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?

  • $\begingroup$ 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? $\endgroup$
    – Yemon Choi
    Mar 30, 2017 at 16:09
  • $\begingroup$ What is specific name for these spaces? Is there any paper, reference... $\endgroup$ Mar 31, 2017 at 10:20
  • 1
    $\begingroup$ 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 $\endgroup$
    – Yemon Choi
    Mar 31, 2017 at 16:16


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