Besov space

Remarks

A different choice of $\psi_0$ gives the same space with an equivalent quasi-norm. For $p, q \ge 1$ the space is a Banach space; otherwise it is a complete quasi-normed space. Here $L^p(\R^n)$ is the Lp space, with the same formula when $0 < p < 1$.

Example: $B^s_{2,2}(\R^n)$ is the space $H^s(\R^n)$ of tempered distributions with $(1 + \abs{\xi}^2)^{s/2} \hat f \in L^2$. Indeed, by Plancherel, $\norm{\Delta_j f}_{L^2}^2 = \int \psi_j^2 \abs{\hat f}^2 \, d\xi$ up to a constant factor, and $2^{2js}$ is comparable with $(1 + \abs{\xi}^2)^s$ on the support of $\psi_j$.

Non-example: the Dirac delta $\delta$ on $\R^n$ is not in $B^0_{1,q}(\R^n)$ for $q < \infty$. Here $\Delta_j \delta = \mathcal{F}^{-1} \psi_j$, and for $j \ge 1$ this function is $x \mapsto 2^{(j-1)n} (\mathcal{F}^{-1} \psi_1)(2^{j-1} x)$. So $\norm{\Delta_j \delta}_{L^1}$ is the same nonzero number for all $j \ge 1$, and the sum defining the norm diverges. The same computation shows that $\delta \in B^0_{1,\infty}(\R^n)$.

For $s > 0$ not an integer, $B^s_{\infty,\infty}(\R^n)$ is the Hölder space $C^{k,\alpha}(\R^n)$ with $k = \floor{s}$ and $\alpha = s - k$.