La théorie de Littlewood-Paley pour la transformation de Fourier- Bessel. (The Littlewood-Paley theory for the Fourier-Bessel transform). (French) Zbl 0591.42014

Summary: We consider the harmonic analysis of the operator \(L=d^ 2/dx^ 2+(r/x)d/dx\), \(r>0\). We prove that a maximal function closely related to the convolution structure in \(L^ 1((0,\infty),x^ rdx)\) is of weak type (1,1). As a consequence, the almost everywhere convergence in various summability methods is established. We obtain also the \(L^ p\)- inequalities for the g-functions g and \(g^*_{\lambda}\), \(\lambda >1\) and this allow us to prove a multiplier theorem of Hörmander-Mihlin type. Finally, we study the maximal function \(f^*\) and, applying a technique of Kurtz and Wheeden, we prove the weighted multiplier theorem for the Fourier-Bessel transform.


42B25 Maximal functions, Littlewood-Paley theory
47E05 General theory of ordinary differential operators