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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:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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