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Hardy spaces on homogeneous groups and Littlewood-Paley functions

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 Added by Shuichi Sato
 Publication date 2019
  fields
and research's language is English
 Authors Shuichi Sato




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We establish a characterization of the Hardy spaces on the homogeneous groups in terms of the Littlewood-Paley functions. The proof is based on vector-valued inequalities shown by applying the Peetre maximal function.



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We consider Littlewood-Paley functions associated with non-isotropic dilations. We prove that they can be used to characterize the parabolic Hardy spaces of Calder{o}n-Torchinsky.
78 - Shuichi Sato 2016
We consider certain Littlewood-Paley operators and prove characterization of some function spaces in terms of those operators. When treating weighted Lebesgue spaces, a generalization to weighted spaces will be made for Hormanders theorem on the invertibility of homogeneous Fourier multipliers. Also, applications to the theory of Sobolev spaces will be given.
176 - Guozhen Lu , Qiaohua Yang 2019
Using the Fourier analysis techniques on hyperbolic spaces and Greens function estimates, we confirm in this paper the conjecture given by the same authors in [43]. Namely, we prove that the sharp constant in the $frac{n-1}{2}$-th order Hardy-Sobolev-Mazya inequality in the upper half space of dimension $n$ coincides with the best $frac{n-1}{2}$-th order Sobolev constant when $n$ is odd and $ngeq9$ (See Theorem 1.6). We will also establish a lower bound of the coefficient of the Hardy term for the $k-$th order Hardy-Sobolev-Mazya inequality in upper half space in the remaining cases of dimension $n$ and $k$-th order derivatives (see Theorem 1.7). Precise expressions and optimal bounds for Greens functions of the operator $ -Delta_{mathbb{H}}-frac{(n-1)^{2}}{4}$ on the hyperbolic space $mathbb{B}^n$ and operators of the product form are given, where $frac{(n-1)^{2}}{4}$ is the spectral gap for the Laplacian $-Delta_{mathbb{H}}$ on $mathbb{B}^n$. Finally, we give the precise expression and optimal pointwise bound of Greens function of the Paneitz and GJMS operators on hyperbolic space, which are of their independent interest (see Theorem 1.10).
93 - Quanhua Xu 2021
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We establish a connection between the function space BMO and the theory of quasisymmetric mappings on emph{spaces of homogeneous type} $widetilde{X} :=(X,rho,mu)$. The connection is that the logarithm of the generalised Jacobian of an $eta$-quasisymmetric mapping $f: widetilde{X} rightarrow widetilde{X}$ is always in $rm{BMO}(widetilde{X})$. In the course of proving this result, we first show that on $widetilde{X}$, the logarithm of a reverse-H{o}lder weight $w$ is in $rm{BMO}(widetilde{X})$, and that the above-mentioned connection holds on metric measure spaces $widehat{X} :=(X,d,mu)$. Furthermore, we construct a large class of spaces $(X,rho,mu)$ to which our results apply. Among the key ingredients of the proofs are suitable generalisations to $(X,rho,mu)$ from the Euclidean or metric measure space settings of the Calder{o}n--Zygmund decomposition, the Vitali Covering Theorem, the Radon--Nikodym Theorem, a lemma which controls the distortion of sets under an $eta$-quasisymmetric mapping, and a result of Heinonen and Koskela which shows that the volume derivative of an $eta$-quasisymmetric mapping is a reverse-H{o}lder weight.
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