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As I am relatively new to these matters, I would like to know if you could provide me a reference for Besov spaces on unbounded domains, because when I checked the first tome of Triebel's Theory of Function Spaces, I only found the case of a smooth bounded set aside of whole or half space (in Bergh's Interpoation spaces, there is no mention neither of these spaces).

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3 Answers 3

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Hormander: The Analysis of Linear Partial Differential Operators II, 1983, page 13 ff.

These spaces are $B_{k,p}(\mathbb R^n)\cap \mathcal E'(X)$, where $X$ is open in $\mathbb R^n$.

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O.V Besov, V.P Il'in, S.M Nikolskii. Integral Representations of Functions and Embedding Theorems. Most of results are stated for arbitrary domains $G\subset \mathbb R^n$.

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  • $\begingroup$ Thank you very much Andrew, I didn't accept your answer because I can only accept one, but I really appreciate the extra reference. I didn't know Besov himself wrote about the subject; sometimes in Analysis, there are pioneers who don't write about their own work, e.g. Calderón. $\endgroup$
    – Paul-Benjamin
    Commented Apr 18, 2015 at 17:20
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The following papers may help:

  • "Besov Spaces on Domains in $\mathbb{R}^d$" by R. Devore and R. Sharpley

This paper is for Besov spaces $B^{\alpha}_{q}(L_p(\Omega)),~p,q,\alpha \in (0,\infty )$ on domains $\Omega\subset \mathbb{R}^d$

http://www.ams.org/journals/tran/1993-335-02/S0002-9947-1993-1152321-6/S0002-9947-1993-1152321-6.pdf

  • "Elliptic and parabolic problems in unbounded domains" by Patrick Guidotti

This examines general existence and regularity results in Besov spaces (specifically, on unbounded domains $\mathbb{R}^n\times\Omega$).

http://www.math.uci.edu/~gpatrick/source/papers/G041.pdf

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  • $\begingroup$ Thank you very much Mr Simpson, this is a good complementary reference that you added here, as Hörmander's treatment of the subject does not include the Besov spaces of three parameters. $\endgroup$
    – Paul-Benjamin
    Commented Jul 24, 2015 at 10:56
  • $\begingroup$ No problem. I am quite interested in Besov spaces. $\endgroup$
    – user71046
    Commented Jul 24, 2015 at 10:57

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