On various approaches to Besov-type spaces of variable smoothness

Abstract

The paper is concerned with Besov spaces of variable smoothness B ϕ 0 p,q (R n , {t k }), in which the norms are defined in terms of convolutions with smooth functions. A relation is found between the spaces B ϕ 0 p,q (R n , {t k }) and the spaces B l p,q,r (R n , {t k }), which were introduced earlier by the author.

Besov-type spaces of variable smoothness on rough domains

“…The Besov spaces of [34] extend (in the case of constant exponents p, q) both the wellknown 2-microlocal Besov spaces, which were first studied by H. Kempka in [19], and the weighted Besov spaces from [27]. Besides, a more subtle method was introduced in [33], [34] for the analysis of a weight sequence {t k } (which specifies the variable smoothness). This methods was found to be instrumental in refining many available theorems on Besov spaces of variable smoothness by relaxing unnatural pointwise constraints on a variable smoothness {t k }.…”

“…Besov spaces of variable smoothness (and various extensions thereof) have been intensively studied in the last 30 years. Among the fundamental works we mention [1], [21], [18], [17], [19], [20], [10], [11], [12], [13], [14], [3], [4], [33], [34], [22], [15], and [29]. We also note the recent paper [35], which introduces a new Besov-type space, which has not only variables smoothness, but also in a sense a 'variable structure'.…”

“…Recently, the author [33] proposed a new nonlinear approximation approach to the study of Besov spaces of variable smoothness. In [34] a new approach involving convolutions with smooth kernels (instead of the Fourier transform) was proposed to determine the norm in the corresponding space. The Besov spaces of [34] extend (in the case of constant exponents p, q) both the wellknown 2-microlocal Besov spaces, which were first studied by H. Kempka in [19], and the weighted Besov spaces from [27].…”

“…In [34] a new approach involving convolutions with smooth kernels (instead of the Fourier transform) was proposed to determine the norm in the corresponding space. The Besov spaces of [34] extend (in the case of constant exponents p, q) both the wellknown 2-microlocal Besov spaces, which were first studied by H. Kempka in [19], and the weighted Besov spaces from [27]. Besides, a more subtle method was introduced in [33], [34] for the analysis of a weight sequence {t k } (which specifies the variable smoothness).…”

“…The purpose of the present paper is to provide an exact description of the traces of Besov spaces of variable smoothness on domains with nonsmooth boundary. For the spaces B ϕ0 p,q (R n , {t k }), which were first introduced in [34], we put forward a description of the traces on bounded Lipschitz domains and on special Lipschitz domains (epigraphs of Lipschitz functions). An exact description of the traces on (ε, δ)-domains will be given for the spaces B l p,q,r (R n , {t k }), which were introduced by the author in [33].…”