# zbMATH — the first resource for mathematics

Local growth envelopes of Besov spaces of generalized smoothness. (English) Zbl 1119.46032
The main aim of this paper is to give a unified approach to the question of determining the growth envelope for Besov spaces $$B^{\alpha,\,N}_{p,\,q}(\mathbb{R}^n)$$ of generalized smoothness, where $$0<p,q\leq\infty$$, and $$\alpha=(\alpha_j)_j$$ and $$N=(N_j)_j$$ are admissible sequences, namely, there exist positive constants $$c_0$$ and $$c_1$$ such that for $$\gamma\in \{\alpha,\,N\}$$ and all $$j\in \mathbb{N}\cup\{0\}$$, $$c_0\gamma_j\leq\gamma_{j+1}\leq c_1\gamma_j$$. The local growth envelope function $$\mathcal{E}_{\text{LG}}| B^{\alpha,\,N}_{p,\,q}$$ for $$t>0$$ is defined by $\mathcal{E}_{\text{LG}}| B^{\alpha,\,N}_{p,\,q}(t)\equiv \sup\{f^*(t): \| f| B^{\alpha,\,N}_{p,\,q}\| \leq1\},$ where $$f^*$$ stands for the decreasing rearrangement of $$f$$.
Let $$0<p,\,q\leq\infty$$, and let $$N=\{N_j\}_{j\in\mathbb{N}\cup\{0\}}$$ be an admissible sequence with $$1<c_0\leq c_1$$. Let $$\Lambda$$ be an admissible function such that $$\Lambda(z)\sim \sigma_j$$, $$z\in [N_j,\,N_{j+1}]$$, $$j\in \mathbb{N}_0$$, with equivalence constants independent of $$j$$, and let $$\Phi_{q'}$$ for $$t\in (0,\,N_{J_0}^{-n}]$$ be defined by $\Phi_{q'}(t)\equiv\left(\int^1_{t^{\frac 1{n}}}y^{-\frac n{p}q'} \Lambda(y^{-1})^{-q'}\,\frac{dy}{y}\right)^{1/q'},\;\text{if}\;0<q'<\infty,$ and $$\Phi_{q'}(t)\equiv \sup_{t^{\frac 1{n}}\leq y\leq1}y^{-\frac n{p}}\Lambda(y^{-1})^{-1},$$ if $$q'=\infty$$, where $$1/q+1/q'=1$$, and $$J_0\in\mathbb{N}$$ is chosen such that $$N_{J_0}>1$$. Recall that $$\mathfrak{E}_{\text{LG}}B^{\alpha,\,N}_{p,\,q}\equiv([\mathcal{E}_{\text{LG}}| B^{\alpha,\,N}_{p,\,q}], u)$$ is called the local growth envelope of $$B^{\alpha,\,N}_{p,\,q}$$, if $$u$$ is the minimum (assuming that it exists) of all $$\nu>0$$ such that there exists $$c(\nu)>0$$:
$\forall f\in B^{\alpha,\,N}_{p,\,q},\;\;\left(\int_{(0,\,\varepsilon]}\left(\frac{f^*(t)}{h(t)}\right)^\nu\, \mu_H(dt)\right)^{1/\nu}\leq c(\nu)\| f| B^{\alpha,\,N}_{p,\,q}\| ,$ where $$h(t)$$ is a continuous representative in $$[\mathcal{E}_{\text{LG}}| B^{\alpha,\,N}_{p,\,q}]$$ with domain $$(0,\,\varepsilon]$$, $$0<\varepsilon<1$$, $$H(t)\equiv -\log_2\,\mathcal{E}_{\text{LG}}| B^{\alpha,\,N}_{p,\,q}$$ and $$\mu_H$$ is the only Borel measure such that $$\mu_H([a,\,b])=H(b)-H(a)$$ for any $$[a,\,b]\subset (0,\,\varepsilon]$$.
The main result of this paper is that under the above assumptions, $$\mathfrak{E}_{\text{LG}}B^{\alpha,\,N}_{p,\,q}(\mathbb R^n)=(E_{q'},\,q)$$.

##### MSC:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: