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I am trying to understand Besov spaces; and I am eager to see why the various norms are equivalent on it.

Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with

$ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1$ if $|\xi|\leq 1.$

Let $j\in \mathbb N$ and $$\phi_{j}(\xi)= \phi(2^{-j}\xi)- \phi(2^{-j+1}\xi), (\xi \in \mathbb R^{n})$$

Then we have $$\operatorname{supp} \phi_{j} \subset \{\xi\in \mathbb R^{n}: 2^{j-1}\leq |\xi| \leq 2^{j+1} \}, j\in \mathbb N $$ and, with $\phi_{0}=\phi,$ $$\sum_{k=0}^{\infty} \phi_{k}(\xi)=1, \text{if} \ \xi\in \mathbb R^{n}.$$ In other words, $\{\phi_{k}\}_{k\in\mathbb N_{0}}$ is a resolution of unity. Let $f\in \mathcal{S'}$ then $\phi_{k}(D)f(x)= (\phi_{k}\hat{f})^{\vee}(x), k\in \mathbb N_{0}, x \in \mathbb R^{n}$ is an entire analytic function, (by Paley-Wiener-Schwartz theorem) and we have

$$f= \sum_{k=0}^{\infty} \phi_{k}(D)f;$$ convergence in $\mathcal{S'}.$ In other words, we decompose $f$ in entire analytic functions, and we introduce spaces by checking the behaviour of these analytic functions with respect to $x\in \mathbb R^{n}$ and $k\in \mathbb N_{0}.$

Let $0<p\leq \infty, 0 < q\leq \infty$ and $s\in \mathbb R$ then $$B^{s}_{p,q}(\mathbb R^{n})=\{f\in \mathcal{S'}(\mathbb R^{n}):\|f\|_{B^{s}_{p,q}}:=\left(\sum_{k=0}^{\infty} 2^{ksq} \|(\phi_{k}\hat{f})^{\vee}\|_{L^{p}}^{q}\right)^{1/q}<\infty \}.$$

To begin with we deal with integral means of differences. Let

$\triangle_{h}f(x):= f(x+h)-f(x), \forall h, x \in \mathbb R, \omega_{p}(f,h) := \left(\int_{\mathbb R} |\triangle_{h}f(x)|^{p} dx \right)^{1/p}$

Assume that $m<s<m+1$ for some $m\in \mathbb N.$

My Question is:

(1) Why the following expression

$$\|f\|_{L^{p}}+\left(\int_{\mathbb R} \left(\frac{\omega_{p}(f^{(m)},h)}{|h|^{s-m}}\right)^{q}\frac{dh}{|h|}\right)^{1/q},$$

is an equivalent norm in $B^{s}_{p,q}(\mathbb R).$

(2) Put $\Omega_{p}(f,t)=\left( \int_{\mathbb R} \sup_{|h|\leq t} |\triangle_{h}f(x)|^{p} \right)^{1/p} .$ Why the following expression is $$\|f\|_{L^{p}}+\left(\int_{0}^{\infty} \left(\frac{\Omega_{p}(f,t)}{t^{s}} \right)^{q}\frac{dt}{t}\right)^{1/q}$$ is an equivalent norm in $B^{s}_{p,q}(\mathbb R) (1/p<s<1).$

(3)Bit rough: Can you give me some feeling(underneath ideas) so that above norms are equivalent ? How much dyadic decomposition plays a role in proving so ? If possible, Historically who has observed these first time ? Can you suggests some original papers where I can find the proofs?

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    $\begingroup$ I am pretty sure you can find your answers in Adams and Fournier's Sobolev Spaces. The books has a reference section so you can probably also find the original papers there. // Additionally you may want to look at Hans Triebel's Theory of function spaces I and II. $\endgroup$ Nov 19 '14 at 9:24
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    $\begingroup$ Another book that is quite detailed is Runst and Sickel's Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. However, I guess for the equivalence of the norms they refer to Triebel… $\endgroup$
    – Dirk
    Nov 19 '14 at 9:30
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    $\begingroup$ Fourier transform exchanges smoothness of a function $f$ and the decay at infinity of $\hat{f}$. The original definition is on the Fourier side, which measures the decay at $\infty$, the second definition emphasize the smoothness of $f$ $\endgroup$
    – Tomas
    Nov 19 '14 at 14:49

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