Entropy and Approximation Numbers of Embeddings of Function Spaces with Muckenhoupt Weights, I

We study compact embeddings for weighted spaces of Besov and Triebel-Lizorkin type where the weight belongs to some Muckenhoupt Ap class. For weights of purely polynomial growth, both near some singular point and at infinity, we obtain sharp asymptotic estimates for the entropy numbers and approximation numbers of this embedding. The main tool is a discretization in terms of wavelet bases.We study compact embeddings for weighted spaces of Besov and Triebel-Lizorkin type where the weight belongs to some Muckenhoupt Ap class. For weights of purely polynomial growth, both near some singular point and at infinity, we obtain sharp asymptotic estimates for the entropy numbers and approximation numbers of this embedding. The main tool is a discretization in terms of wavelet bases

Nuclear embeddings of Morrey sequence spaces and smoothness Morrey spaces

We study nuclear embeddings for spaces of Morrey type, both in its sequence space version and as smoothness spaces of functions defined on a bounded domain $\Omega \subset {\mathbb R}^d$. This covers, in particular, the meanwhile well-known and completely answered situation for spaces of Besov and Triebel-Lizorkin type defined on bounded domains which has been considered for a long time. The complete result was obtained only recently. Compact embeddings for function spaces of Morrey type have already been studied in detail, also concerning their entropy and approximation numbers. We now prove the first and complete nuclearity result in this context. The concept of nuclearity has already been introduced by Grothendieck in 1955. Again we rely on suitable wavelet decomposition techniques and the famous Tong result (1969) which characterises nuclear diagonal operators acting between sequence spaces of $\ell_r$ type, $1 \leq r \leq\infty$

Traces of some weighted function spaces and related non‐standard real interpolation of Besov spaces

We study traces of weighted Triebel–Lizorkin spaces F p , q s ( R n , w ) $F^s_{p,q}(\mathbb {R}^n,w)$ on hyperplanes R n − k $\mathbb {R}^{n-k}$ , where the weight is of Muckenhoupt type. We concentrate on the example weight w α ( x ) = | x n | α $w_\alpha (x) = {\big\vert x_n\big\vert }^\alpha$ when | x n | ≤ 1 $\big\vert x_n\big\vert \le 1$ , x ∈ R n $x\in \mathbb {R}^n$ , and w α ( x ) = 1 $w_\alpha (x)=1$ otherwise, where α > − 1 $\alpha >-1$ . Here we use some refined atomic decomposition argument as well as an appropriate wavelet representation in corresponding (unweighted) Besov spaces. The second main outcome is the description of the real interpolation space ( B p 1 , p 1 s 1 ( R n − k ) , B p 2 , p 2 s 2 ( R n − k ) ) θ , r $\big (B^{s_1}_{p_1,p_1}\big (\mathbb {R}^{n-k}\big ), B^{s_2}_{p_2,p_2}{\big (\mathbb {R}^{n-k}\big )\big )}_{\theta ,r}$ , 0 0$sufficiently large, 0 < θ < 1$0<\theta <1$, 0 < r ≤ ∞$

Embeddings of generalised Morrey smoothness spaces

We study embeddings between generalised Triebel-Lizorkin-Morrey spaces ${\mathcal E}^{s}_{\varphi,p,q}({\mathbb R}^d)$ and within the scales of further generalised Morrey smoothness spaces like ${\mathcal N}^{s}_{\varphi,p,q}({\mathbb R}^d)$, ${B}_{p,q}^{s,\varphi}({\mathbb R}^d)$ and ${F}_{p,q}^{s,\varphi}({\mathbb R}^d)$. The latter have been investigated in a recent paper by the first two authors (2023), while the embeddings of the scale ${\mathcal N}^{s}_{\varphi,p,q}({\mathbb R}^d)$ were mainly obtained in a paper of the first and last two authors (2022). Now we concentrate on the characterisation of the spaces ${\mathcal E}^{s}_{\varphi,p,q}({\mathbb R}^d)$. Our approach requires a wavelet characterisation of those spaces which we establish for the system of Daubechies' wavelets. Then we prove necessary and sufficient conditions for the embedding ${\mathcal E}^{s_1}_{\varphi_1,p_1,q_1}({\mathbb R}^d)\hookrightarrow {\mathcal E}^{s_2}_{\varphi_2,p_2,q_2}({\mathbb R}^d)$. We can also provide some almost final answer to the question when ${\mathcal E}^{s}_{\varphi,p,q}({\mathbb R}^d)$ is embedded into $C({\mathbb R}^d)$, complementing our recent findings in case of ${\mathcal N}^{s}_{\varphi,p,q}({\mathbb R}^d)$