## 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:

 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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