# Measure density and embeddings of Hajłasz-Besov and Hajłasz-Triebel-Lizorkin spaces

@article{Karak2019MeasureDA,
title={Measure density and embeddings of Hajłasz-Besov and Hajłasz-Triebel-Lizorkin spaces},
author={Nijjwal Karak},
journal={Journal of Mathematical Analysis and Applications},
year={2019}
}
• Nijjwal Karak
• Published 1 March 2018
• Mathematics
• Journal of Mathematical Analysis and Applications
In this paper, we investigate the relation between Sobolev-type embeddings of Haj{\l}asz-Besov spaces (and also Haj{\l}asz-Triebel-Lizorkin spaces) defined on a metric measure space $(X,d,\mu)$ and lower bound for the measure $\mu.$ We prove that if the measure $\mu$ satisfies $\mu(B(x,r))\geq cr^Q$ for some $Q>0$ and for any ball $B(x,r)\subset X,$ then the Sobolev-type embeddings hold on balls for both these spaces. On the other hand, if the Sobolev-type embeddings hold in a domain $\Omega… 7 Citations Duality and distance formulas in Lipschitz–Hölder spaces • Mathematics • 2019 For a compact metric space$(K, \rho)$, the predual of$Lip(K, \rho)$can be identified with the normed space$M(K)$of finite (signed) Borel measures on$K$equipped with the Kantorovich-Rubinstein Sobolev embedding for M1, spaces is equivalent to a lower bound of the measure • Mathematics • 2019 It has been known since 1996 that a lower bound for the measure,$\mu(B(x,r))\geq br^s$, implies Sobolev embedding theorems for Sobolev spaces$M^{1,p}$defined on metric-measure spaces. We prove Lower bound of measure and embeddings of Sobolev, Besov and Triebel–Lizorkin spaces In this article, we study the relation between Sobolev-type embeddings for Sobolev spaces or Besov spaces or Triebel-Lizorkin spaces defined either on a doubling or on a geodesic metric measure space Characterization of trace spaces on regular trees via dyadic norms In this paper, we study the traces of Orlicz-Sobolev spaces on a regular rooted tree. After giving a dyadic decomposition of the boundary of the regular tree, we present a characterization on the A necessary condition on domains for optimal Orlicz-Sobolev embeddings We provide a necessary condition on the regularity of domains for the optimal embeddings of first order (and higher order) Orlicz-Sobolev spaces into Orlicz spaces in the sense of \cite{Cia96} (and Besov type function spaces defined on metric-measure spaces Abstract The purpose of this article is to study the Besov type function spaces for maps which are defined on abstract metric-measure spaces. We extend some of the embedding theorems of the classical ## References SHOWING 1-10 OF 29 REFERENCES Pointwise Characterizations of Besov and Triebel-Lizorkin Spaces and Quasiconformal Mappings • Mathematics • 2010 In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces$\dot B^s_{p,\,q}$and Triebel-Lizorkin spaces$\dot F^s_{p,\,q}$for all$s\in(0,\,1)$and Measure Density and Extension of Besov and Triebel–Lizorkin Functions • Mathematics • 2014 We show that a domain is an extension domain for a Hajłasz–Besov or for a Hajłasz–Triebel–Lizorkin space if and only if it satisfies a measure density condition. We use a modification of the Whitney Besov and Triebel–Lizorkin spaces on metric spaces: Embeddings and pointwise multipliers Abstract In this paper, we obtain the Franke–Jawerth embedding property of Hajlasz–Besov and Hajlasz–Triebel–Lizorkin spaces on a measure metric space ( X , d , μ ) which is Ahlfors regular with Interpolation properties of Besov spaces defined on metric spaces • Mathematics • 2010 Let X = (X, d, μ)be a doubling metric measure space. For 0 < α < 1, 1 ≤p, q < ∞, we define semi-norms When q = ∞ the usual change from integral to supremum is made in the definition. The In Metric-measure Spaces Sobolev Embedding is Equivalent to a Lower Bound for the Measure We study Sobolev inequalities on doubling metric measure spaces. We investigate the relation between Sobolev embeddings and lower bound for measure. In particular, we prove that if the Sobolev Sobolev embeddings, extensions and measure density condition • Mathematics • 2008 There are two main results in the paper. In the first one, Theorem 1, we prove that if the Sobolev embedding theorem holds in Ω, in any of all the possible cases, then Ω satisfies the measure density Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces • Mathematics • 2001 Our main result is that, when$f$is smooth and has bounded derivatives, and when$u$belongs to the spaces$W^{s,p}$and$W^{1,sp}$, the map$f(u)$is in$W^{s,p}\$.
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