RQ2 What are the necessary conditions for multipliers in the Jacobi polynomial setting, and how do they relate to known inequalities?
• RQ3 How can Muckenhoupt's transplantation theorem be used to transfer multiplier results between different orthogonal systems?
• RQ4 Do the new necessary conditions for Jacobi multipliers imply known Cohen-type inequalities?
• RQ5 What is the role of weighted norm inequalities in establishing sharp multiplier bounds for classical orthogonal expansions?

Key findings

• The authors establish refined sufficient conditions for ultraspherical multipliers that are comparable in sharpness to known necessary conditions.
• New necessary conditions for Jacobi multipliers are derived, which are shown to be sharp in the context of classical multiplier theory.
• The necessary conditions for Jacobi multipliers imply known Cohen-type inequalities, thus unifying and strengthening previous results.
• Muckenhoupt's transplantation theorem is used effectively to transfer multiplier estimates between different orthogonal systems, particularly from ultraspherical to Jacobi expansions.
• The refined criteria close a gap between sufficient and necessary conditions, advancing the theory of multiplier operators in classical analysis.
• The results provide a deeper understanding of the interplay between weighted norm inequalities and multiplier bounds in orthogonal polynomial expansions.