It is known that theLp-norms of the sums of power series can be estimated from below and above by means of their coefficients, provided these coefficients
are nonnegative. In the paper we prove analogous estimates for theLp-norms of the sums of Dirichlet series. Our main result gives exact lower and upper estimates for the BMO-norm of the sums
of power series and Dirichlet series, respectively, by means of their coefficients.
Moduli of p-continuity provide a measure of fractional smoothness of functions via p-variation. We prove a sharp estimate of the modulus of p-continuity in terms of the modulus of q-continuity (1<p<q<∞).
Authors:J. Garcia-Cuerva, K. Kazarian and V. Kolyada
We investigate the extension to Banach-space-valued functions of the classical inequalities due to Paley for the Fourier coefficients
with respect to a general orthonormal system Φ. This leads us to introduce the notions of Paley Φ-type and Φ-cotype for a
Banach space and some related concepts. We study the relations between these notions of type and cotype and those previously
defined. We also analyze how the interpolation spaces inherit these characteristics from the original spaces, and use them
to obtain sharp coefficient estimates for functions taking values in Lorentz spaces.