Harmonic analysis associated with the one-dimensional Dunkl transform
From the summary: The purpose of the present paper is to develop the harmonic analysis associated with the Dunkl operator and the Dunkl transform on the line \(\mathbb{R}\), generalizing the results of Muckenhoupt and Stein to the whole half-plane \(\mathbb{R}^2_+\). The main body of the paper consists of three parts. The first part is devoted to study the \(\lambda\)-subharmonic functions and the associated Hardy spaces. The \(\lambda\)-Poisson integral is treated in a compact argument. The second part is about the conjugate \(\lambda\)-Poisson integral and the \(\lambda\)-Hilbert transform. The main conclusions are the strong \((p, p)\) boundedness for \(1< p<\infty\) and weak \((1,1)\) boundedness of the associated maximal functions and the existence of the \(\lambda\)-Hilbert transform for functions in \(L^p_\lambda(\mathbb{R})\), \(1\leq p<\infty\). The final part is devoted to the theory about the Hardy space \(H^p_\lambda(\mathbb{R}^2_+)\). The fundamental result is the existence of boundary values of functions in \(H^p_\lambda(\mathbb{R}^2_+)\) when \(p\geq p_0= 2\lambda/(2\lambda+1 )\). Various properties of \(H^p_\lambda(\mathbb{R}^2_+)\) are studied. In particular, for \(p=1\), a generalization of \(F.\) and M. Riesz's theorem supplies a new criterion for the absolute continuity of a measure; the \(\lambda\)-Hilbert transform \(H_\lambda\) is shown to be a bounded mapping from \(H_\lambda'(\mathbb{R})\) to \(L_\lambda'(\mathbb{R})\).
scientific article; zbMATH DE number 6150046
scientific article