Hardy-littlewood Type Inequalities for Laguerre Series

Let {cj} be a null sequence of bounded variation. We give appreciate smoothness and growth conditions on {cj} to obtain the pointwise convergence as well as Lr -convergence of Laguerre series ∑ cj a j . Then, we prove aHardy-Littlewood type inequality ∫∞ 0 |f(t)|r dt≤ C ∑∞ j=0 |cj| j̄1−r/2 for certain r ≤ 1, where f is the limit function of ∑ cj a j . Moreover, we show that if f(x) ∼ ∑cj a j is in Lr , r ≥ 1, we have the converse Hardy-Littlewood type inequality ∑∞ j=0 |cj| j̄β ≤ C ∫∞ 0 |f(t)|r dt for r ≥ 1 and β <−r/2. 2000 Mathematics Subject Classification: 42C10, 42C15.