Almost-Everywhere Convergence of Fourier Integrals for Functions in Sobolev Spaces, and an $L^2$-Localisation Principle

Almost Everywhere Convergence of Poisson Integrals on Generalized Half-planes

Mean and Almost Everywhere Convergence of Fourier-neumann Series

Let Jμ denote the Bessel function of order μ. The functions xJα+β+2n+1(x 1/2), n = 0, 1, 2, . . . , form an orthogonal system in L2((0,∞), xα+βdx) when α+ β > −1. In this paper we analyze the range of p, α and β for which the Fourier series with respect to this system converges in the Lp((0,∞), xαdx)-norm. Also, we describe the space in which the span of the system is dense and we show some of ...

On radial Fourier multipliers and almost everywhere convergence

We study a.e. convergence on L, and Lorentz spaces L, p > 2d d−1 , for variants of Riesz means at the critical index d( 1 2 − 1 p )− 1 2 . We derive more general results for (quasi-)radial Fourier multipliers and associated maximal functions, acting on L spaces with power weights, and their interpolation spaces. We also include a characterization of boundedness of such multiplier transformation...

The surprising almost everywhere convergence of Fourier-Neumann series

For most orthogonal systems and their corresponding Fourier series, the study of the almost everywhere convergence for functions in L requires very complicated research, harder than in the case of the mean convergence. For instance, for trigonometric series, the almost everywhere convergence for functions in L is the celebrated Carleson theorem, proved in 1966 (and extended to L by Hunt in 1967...

Almost Everywhere Convergence of Series in L

We answer positively a question of J. Rosenblatt (1988), proving the existence of a sequence (ci) with ∑∞ i=1 |ci| = ∞, such that for every dynamical system (X,Σ, m, T ) and f ∈ L1(X), ∑∞i=1 cif(T ix) converges almost everywhere. A similar result is obtained in the real variable context.