# Space of analytic function and sequence space $l^p$

Let $\mathbb{D} = \{z:|z|<1\}$ be open unit disc in complex plane. Define space of analytic function: $N^p=\{f:\mathbb{D} \to \mathbb{C} | f(z)=\sum_{n=0}^{\infty} a_n z^n, \sum_{n=0}^{\infty}|a_n|^p < \infty\}$. For $p=1$ we have Wiener algebra, and for $p=2$ we have Hardy space $H^2$. What we know about this space of analytic function for $p \in (1,2) \cup (2,\infty]$. Describe relationship between Hardy spaces and $N^p$ spaces? Is there any paper or reference for this kind of spaces?

• I think that rather than ask "where do I find a book about these spaces" one should ask "what do I want to know about these spaces"? That is: what kinds of questions do you have about how the summability of the Taylor series of an analytic function affects properties of the function? – Yemon Choi Mar 30 '17 at 16:09
• What is specific name for these spaces? Is there any paper, reference... – Nebojša Đurić Mar 31 '17 at 10:20
• OK, if you want to know how $N^p$ is related or unrelated to the scale of Hardy spaces, then you might find some things in Duren's book Theory of $H^p$-spaces. However, without further details of what kinds of question you want to answer or investigate, I'm not sure if that book will turn out to be helpful – Yemon Choi Mar 31 '17 at 16:16