$L^{p}$-behavior of power series with positive coefficients and Hardy spaces

Abstract:

For the power series $f(x) = \sum \nolimits _1^\infty {{a_n}{x^n}}$ with ${a_n} \geqslant 0$, certain weighted ${L^p}$-norms of $f$ on $[0,1]$ are estimated from above and below in terms of the coefficients ${a_n}$. Some consequences of this are obtained. For example, some known results concerning Hardy spaces may be extended to a wider class of spaces.