LINEAR SPACE PROPERTIES OF Hp SPACES OF DIRICHLET SERIES

We study H-p spaces of Dirichlet series, called H-p, for the range 0 < p < infinity. We begin by showing that two natural ways to define H-p coincide. We then proceed to study some linear space properties of H-p. More specifically, we study linear functionals generated by fractional primitives of the Riemann zeta function; our estimates rely on certain Hardy-Littlewood inequalities and display an interesting phenomenon, called contractive symmetry between H-p and H-4/p,H- contrasting with the usual L-p duality. We next deduce general coefficient estimates, based on an interplay between the multiplicative structure of H-p and certain new one variable bounds. Finally, we deduce general estimates for the norm of the partial sum operator Sigma(infinity)(n=1) a(n)(n-s) bar right arrow on Sigma(N)(n=1) a(n)(n-s) H-p with 0 < p <= 1, supplementing a classical result of Helson for the range 1 < p < infinity. The results for the coefficient estimates and for the partial sum operator exhibit the traditional schism between the ranges 1 <= p <= infinity and 0 < p < 1.