arXiv subjects

Leszek Skrzypczak

Publications and source records attributed to Leszek Skrzypczak.

At least 19 recordsLinked to original sources

arXiv2026

Compact embeddings of generalised Morrey smoothness spaces on bounded domains

We study embeddings within different scales of generalised smoothness Morrey spaces defined on bounded smooth domains, i.e., in $\mathcal{N}^s_{φ,p,q}(Ω)$, $\mathcal{E}^s_{φ,p,q}(Ω)$, $B^{s,φ}_{p,q}(Ω)$ and $F^{s,φ}_{p,q}(Ω)$ spaces. We prove sufficient conditions for continuity and compactness of the embeddings. In some cases the conditions are also necessary. We generalise and even improve some earlier results known for the classical smoothness Morrey spaces. Our approach is based on wavelet characterisation of the function spaces.

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arXiv2025

Generalised Morrey sequence spaces

Generalised Morrey (function) spaces enjoyed some interest recently and found applications to PDE. Here we turn our attention to their discrete counterparts. We define generalised Morrey sequence spaces $m_{φ,p}=m_{φ,p}(\mathbb{Z}^d)$. They are natural generalisations of the classical Morrey sequence spaces $m_{u,p}$, $0<p\le u<\infty$, which were studied earlier. We consider some basic features of the spaces as well as embedding properties such as continuity, compactness and strict singularity.

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arXiv2023

Embeddings of block-radial functions -- approximation properties and nuclearity

Let $ R_γB^{s}_{p,q}(\mathbb{R}d)$ be a subspace of the Besov space $B^{s}_{p,q}(\mathbb{R}^d)$ that consists of block-radial (multi-radial) functions. We study an asymptotic behaviour of approximation numbers of compact embeddings $id: R_γB^{s_1}_{p_1,q_1}(\mathbb{R}^d) \rightarrow R_γB^{s_2}_{p_2,q_2}(\mathbb{R}^d)$. Moreover we find the sufficient and necessary condition for nuclearity of the above embeddings. Analogous results are proved for fractional Sobolev spaces $R_γH^s_p(\mathbb{R}^d)$.

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arXiv2023

Embeddings of generalised Morrey smoothness spaces

We study embeddings between generalised Triebel-Lizorkin-Morrey spaces ${\mathcal E}^{s}_{φ,p,q}({\mathbb R}^d)$ and within the scales of further generalised Morrey smoothness spaces like ${\mathcal N}^{s}_{φ,p,q}({\mathbb R}^d)$, ${B}_{p,q}^{s,φ}({\mathbb R}^d)$ and ${F}_{p,q}^{s,φ}({\mathbb R}^d)$. The latter have been investigated in a recent paper by the first two authors (2023), while the embeddings of the scale ${\mathcal N}^{s}_{φ,p,q}({\mathbb R}^d)$ were mainly obtained in a paper of the first and last two authors (2022). Now we concentrate on the characterisation of the spaces ${\mathcal E}^{s}_{φ,p,q}({\mathbb R}^d)$. 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 ${\mathcal E}^{s_1}_{φ_1,p_1,q_1}({\mathbb R}^d)\hookrightarrow {\mathcal E}^{s_2}_{φ_2,p_2,q_2}({\mathbb R}^d)$. We can also provide some almost final answer to the question when ${\mathcal E}^{s}_{φ,p,q}({\mathbb R}^d)$ is embedded into $C({\mathbb R}^d)$, complementing our recent findings in case of ${\mathcal N}^{s}_{φ,p,q}({\mathbb R}^d)$.

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arXiv2023

Mapping properties of Fourier transforms, revisited

The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished Besov spaces $B^s_p(\mathbb{R}^n) = B^s_{p,p}(\mathbb{R}^n)$, $1\le p \le \infty$, and between Sobolev spaces $H^s_p(\mathbb{R}^n)$, $1<p< \infty$. In contrast to the paper {\em H. Triebel, Mapping properties of Fourier transforms. Z. Anal. Anwend. 41 (2022), 133--152}, based mainly on embeddings between related weighted spaces, we rely on wavelet expansions, duality and interpolation of corresponding (unweighted) spaces, and (appropriately extended) Hausdorff-Young inequalities. The degree of compactness will be measured in terms of entropy numbers and approximation numbers, now using the symbiotic relationship to weighted spaces.

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arXiv2023

On a bridge connecting Lebesgue and Morrey spaces in view of their growth properties

We study unboundedness properties of functions belonging to generalised Morrey spaces ${\mathcal M}_{φ,p}({\mathbb R}^d)$ and generalised Besov-Morrey spaces ${\mathcal N}^{s}_{φ,p,q}({\mathbb R}^d)$ by means of growth envelopes. For the generalised Morrey spaces we arrive at the same three possible cases as for classical Morrey spaces $\mathcal{M}_{u,p}({\mathbb R}^d)$, i.e., boundedness, the $L_p$-behaviour or the proper Morrey behaviour for $p<u$, but now those cases are characterised in terms of the limit of $φ(t)$ and $t^{-d/p} φ(t)$ as $t \to 0^+$ and $t\to\infty$, respectively. For the generalised Besov-Morrey spaces the limit of $t^{-d/p} φ(t)$ as $t \to 0^+$ also plays a rôle and, once more, we are able to extend to this generalised spaces the known results for classical Besov-Morrey spaces, although some cases are not completely solved. In this context we can completely characterise the situation when ${\mathcal N}^{s}_{φ,p,q}({\mathbb R}^d)$ consists of essentially bounded functions only, and when it contains regular distributions only.

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arXiv2022

Nuclear and compact embeddings in function spaces of generalised smoothness

We study nuclear embeddings for function spaces of generalised smoothness defined on a bounded Lipschitz domain $Ω\subset\mathbb{R}^d$. This covers, in particular, the well-known situation for spaces of Besov and Triebel-Lizorkin spaces defined on bounded domains as well as some first results for function spaces of logarithmic smoothness. In addition, we provide some new, more general approach to compact embeddings for such function spaces, which also unifies earlier results in different settings, including also the study of their entropy numbers. Again we rely on suitable wavelet decomposition techniques and the famous Tong result (1969) about nuclear diagonal operators acting in $\ell_r$ spaces, which we could recently extend to the vector-valued setting needed here.

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arXiv2022

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 $Ω\subset {\mathbb R}^d$. 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, $1 \leq r \leq\infty$.

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arXiv2022

Nuclear Fourier transforms

The paper deals with the problem under which conditions for the parameters $s_1,s_2\in\mathbb{R}$, $1\leq p,q_1,q_2\leq\infty$ the Fourier transform $\mathcal{F}$ is a nuclear mapping from $A^{s_1}_{p,q_1}(\mathbb{R}^n)$ into $A^{s_2}_{p,q_2}(\mathbb{R}^n)$, where $A\in\{B,F\}$ stands for a space of Besov or Triebel-Lizorkin type, and $n\in\mathbb{N}$. It extends the recent paper arXiv:2112.04896 where the compactness of $\mathcal{F}$ acting in the same type of spaces was studied.

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arXiv2021

Limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces on domains and an extension operator

In this paper, we study limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces, $id_τ: B_{p_1,q_1}^{s_1,τ_1}(Ω) \rightarrow B_{p_2,q_2}^{s_2,τ_2}(Ω)$ and $id_τ: F_{p_1,q_1}^{s_1,τ_1}(Ω) \rightarrow F_{p_2,q_2}^{s_2,τ_2}(Ω)$, where $Ω\subset \mathbb{R}^d$ is a bounded domain, obtaining necessary and sufficient conditions for the continuity of $id_τ$. This can also be seen as the continuation of our previous studies of compactness of the embeddings in the non-limiting case. Moreover, we also construct Rychkov's linear, bounded universal extension operator for these spaces.

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arXiv2021

Nuclear embeddings in general vector-valued sequence spaces with an application to Sobolev embeddings of function spaces on quasi-bounded domains

We study nuclear embeddings for spaces of Besov and Triebel-Lizorkin type defined on quasi-bounded domains $Ω\subset {\mathbb R}^d$. The counterpart for such function spaces defined on bounded domains has been considered for a long time and the complete answer was obtained only recently. Compact embeddings for function spaces defined on quasi-bounded domains have been studied in detail already, also concerning their entropy and $s$-numbers. We now prove the first and complete nuclearity result in this context. The concept of nuclearity has been introduce by Grothendieck in 1955 already. Our second main contribution is the generalisation of the famous Tong result (1969) which characterises nuclear diagonal operators acting between sequence spaces of $\ell_r$ type, $1\leq r\leq\infty$. We can now extend this to the setting of general vector-valued sequence spaces of type $\ell_q(β_j \ell_p^{M_j})$ with $1\leq p,q\leq\infty$, $M_j\in {\mathbb N}_0$ and weight sequences with $β_j>0$. In particular, we prove a criterion for the embedding $id_β: \ell_{q_1}(β_j \ell_{p_1}^{M_j}) \hookrightarrow \ell_{q_2}(\ell_{p_2}^{M_j})$ to be nuclear.

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arXiv2020

Wavelet decomposition and embeddings of generalised Besov-Morrey spaces

We study embeddings between generalised Besov-Morrey spaces. Both sufficient and necessary conditions for the embeddings are proved. Embeddings of the Besov-Morrey spaces into the Lebesgue spaces are also considered. Our approach requires a wavelet characterisation of the spaces which we establish for the system of Daubechies wavelets.

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arXiv2020

Entropy numbers of compact embeddings of Smoothness Morrey spaces on bounded domains

We study the compact embedding between smoothness Morrey spaces on bounded domains and characterise its entropy numbers. Here we discover a new phenomenon when the difference of smoothness parameters in the source and target spaces is rather small compared with the influence of the fine parameters in the Morrey setting. In view of some partial forerunners this was not to be expected till now. Our argument relies on wavelet decomposition techniques of the function spaces and a careful study of the related sequence space setting.

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arXiv2020

On compact subsets of Sobolev spaces on manifolds

It is common that a Sobolev space defined on $\mathbb{R}^m$ has a non-compact embedding into an $L^p$-space, but it has subspaces for which this embedding becomes compact. There are three well known cases of such subspaces, the Rellich compactness, for a subspace of functions on a bounded domain (or an unbounded domain, sufficiently thin at infinity), the Strauss compactness, for a subspace of radially symmetric functions in $\mathbb{R}^m$, and the weighted Sobolev spaces. Known generalizations of Strauss compactness include subspaces of functions with block-radial symmetry, subspaces of functions with certain symmetries on Riemannian manifolds, as well as similar subspaces of more general Besov and Triebel-Lizorkin spaces. Presence of symmetries can be interpreted in terms of the rising critical Sobolev exponent corresponding to the smaller effective dimension of the quotient space.

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arXiv2020

Nuclear embeddings in weighted function spaces

We study nuclear embeddings for weighted spaces of Besov and Triebel-Lizorkin type where the weight belongs to some Muckenhoupt class and is essentially of polynomial type. Here we can extend our previous results [17,19] where we studied the compactness of corresponding embeddings. The concept of nuclearity goes back to Grothendieck who defined it in [14]. Recently there is a refreshed interest to study such questions [5-8,49]. This led us to the investigation in the weighted setting. We obtain complete characterisations for the nuclearity of the corresponding embedding. Essential tools are a discretisation in terms of wavelet bases, operator ideal techniques, as well as a very useful result of Tong [43] about the nuclearity of diagonal operators acting in $\ell_p$ spaces. In that way we can further contribute to the characterisation of nuclear embeddings on domains obtained in [5,33,34,49].

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arXiv2020

Compact embeddings in Besov-type and Triebel-Lizorkin-type Spaces on bounded domains

We study embeddings of Besov-type and Triebel-Lizorkin-type spaces, $id_τ: {B}_{p_1,q_1}^{s_1,τ_1}(Ω) \hookrightarrow {B}_{p_2,q_2}^{s_2,τ_2}(Ω)$ and $id_τ: {F}_{p_1,q_1}^{s_1,τ_1}(Ω) \hookrightarrow {F}_{p_2,q_2}^{s_2,τ_2}(Ω) $, where $Ω\subset {\mathbb R}^d$ is a bounded domain, and obtain necessary and sufficient conditions for the compactness of $id_τ$. Moreover, we characterise its entropy and approximation numbers. Surprisingly, these results are completely obtained via embeddings and the application of the corresponding results for classical Besov and Triebel-Lizorkin spaces as well as for Besov-Morrey and Triebel-Lizorkin-Morrey spaces.

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arXiv2019

Some embeddings of Morrey spaces with critical smoothness

We study embeddings of Besov-Morrey spaces ${\cal N}^{s}_{u,p,q}}({\mathbb R}^d)$ and of Triebel-Lizorkin-Morrey spaces ${\cal E}^{s}_{u,p,q}}({\mathbb R}^d)$ in the limiting cases when the smoothness $s$ equals $s_0=d\max(1/u-p/u,0)$ or $s_{\infty}=d/u$, which is related to the embeddings in $L_1^{loc}({\mathbb R}^d)$ or in $L_\infty({\mathbb R}^d)$, respectively. When $s=s_0$ we characterise the embeddings in $L_1^{loc}({\mathbb R}^d)$ and when $s=s_{\infty}$ we obtain embeddings into Orlicz-Morrey spaces of exponential type and into generalised Morrey spaces.

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arXiv2018

Some properties of block-radial functions and Schrödinger type operators with block-radial potentials

Let $ R_γB^{s}_{p,q}(\Rd)$ be a subspace of the Besov space $B^{s}_{p,q}(\Rd)$ that consists of block-radial functions. We prove that the asymptotic behaviour of the entropy numbers of compact embeddings $\id: \: R_γB^{s_1}_{p_1,q_1}(\R^d) \rightarrow R_γB^{s_2}_{p_2,q_2}(\R^d)$ depends on the number of blocks of the lowest dimension, the parameters $p_1$ and $p_2$, but is independent of the smoothness parameters $s_1$, $s_2$. We apply the asymptotic behaviour to estimation of powers of a negative spectra of Schrödinger type operators with block-radial potentials. This part essentially relies on the Birman-Schwinger principle.

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