António M. Caetano

Continuity envelopes of spaces of generalised smoothness: a limiting case; embeddings and approximation numbers

Continuity envelopes for the spaces of generalised smoothness Bpq(s,Ψ)(ℝn) and Fpq(s,Ψ)(ℝn) are studied in the so-called supercritical s=1

Weyl Numbers in Function Spaces II

Using a method introduced by Edmunds and Triebel, we obtain in a unified way old and new estimates for the Weyl numbers of embeddings between function spaces belonging to the scales of spaces Bptq and F£q. 1980 Mathematics Subject Classification (1985 Revision): 46E35, 46F99, 47A67, 47B05, 47B38.

On 2-microlocal spaces with all exponents variable

Abstract In this paper we study various key properties for 2-microlocal Besov and Triebel–Lizorkin spaces with all exponents variable, including the lifting property, embeddings and Fourier multipliers. We also clarify and improve some statements recently published.

Atomic and molecular decompositions in variable exponent 2-microlocal spaces and applications ☆

Abstract In this article we study atomic and molecular decompositions in 2-microlocal Besov and Triebel–Lizorkin spaces with variable integrability. We show that, in most cases, the convergence implied in such decompositions holds not only in the distributions sense, but also in the function spaces themselves. As an application, we give a simple proof for the denseness of the Schwartz class in such spaces. Some other properties, like Sobolev embeddings, are also obtained via atomic representations.

Weyl numbers in sequence spaces and sections of unit balls

Using techniques of a geometrical nature, the following result is proved: denoting by Iqp(M): lpM → lqM the natural embedding and by xk the kth Weyl number, if 0 < p ⩽ q ⩽ 2 there are positive constants c1 and c2 such that, for all k, MϵN with k ⩽ M2, c1k1q − 1p ⩽ xk(Iqp(M)) ⩽ c2k1q − 1p, the upper estimate being even valid for all k ϵ {1,…, M}. As a consequence of the approach used, some results about sections of unit balls are also derived, namely VolH(H ∩ BpM) ⩽ Volk(Bpk) for 0 < p ⩽ 2, where BpM, Bpk are the closed unit balls centred at zero of the spaces lpM and lpk, respectively, H is a k-dimensional subspace of lpM, and Volk, VolH denote Lebesgue measures in Rk and H, respectively.

Embeddings and the growth envelope of Besov spaces involving only slowly varying smoothness

Abstract We characterize local embeddings of Besov spaces B p , r 0 , b involving only a slowly varying smoothness b into classical Lorentz spaces. These results are applied to establish sharp local embeddings of the Besov spaces in question into Lorentz–Karamata spaces. As a consequence of these results, we are able to determine growth envelopes of spaces B p , r 0 , b and to show that we cannot describe all local embeddings of Besov spaces B p , r 0 , b into Lorentz–Karamata spaces in terms of growth envelopes.

A homogeneity property for Besov spaces

A homogeneity property for some Besov spaces Bp,qs is proved. An analogous property for some Fp,qs spaces is already known.

Sharp Estimates of Approximation Numbers via Growth Envelopes

We give sharp asymptotic estimates for the approximation numbers of the compact embedding id: \(L_p (\log L)_a (\Omega ) \to B_{\infty ,\infty }^{ - 1} (\Omega ),\;a > 0,\;n < p < \infty \) applying the newly developed tool of (growth) envelopes for function spaces.

Traces of Besov spaces on fractal

We study the existence of traces of Besov spaces on fractal

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