Paley and Hardy's inequalities for the Fourier-Dunkl expansions

Abstract

Purpose – Paley's and Hardy's inequality are proved on a Hardy-type space for the Fourier–Dunkl expansions based on a complete orthonormal system of Dunkl kernels generalizing the classical exponential system defining the classical Fourier series. Design/methodology/approach – Although the difficulties related to the Dunkl settings, the techniques used by K. Sato were still efficient in this case to establish the inequalities which have expected similarities with the classical case, and Hardy and Paley theorems for the Fourier–Bessel expansions due to the fact that the Bessel transform is the even part of the Dunkl transform. Findings – Paley's inequality and Hardy's inequality are proved on a Hardy-type space for the Fourier–Dunkl expansions. Research limitations/implications – This work is a participation in extending the harmonic analysis associated with the Dunkl operators and it shows the utility of BMO spaces to establish some analytical results. Originality/value – Dunkl theory is a generalization of Fourier analysis and special function theory related to root systems. Establishing Paley and Hardy's inequalities in these settings is a participation in extending the Dunkl harmonic analysis as it has many applications in mathematical physics and in the framework of vector valued extensions of multipliers.

Keywords

• Fourier–Dunkl expansions
• Hardy spaces
• BMO spaces
• Paley's inequality
• Hardy's inequality