Embeddings of Besov spaces on fractal $h$-sets

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.

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“…If 1 < q < ∞, then it follows from the Landau theorem, that remains valid in the quasi-Banach setting, cf. [4], that there exists a sequence…”

Section: ) Thus It Follows Frommentioning

confidence: 99%

Unboundedness properties of smoothness Morrey spaces of regular distributions on domains

et al. 2017

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“…1203-1224 , DOI: 10.1515/fca-2019-0064 such spaces. We refer, for instance, to the books [15], [26] and [28] and papers [3], [4], [5], [6], [9], [12], [16], [20], [22] and references therein. In this paper we consider weighted generalized Morrey spaces on underlying quasi-metric measure sets.…”

Section: Introductionmentioning

confidence: 99%

Embeddings of Weighted Generalized Morrey Spaces Into Lebesgue Spaces on Fractal Sets

Samko

2019

FCAA

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We study embeddings of weighted local and consequently global generalized Morrey spaces defined on a quasi-metric measure set (X, d, μ) of general nature which may be unbounded, into Lebesgue spaces Ls(X), 1 ≤ s ≤ p < ∞. The main motivation for obtaining such an embedding is to have an embedding of non-separable Morrey space into a separable space. In the general setting of quasi-metric measure spaces and arbitrary weights we give a sufficient condition for such an embedding. In the case of radial weights related to the center of local Morrey space, we obtain an effective sufficient condition in terms of (fractional in general) upper Ahlfors dimensions of the set X. In the case of radial weights we also obtain necessary conditions for such embeddings of local and global Morrey spaces, with the use of (fractional in general) lower and upper Ahlfors dimensions. In the case of power-logarithmic-type weights we obtain a criterion for such embeddings when these dimensions coincide.

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Besov Spaces, Triebel–Lizorkin Spaces and Modulation Spaces

Sawano

2018

Theory of Besov Spaces

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Embeddings of Besov spaces on fractal $h$-sets

Cited by 5 publications

References 30 publications

Unboundedness properties of smoothness Morrey spaces of regular distributions on domains

Unboundedness properties of smoothness Morrey spaces of regular distributions on domains

Embeddings of Weighted Generalized Morrey Spaces Into Lebesgue Spaces on Fractal Sets

Besov Spaces, Triebel–Lizorkin Spaces and Modulation Spaces

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