Hardy-Littlewood type inequalities for laguerre series

Let { cj} be a null sequence of bounded variation. We give appreciate smoothness and growth conditions on {c j} to obtain the pointwise convergence as well as L r-convergence of Laguerre series Σcj&Lfr;ja. Then, we prove a Hardy-Littlewood type inequality ∫0_∞ | f (t)| rdt≤CΣ j=0_∞ | cj| rj^- 1-r/2 for certain r≤1, where f is the limit function of Σcj&Lfr;ja. Moreover, we show that if f (x)∼Σcj&Lfr;ja is in L r, r≥1, we have the converse Hardy-Littlewood type inequality Σ j=0_∞ | cj| rj ^-β≤ C∫0_∞ | f (t)| rdt for r≥1 and β<-r/2.