Weighted weak behaviour of Fourier–Jacobi series

Abstract. Let w(x) = (1−x)α(1+x)β be a Jacobi weight on the interval [−1, 1] and 1 < p < ∞. If either α> −1/2 or β> −1/2 and p is an endpoint of the interval of mean convergence of the associated Fourier-Jacobi series, we show that the partial sum operators Sn are uniformly bounded from Lp,1 to Lp,∞, thus extending a previous result for the case that both α, β> −1/2. For α, β> −1/2, we study the weak and restricted weak (p, p)-type of the weighted operators f − → uSn(u−1f), where u is also a Jacobi weight. §1. Introduction and main results. Let w be a Jacobi weight on the interval [−1, 1], that is, w(x) = (1 − x)α(1 + x)β, α, β> −1 and let 1 < p < ∞; Snf stands for the n-th partial sum of the Fourier series associated to the Jacobi polynomials, orthonormal on [−1, 1] with respect to w. It is well known that Sn