Papers by Leszek Skrzypczak
Journal of Approximation Theory, Apr 1, 2011
We investigate compactness and asymptotic behaviour of the entropy numbers of embeddings B s 1 ,s... more We investigate compactness and asymptotic behaviour of the entropy numbers of embeddings B s 1 ,s ′ 1 p 1 ,q 1 (R n , U) ↩→ B s 2 ,s ′ 2 p 2 ,q 2 (R n , U). Here B s,s′ p,q (R n , U) denotes a 2-microlocal Besov space with a weight given by the distance to a fixed set U ⊂ R n. c
Journal of Approximation Theory, 2020
We study the compact embedding between smoothness Morrey spaces on bounded domains and characteri... more 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.
Institute of Mathematics, Polish Academy of Sciences eBooks, 2014
arXiv (Cornell University), Feb 13, 2019
We study the compact embedding between smoothness Morrey spaces on bounded domains and characteri... more 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.
Journal of Mathematical Analysis and Applications, Nov 1, 2010
This paper is based on our previous results (Haroske and Skrzypczak (2008) [23], Haroske and Skrz... more This paper is based on our previous results (Haroske and Skrzypczak (2008) [23], Haroske and Skrzypczak (in press) [25]) on compact embeddings of Muckenhoupt weighted function spaces of Besov and Triebel-Lizorkin type with example weights of polynomial growth near infinity and near some local singularity. Our approach also extends (Haroske and Triebel (1994) [21]) in various ways. We obtain eigenvalue estimates of degenerate pseudodifferential operators of type b 2 • p(x, D) • b 1 where b i ∈ L r i (R n , w i), w i ∈ A ∞ , i = 1, 2, and p(x, D) ∈ Ψ − 1,0 , > 0. Finally we deal with the 'negative spectrum' of some operator H γ = A − γ V for γ → ∞, where the potential V may have singularities (in terms of Muckenhoupt weights), and A is a positive elliptic pseudodifferential operator of order > 0, self-adjoint in L 2 (R n). This part essentially relies on the Birman-Schwinger principle. We conclude this paper with a number of examples, also comparing our results with preceding ones.
Journal of Function Spaces and Applications, 2011
We study compact embeddings for weighted spaces of Besov and Triebel-Lizorkin type where the weig... more We study compact embeddings for weighted spaces of Besov and Triebel-Lizorkin type where the weight belongs to some Muckenhoupt Ap class. This extends our previous results [25] to more general weights of logarithmically disturbed polynomial growth, both near some singular point and at infinity. We obtain sharp asymptotic estimates for the entropy numbers of this embedding. Essential tools are a discretisation in terms of wavelet bases, as well as a refined study of associated embeddings in sequence spaces and interpolation arguments in endpoint situations.
arXiv (Cornell University), Jul 3, 2018
Morrey (function) spaces and, in particular, smoothness spaces of Besov-Morrey or Triebel-Lizorki... more Morrey (function) spaces and, in particular, smoothness spaces of Besov-Morrey or Triebel-Lizorkin-Morrey type enjoyed a lot of interest recently. Here we turn our attention to Morrey sequence spaces mu,p = mu,p(Z d), 0 < p ≤ u < ∞, which have yet been considered almost nowhere. They are defined as natural generalisations of the classical p spaces. We consider some basic features, embedding properties, the pre-dual, a corresponding version of Pitt's compactness theorem, and can further characterise the compactness of embeddings of related finite-dimensional spaces.
Revista Matematica Complutense, Jul 16, 2020
We study embeddings of Besov-type and Triebel-Lizorkin-type spaces, id τ : B s 1 ,τ 1 p 1 ,q 1 (Ω... more We study embeddings of Besov-type and Triebel-Lizorkin-type spaces, id τ : B s 1 ,τ 1 p 1 ,q 1 (Ω) → B s 2 ,τ 2 p 2 ,q 2 (Ω) and id τ : F s 1 ,τ 1 p 1 ,q 1 (Ω) → F s 2 ,τ 2 p 2 ,q 2 (Ω), where Ω ⊂ R d is a bounded domain, and obtain necessary and sufficient conditions for the compactness of id τ. Moreover, we characterize 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.
Science China-mathematics, Sep 7, 2017
We study unboundedness of smoothness Morrey spaces on bounded domains Ω ⊂ R n in terms of growth ... more We study unboundedness of smoothness Morrey spaces on bounded domains Ω ⊂ R n in terms of growth envelopes. It turns out that in this situation the growth envelope function is finite-in contrast to the results obtained by Haroske et al. (2016) for corresponding spaces defined on R n. A similar effect was already observed by Haroske et al. (2017), where classical Morrey spaces Mu,p(Ω) were investigated. We deal with all cases where the concept is reasonable and also include the tricky limiting cases. Our results can be reformulated in terms of optimal embeddings into the scale of Lorentz spaces Lp,q(Ω).
Communications in Contemporary Mathematics, Feb 1, 2012
We deal with decay and boundedness properties of radial functions belonging to Besov and Lizorkin... more We deal with decay and boundedness properties of radial functions belonging to Besov and Lizorkin-Triebel spaces. In detail we investigate the surprising interplay of regularity and decay. Our tools are atomic decompositions in combination with trace theorems.
Journal of Approximation Theory, Apr 1, 2015
In this paper, the authors prove some Franke-Jawerth embedding for the Besovtype spaces B s,τ p,q... more In this paper, the authors prove some Franke-Jawerth embedding for the Besovtype spaces B s,τ p,q (R n) and the Triebel-Lizorkin-type spaces F s,τ p,q (R n). By using some limiting embedding properties of these spaces and the Besov-Morrey spaces N s u,p,q (R n), the continuity envelopes in B s,τ p,q (R n), F s,τ p,q (R n) and N s u,p,q (R n) are also worked out. As applications, the authors present some Hardy type inequalities in the scales of B s,τ p,q (R n), F s,τ p,q (R n) and N s u,p,q (R n), and also give the estimates for approximation numbers of the embeddings from B s,τ p,q (Ω), F s,τ p,q (Ω) and N s u,p,q (Ω) into C(Ω), where Ω denotes the unit ball in R n .
Journal of Fourier Analysis and Applications, Nov 24, 2011
Journal of Approximation Theory, Apr 1, 2003
Let RB s p;q ðR n Þ be the radial subspace of the Besov space B s p;q ðR n Þ: We prove the indepe... more Let RB s p;q ðR n Þ be the radial subspace of the Besov space B s p;q ðR n Þ: We prove the independence of the asymptotic behavior of the entropy numbers Ànð 1 p0 À 1 p1 Þ : This is in a certain contrast to earlier results on entropy numbers in the context of Besov spaces B s p;q ðOÞ on bounded domains O:
arXiv (Cornell University), Apr 21, 2018
Defect of compactness, relative to an embedding of two Banach spaces E ֒→ F , is the difference b... more Defect of compactness, relative to an embedding of two Banach spaces E ֒→ F , is the difference between a weakly convergent sequence in E and its weak limit, taken up to a remainder that vanishes in the norm of F. For a number of known embeddings, Sobolev embeddings in particular, defect of compactness takes form of a profile decomposition-a sum of clearly structured terms with asymptotically disjoint supports, called elementary concentrations. In this paper we construct a profile decomposition for the Sobolev space H 1,2 (M) of a Riemannian manifold with bounded geometry, in the form of a sum of elementary concentrations associated with concentration profiles defined on manifolds induced by a limiting procedure at infinity, and thus different from M. The profiles satisfy an inequality of Plancherel type: the sum of the quadratic forms of Laplace-Beltrami operators for the profiles on their respective manifolds is bounded by the quadratic form of the Laplace-Beltrami operator of the sequence. A similar relation, related to the Brezis-Lieb Lemma, holds for the L p-norms of profiles on the respective manifolds.
arXiv (Cornell University), Feb 8, 2020
We study nuclear embeddings for weighted spaces of Besov and Triebel-Lizorkin type where the weig... more 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 p spaces. In that way we can further contribute to the characterisation of nuclear embeddings on domains obtained in [5, 33, 34, 49].
Proceedings of the Edinburgh Mathematical Society, May 30, 2006
We investigate the asymptotic behaviour of the entropy numbers of the compact embedding B s 1 p 1... more We investigate the asymptotic behaviour of the entropy numbers of the compact embedding B s 1 p 1 ,q 1 (R d , w 1) → B s 2 p 2 ,q 2 (R d , w 2). Here B s p,q (R d , w) denotes a weighted Besov space. We present a general approach which allows us to work with a large class of weights.
arXiv (Cornell University), Apr 28, 2023
We study unboundedness properties of functions belonging to generalised Morrey spaces Mϕ,p(R d) a... more We study unboundedness properties of functions belonging to generalised Morrey spaces Mϕ,p(R d) and generalised Besov-Morrey spaces N s ϕ,p,q (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 Mu,p(R d), i.e., boundedness, the Lp-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 → 0 + and t → ∞, respectively. For the generalised Besov-Morrey spaces the limit of t −d/p ϕ(t) as t → 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 N s ϕ,p,q (R d) consists of essentially bounded functions only, and when it contains regular distributions only.
arXiv (Cornell University), Sep 24, 2021
In this paper, we study limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces, idτ :... more In this paper, we study limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces, idτ : B s 1 ,τ 1 p 1 ,q 1 (Ω) ֒→ B s 2 ,τ 2 p 2 ,q 2 (Ω) and idτ : F s 1 ,τ 1 p 1 ,q 1 (Ω) ֒→ F s 2 ,τ 2 p 2 ,q 2 (Ω), where Ω ⊂ R d is a bounded domain, obtaining necessary and sufficient conditions for the continuity of idτ. 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.
arXiv (Cornell University), Jan 13, 2016
The paper provides weighted Sobolev inequalities of the Caffarelli-Kohn-Nirenberg type for functi... more The paper provides weighted Sobolev inequalities of the Caffarelli-Kohn-Nirenberg type for functions with multi-radial symmetry. An elementary example of such inequality is the following inequality of Hardy type for functions u = u(r 1 (x), r 2 (x)), where r 1 (x) = x 2 1 + x 2 2 and r 2 (x) = x 2 3 + x 2 4 from the subspaceḢ 1,2 (2,2) (R 4) of the Sobolev spaceḢ 1,2 (R 4), radially symmetric in variables (x 1 , x 2) and in variables (x 3 , x 4): R 4 u 2 r 1 (x)r 2 (x) dx ≤ C R 4 |∇u| 2 dx, Similarly to the previously studied radial case, the range of parameters in CKN inequalities can be extended, sometimes to infinity, providing a pointwise estimate similar to the radial estimate in [25]. Furthermore, the "multi-radial" weights are a stronger singularity than radial weights of the same homogeneity, e.g. 1 r1(x)r2(x) ≥ 1 2|x| 2 .
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Papers by Leszek Skrzypczak