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Embedding theorems for anisotropic Lipschitz spaces. (English) Zbl 1079.46025

Assume that \(1\leq p<q<\infty\), \(n\) is a natural number greater than \(1\), and \(0<r_i<\infty\) for \(i=1,\dots,n\). Let \[ r=\Big(\sum_{i=1}^n {1\over r_i}\Big)^{-1}, \;r'=n\Big(\sum_{i:r_i\in{\mathbb N}} {1\over r_i}\Big)^{-1},\;s={r'p\over r}, q^*={np\over n-rp},\;\kappa=1-{n\over r}\Big({1\over p}-{1\over q}\Big),\;\alpha_i=\kappa r_i, \] and put \(1/\gamma_i=(1-\kappa)/s+\kappa/p\) if \(r_i\in{\mathbb N}\) and \(1/\gamma_i=(1-\kappa)/s\) if \(r_i\notin {\mathbb N}\). The author proves three theorems about embeddings of anisotropic Lipschitz classes \(\Lambda_p^{r_1,\dots,r_n}({\mathbb R}^n)\) with the norm \(\| f\| _{\Lambda_p^{r_1,\dots,r_n}({\mathbb R}^n)}=\| f\| _p+\| f\| _{\lambda_p^{r_1,\dots,r_n}}\), where \(\| f\| _{\lambda_p^{r_1,\dots,r_n}}= \sum_{j=1}^n\sup\limits_{\delta>0}\delta^{-r_j}\omega_j^{\overline{r}_j}(f;\delta)_p\), \(\overline{r}_j\) is the least integer such that \(\overline{r}_j\geq r_j\), and \(\omega_j^k(f;\delta)_p\) is the partial modulus of continuity of \(f\) of order \(k\) in \(L^p({\mathbb R}^n)\) with respect to the \(j\)th variable.
Consider the following Lorentz type spaces. Let \({\mathcal R}_k f\) be the non-increasing rearrangement of \(f\) with respect of the \(k\)th variable. Let \({\mathcal P}_n\) be the set of all permutations \(\sigma=\{k_1,\dots,k_n\}\) of \(\{1,\dots,n\}\) and let \({\mathcal R}_\sigma f={\mathcal R}_{k_n}\dots{\mathcal R}_{k_1} f\). For \(t\in{\mathbb R}_+^n\), put \(\pi(t)=\prod_{k=1}^n t_k\) and consider the space \(L_{{\mathcal R}}^{p,q}({\mathbb R}^n)\) of all measurable functions \(f\) satisfying \[ \| f\| _{p,q;{\mathcal R}}=\sum_{\sigma\in{\mathcal P}_n}\Big(\int_{{\mathbb R}_+^n} [\pi(t)^{1/q}{\mathcal R}_\sigma f(t)]^p{dt\over \pi(t)}\Big)^{1/p}<\infty. \] A. A. Yatsenko [Izv. Vyssh. Uchebn. Zaved. Mat. 1998, No. 5, 73–77; translation in Russian Math. (Iz. VUZ) 42, No. 5, 71–75 (1998; MR 99i:46019)] proved that \(L_{{\mathcal R}}^{p,q}({\mathbb R}^n)\) is properly embedded into the classical Lorentz space \(L^{p,q}({\mathbb R}^n)\) whenever \(q>p\). The first result of the paper under review (Theorem 1) is the following Sobolev type inequality extending previous results by V. I. Kolyada and Yu. V. Netrusov. Suppose that \(1\leq p<n/r\). Then \(\| f\| _{q^*,s;{\mathcal R}}\leq c\| f\| _{\lambda_p^{r_1,\dots,r_n}}\) for all \(f\in\Lambda_p^{r_1,\dots,r_n}({\mathbb R}^n)\), where \(c\) is a constant independent of \(f\). This result is complemented by Theorem 3. If \(\kappa>0\) and \(0<\xi<\infty\), then \(\| f\| _{q,s;{\mathcal R}}\leq c\| f\| _{\Lambda_p^{r_1,\dots,r_n}}\) for all \(f\in\Lambda_p^{r_1,\dots,r_n}({\mathbb R}^n)\), where \(c\) is a constant independent of \(f\). Theorem 2 is the following Il’in type inequality. If \(\kappa>0\), then for any \(f\in\Lambda_p^{r_1,\dots,r_n}({\mathbb R}^n)\), \[ \sum_{i=1}^n \Big(\int_0^\infty[h^{-\alpha_i}\| \Delta_i^{\overline{r}_i}(h)f\| _{q,1;{\mathcal R}}]^{\gamma_i} {dh\over h}\Big)^{1/\gamma_i} \leq c\| f\| _{\lambda_p^{r_1,\dots,r_n}}, \] where \(c\) is a constant independent of \(f\) and \(\Delta_j^k(h)f(x)\) is the difference of order \(k\) with respect to the \(j\)th variable. In particular, this result implies embeddings of anisotropic Lipschitz classes into anisotropic Besov spaces. The proofs are based on clever estimates of iterative non-increasing rearrangements \({\mathcal R}_\sigma f\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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