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Register a new userThe Haar System in Besov-type Spaces
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Some Besov-type spaces $B^{s,tau}_{p,q}(mathbb{R}^n)$ can be characterized in terms of the behavior of the Fourier--Haar coefficients. In this article, the authors discuss some necessary restrictions for the parameters $s$, $tau$, $p$, $q$ and $n$ of this characterization. Therefore, the authors measure the regularity of the characteristic function $mathcal X$ of the unit cube in $mathbb{R}^n$ via the Besov-type spaces $B^{s,tau}_{p,q}(mathbb{R}^n)$. Furthermore, the authors study necessary and sufficient conditions such that the operation $langle f, mathcal{X} rangle$ generates a continuous linear functional on $B^{s,tau}_{p,q}(mathbb{R}^n)$.
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We characterize the Schauder and unconditional basis properties for the Haar system in the Triebel-Lizorkin spaces $F^s_{p,q}(Bbb R^d)$, at the endpoint cases $s=1$, $s=d/p-d$ and $p=infty$. Together with the earlier results in [10], [4], this completes the picture for such properties in the Triebel-Lizorkin scale, and complements a similar study for the Besov spaces given in [5].
We use the method of atomic decomposition to build new families of function spaces, similar to Besov spaces, in measure spaces with grids, a very mild assumption. Besov spaces with low regularity are considered in measure spaces with good grids, and results on multipliers and left compositions are obtained.
In a previous work we introduced Besov spaces $mathcal{B}^s_{p,q}$ defined on a measure spaces with a good grid, with $pin [1,infty)$, $qin [1,infty]$ and $0< s< 1/p$. Here we show that classical Besov spaces on compact homogeneous spaces are examples of such Besov spaces. On the other hand we show that even Besov spaces defined by a good grid made of partitions by intervals may differ from a classical Besov space, giving birth to exotic Besov spaces.
In this article, the authors introduce Besov-type spaces with variable smoothness and integrability. The authors then establish their characterizations, respectively, in terms of $varphi$-transforms in the sense of Frazier and Jawerth, smooth atoms or Peetre maximal functions, as well as a Sobolev-type embedding. As an application of their atomic characterization, the authors obtain a trace theorem of these variable Besov-type spaces.
We study a convergence result of Bourgain--Brezis--Mironescu (BBM) using Triebel-Lizorkin spaces. It is well known that as spaces $W^{s,p} = F^{s}_{p,p}$, and $H^{1,p} = F^{1}_{p,2}$. When $sto 1$, the $F^{s}_{p,p}$ norm becomes the $F^{1}_{p,p}$ norm but BBM showed that the $W^{s,p}$ norm becomes the $H^{1,p} = F^{1}_{p,2}$ norm. Naively, for $p eq 2$ this seems like a contradiction, but we resolve this by providing embeddings of $W^{s,p}$ into $F^{s}_{p,q}$ for $q in {p,2}$ with sharp constants with respect to $s in (0,1)$. As a consequence we obtain an $mathbb{R}^N$-version of the BBM-result, and obtain several more embedding and convergence theorems of BBM-type that to the best of our knowledge are unknown.
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