Weighted Weak Behaviour of Fourier-Jacobi Series

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 S_n are uniformly bounded from L^p,1 to L^p,∞, 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→uS_n(u^?1f), where u is also Jacobi weight.