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 −1 f ), where u is also Jacobi weight.