Limits of Besov norms
Let \(\Delta_{h}^{m}f(x)\) with \(x \in\mathbb{R}^n\), \(h\in \mathbb{R}^n\), \(m\in \mathbb{N}\), be the usual differences of complex-valued Lebesgue-measurable functions on \(\mathbb{R}^n\) and let \(\omega_{m}(f, t)_{p}=\sup_{|h| \leq t}\|\Delta_{h}^{m}f|L_{p}(\mathbb{R}^n)\|\), \(0< p \leq \infty\), \(t >0\), be the related moduli of smoothness. Let \(0 < p, q \leq \infty\) and \(0 < s < m \in \mathbb{N}\). Then \[ \|f|\mathbf{B}^{s}_{p,q}(\mathbb{R}^n)\|_{m}=\|f|L_{p}(\mathbb{R}^n)\|+ \biggl(\int_{|h| \leq 1}|h|^{-sq}\|\Delta_{h}^{m}f|L_{p}(\mathbb{R}^n)\|^{q} \displaystyle{\frac{dh}{|h|^n}} \biggr)^{1/q} \] and \[ \|f|\mathbf{B}^{s}_{p,q}(\mathbb{R}^n)\|^{*}_{m}=\|f|L_{p}(\mathbb{R}^n)\|+ \biggl(\int_{0}^{1}t^{-sq}\omega_{m}^{q}(f,t)_{p} \displaystyle{\frac{dt}{t}} \biggr)^{1/q}\tag{1} \] (obviously modified if \(q=\infty\)) are equivalent quasi-norms of the Besov spaces \(\mathbf{B}^{s}_{p,q}(\mathbb{R}^n)\) consisting of those measurable functions for which (1) is finite. Let \(0 < p \leq \infty\) and \(m \in \mathbb{N}\). Then \(\mathbf{W}_{p}^{m}(\mathbb{R}^n)\) is the collection of all \(f \in L_{p}(\mathbb{R}^n)\) for which there exists a function \(0 \leq g \in L_{p}(\mathbb{R}^n)\) such that for all \(h \in \mathbb{R}^n\), \(0<|h| \leq 1,\) \[ |h|^{-m}|\Delta^{m}_{h}f(x)| \leq \sum_{l=0}^{m}g(x+lh) \,\, {\text{ a.e. \;\;in}}\;\;\mathbb{R}^n\tag{2} \] Furthermore, \[ \|f|\mathbf{W}^{m}_{p}(\mathbb{R}^n)\| =\|f|L_{p}(\mathbb{R}^n)\|+\inf\|g|L_{p}(\mathbb{R}^n)\| \] where the infimum is taken over all \(g\) with (2). If \(1 < p< \infty\), then \(\mathbf{W}^{m}_{p}(\mathbb{R}^n)=W^{m}_{p}(\mathbb{R}^n)\) are the classical Sobolev spaces and \[ \|f|\mathbf{W}^{m}_{p}(\mathbb{R}^n)\|\sim \|f|W^{m}_{p}(\mathbb{R}^n)\|= \sum_{|\alpha| \leq m}\|D^{\alpha}f|L_{p}(\mathbb{R}^n)\| \] are equivalent norms. The paper deals with the question what happens if \(s \uparrow m\) and how the outcome is related to the Sobolev spaces \(\mathbf{W}^{m}_{p}(\mathbb{R}^n)\). We observe that \[ \|f|\mathbf{B}^{s}_{p,q}(\mathbb{R}^n)\|_{m} \sim \|f|\mathbf{B}^{s}_{p,q}(\mathbb{R}^n)\|_{m}^{*} \leq c (q(m-s))^{-1/q}\|f|\mathbf{W}^{m}_{p}(\mathbb{R}^n)\| \] (with 1 in place of \((q(m-s))^{-1/q}\) if \(q=\infty\)). The main result of this paper is as follows: Let \(1 < p < \infty\) and \(m \in \mathbb{N}\). Let \(0 < s < m\) and \(0 < q < \infty\). Then there are equivalence constants which are independent of \(s,q\) and \(f \in W^{m}_{p}(\mathbb{R}^n)\) (but may depend on \(p,m\)) such that \[ \lim_{s \uparrow m}(m-s)^{1/q}\|f| \mathbf{B}^{s}_{p,q} (\mathbb{R}^n)\|_{m}^{*} \sim q^{-1/q} \sum_{|\alpha|=m} \|D^{\alpha}f|L_{p}(\mathbb{R}^n)\|. \]
scientific article; zbMATH DE number 5872601