Integrability of Double Power Series with Nonnegative Coefficients

We consider the double power series ` ` f x, y.[ ajkxjyk js0 ks0 with coefficients ajkG0 for all j and k. Among others, we prove exact estimates of certain weighted L p-norms of f on the unit square, in terms of the coefficients ajk . Our results extend those of Askey and Boas, Hardy and Littlewood, Khan, Leindler, Mateljevi´c and Pavlovi´c from single to double power series. The further extension to multiple power series is also indicated.