Linear space properties of H^p spaces of Dirichlet series
Journal article, Peer reviewed
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Original versionTransactions of the American Mathematical Society. 2019, 372 (9), 6677-6702. https://doi.org/10.1090/tran/7898
We study H p spaces of Dirichlet series, called H p , for the range 0 < p < ∞. 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 , contrasting 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 P∞ n=1 ann −s 7→ PN n=1 ann −s on H p with 0 < p ≤ 1, supplementing a classical result of Helson for the range 1 < p < ∞. The results for the coefficient estimates and for the partial sum operator exhibit the traditional schism between the ranges 1 ≤ p ≤ ∞ and 0 < p < 1.