The Bochner–Riesz means for Fourier–Bessel expansions

Abstract

The Bochner-Riesz means for Fourier-Bessel expansions are analyzed. We prove a uniform two-weight inequality, with potential weights, for these means. The result provides necessary and sufficient conditions for boundedness. Moreover, we obtain some corollaries regarding the convergence of these means and the boundedness of other operators related to the Fourier-Bessel series.

Weighted inequalities for the Bochner–Riesz means related to the Fourier–Bessel expansions

“…In [2] we proved the mean convergence of this summation method, and in [3] we showed results about almost everywhere convergence. In both cases we analyzed the results in Lebesgue spaces with power weights.…”

“…At last, T 1 is readily handled by standard facts. To analyze these operators, we must take into account the estimates obtained for the kernel K δ R in [2]. The regions considered in [2] are not exactly the previously given ones but they are similar and the estimates in that paper imply, without additional effort, that…”

“…The conditions given in [2] for (1) are necessary and sufficient. Then, the convergence of the Bochner-Riesz means is deduced by the density of the orthonormal system considered.…”

“…In the present paper, we use several techniques that are different from those introduced in [2], in order to include more general weights. To be precise, we deal with the local A p theory and Hardy's inequalities with weights.…”

Hardy spaces for Fourier-Bessel expansions

“…The operator L has a basis of eigenfunctions φ ν n (see Subsection 2.1) in L 2 ((0, 1), µ), where dµ(x) = x 2ν+1 dx, see [15,Chapter 2]. Therefore, by applying the Fourier method, we see that the solution of the Dirichlet problem (1.1) is u(x, t) = e −t It is known, see [4,16,21,22], that such operator is bounded on L p ((0, 1), µ), for 1 < p < ∞, and of weak type (1,1). Then, as usual, we can get the desired almost everywhere convergence.…”