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It is well-known that any convergence in $L^p$ for $p \in [1,\infty]$ implies convergence in $L^1_{\text{loc}}$ by Hölder's inequality.

It is also known that for $p>1$ it holds that $L^p(\mathbb R)$ is isomorphic to $H^p(\mathbb H),$ the Hardy space on the upper half plane.

So it is tempting to assume that $H^p_{\text{loc}}(\mathbb H)$ convergence implies $H^q_{\text{loc}}(\mathbb H)$ convergence for $q\le p$ just as in the case for $L^p$ spaces.

My question is whether this is true and how $1$ and $\infty$ fit into this scheme.

I am very grateful for any idea/comment on this matter.

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  • $\begingroup$ Out of curiosity, what is meant by $H^p_{loc}(\mathbb{H})$? I am having some difficulty figuring out how to localize the definition of Hardy space on the upper half plane. $\endgroup$
    – Willie Wong
    Commented Mar 30, 2018 at 15:35
  • $\begingroup$ You usually only consider compact sets $[y_1,y_2] \times [x_1,x_2] \subset \mathbb H$ contained in the upper half plane and then you consider the norm on those sets. $\endgroup$
    – Heins Siedentopf
    Commented Mar 30, 2018 at 15:44

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