Lacunary Series in Mixed Norm Spaces on the Ball and the Polydisk

We characterize lacunary series in mixed norm spaces on the unit ball B in C and on the unit polydisk D in C. Introduction and main results Let n be a positive integer. Two domains will be used in the paper: the open unit ball B in C, B = {z ∈ C : |z| < 1 }, and the open unit polydisk D in C, D = {z = (z1, ..., zn) ∈ C : |z1| < 1, ..., |zn| < 1 }. We write D = B = D. Denote by T the Shilov boundary of D, by ∂B the boundary of B, by dσn the normalized surface measure on ∂B, and define the measure dμn on T by dμn(e , . . . , en) = dθ1 · · · dθn. Lacunary series on the unit ball B The mixed norm space H(B), 0 < p, q ≤ ∞ 0 < α < ∞, consists of all functions f holomorphic in B, f ∈ H(B), such that ||f ||p,q,α = ∫ 1 0 (1− r)qα−1Mp(r, f)dr < ∞, if 0 < q < ∞, and ||f ||p,∞,α = sup 0<r<1 (1− r)Mp(r, f) < ∞. 2010 Mathematics Subject Classifications. 32A36, 32A37.