Weighted Weak Behaviour of Fourier-Jacobi Series

✍ Scribed by José J. Guadalupe; Mario Pérez; Juan L. Varona

• Publisher: John Wiley and Sons
• Year: 2006
• Tongue: English
• Weight: 503 KB
• Volume: 158
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19921580111

✦ Synopsis

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 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.