I'm following two "different" approaches to the Besov Spaces, but I don't get if the two definitions given are equivalent.

  1. Victor I. Burenkov - Sobolev Spaces On Domains.

Given $f:\mathbb{R}^n \to \mathbb{R}$ and $h \in \mathbb{R}^n$ we set $\Delta_h f(x) = f(x+h)-f(x)$ and recursively $\Delta_h ^{\sigma} f(x) = \underbrace{\Delta_h (\Delta_h(\dots (\Delta_h}_{\sigma \text{ times}} f )\dots))(x) = \sum_{k=0}^{\sigma} (-1)^{\sigma} \binom{\sigma}{k} f(x+kh) $.

Definition. Let $l>0, \sigma \in \mathbb{N}, \sigma > l, 1 \le p, \theta \le \infty$. The function $f$ belongs to the Besov-Nikol'skij space $B^l _{p,\theta} (\mathbb{R}^n)$ if $f$ is measurable on $\mathbb{R}^n$ and $$\|f\|_{B^l _{p,\theta} (\mathbb{R}^n)} = \|f\|_{L^p (\mathbb{R}^n)} + \|f\|_{b^l _{p,\theta} (\mathbb{R}^n)}<\infty$$ where $$\|f\|_{b^l _{p,\theta} (\mathbb{R}^n)} = \left( \int_{\mathbb{R}^n} \left(\frac{\|\Delta^{\sigma}_h f \|_{L^p(\mathbb{R}^n)}}{|h|^l} \right)^\theta \frac{dh}{|h|^n} \right)^{1/\theta}$$if $1 \le \theta < \infty$ and $$\|f\|_{B^l _{p,\infty} (\mathbb{R}^n)} = \sup_{h \in \mathbb{R}^n \setminus \{0 \}} \frac{\|\Delta^{\sigma}_h f \|_{L^p(\mathbb{R}^n)}}{|h|^l}. $$ Burenkov says also that the definition is independent of $\sigma > l$.

  1. Giovanni Leoni - A First Course in Sobolev Spaces.

Given a function $u: \mathbb{R}^N \to \mathbb{R}$, for every $h \in \mathbb{R}$, $i=1, \dots, N$, and $x \in \mathbb{R}^n$, we define $$\Delta^h _i u(x) := u(x+h e_i)-u(x)$$ where $e_i$ is the ith vector of the canonical basis in $\mathbb{R}^n$.

Definition. Let $1 \le p, \theta \le \infty$ and $0<s<1$. A function $u \in L^1_{\text{loc}} (\mathbb{R}^N)$ belongs to the Besov space $B^{s,p,\theta}(\mathbb{R}^N)$ if $$\|u\|_{B^{s,p,\theta} (\mathbb{R}^N)} = \|u\|_{L^p (\mathbb{R}^N)} + |u|_{B^{s,p,\theta} (\mathbb{R}^N)}<\infty$$ where $$|u|_{B^{s,p,\theta} (\mathbb{R}^N)} := \sum_{i=1}^N \left( \int_0^\infty \| \Delta^h _i u \|^\theta _{L^p(\mathbb{R}^N)} \frac{dh}{h^{1 +s\theta} }\right)^{1/\theta}$$if $\theta < \infty$ and $$|u|_{B^{s,p,\infty}(\mathbb{R}^N)} := \sum_{i=1}^N \sup_{h>0} \frac{1}{h^s} \| \Delta_i ^h u \|_{L^p(\mathbb{R}^N)}.$$

The two definition look like pretty the same but, at least apparently, they aren't. Are they actually equivalent? I don't need a proof, but I just wanted to know if this is a field of Math in which things with the same name are not equivalently defined (Burenkov defines other nine equivalent Besov norms, but there's not the Leoni's one between them).

Thanks in advance!


1 Answer 1


The answer is positive: these norms are equivalent (the second definition was given only for $0< s < 1$, but it can be extended in a natural way to any $s>0$).


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