1,370 research outputs found
Recent developments in the theory of function spaces with dominating mixed smoothness
summary:The aim of these lectures is to present a survey of some results on spaces of functions with dominating mixed smoothness. These results concern joint work with Winfried Sickel and Miroslav Krbec as well as the work which has been done by Jan VybÃral within his thesis. The first goal is to discuss the Fourier-analytical approach, equivalent characterizations with the help of derivatives and differences, local means, atomic and wavelet decompositions. Secondly, on this basis we study approximation with respect to hyperbolic crosses, embeddings and traces. We follow [42], [43], [44], [59], [63], [64], [70], and [94], [95], [96]. Partial results can be found also in [6], [7], [8], [37] and [48]
The quest for the ultimate anisotropic Banach space
We present a new scale (with and ) of
anisotropic Banach spaces, defined via Paley-Littlewood, on which the transfer
operator associated to a hyperbolic dynamical system has good spectral
properties. When and is an integer, the spaces are analogous to the
"geometric" spaces considered by Gou\"ezel and Liverani. When and
, the spaces are somewhat analogous to the geometric
spaces considered by Demers and Liverani. In addition, just like for the
"microlocal" spaces defined by Baladi-Tsujii, the spaces are
amenable to the kneading approach of Milnor-Thurson to study dynamical
determinants and zeta functions.
In v2, following referees' reports, typos have been corrected (in particular
(39) and (43)). Section 4 now includes a formal statement (Theorem 4.1) about
the essential spectral radius if (its proof includes the content of
Section 4.2 from v1). The Lasota-Yorke Lemma 4.2 (Lemma 4.1 in v1) includes the
claim that is compact.
Version v3 contains an additional text "Corrections and complements" showing
that s> t-(r-1) is needed in Section 4.Comment: 31 pages, revised version following referees' reports, with
Corrections and complement
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