Hardy-Littlewood inequalities and \(Q_p\)-spaces
type: Item
instance of: scholarly article
title: Hardy-Littlewood inequalities and \(Q_p\)-spaces
zbMATH Open document ID: 1096.30044
DOI: 10.4171/ZAA/1246
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2006-03-10T00:00:00Z
review text: Let \(n\) be a positive integer, let \(N = \{1, 2, \cdot\cdot\cdot, n\}\), let \(e_{1}, e_{2}, \cdot\cdot\cdot, e_{n}\) be an orthonormal basis of \(\mathbb{R}^{n}\), and let multiplication be defined such that the combination \(e_{i}e_{j} + e_{j}e_{i} = -2\) if \(i = j\) and \(0\) otherwise. The Clifford Algebra \(CL_{n}\) is the \(2^{n}\)--dimensional real associative algebra with basis given by 1 and all products of the form \(e_{h_{1}}e_{h_{2}} \cdot\cdot\cdot e_{h_{k}}, 1 \leq h_{1} < h_{2} < \cdot\cdot\cdot < h_{k} \leq n.\) Define paravectors \[ z = x_{0} + x_{1}e_{1} + \cdot\cdot\cdot + x_{n}e_{n} \;\;\;\text{and} \;\;\;\bar{z} = x_{0} - x_{1}e_{1} - \cdot\cdot\cdot - x_{n}e_{n} \;, \] where each \(x_{j}\) is a real number. Then \[ | z| ^{2} = z \bar{z} = x_{0}^{2} + x_{1}^{2} + \cdot\cdot\cdot + x_{n}^{2} \;. \] A Clifford valued function \(w(z)\) has the form \[ w(z) = \sum_{A \subset N} w_{A}(z) e_{A} \;, \] where each component function \(w_{A}(z)\) is real valued. Define \[ \partial w = \sum_{i=1}^{n} e_{i} \frac {\partial w} {\partial x_{i}} \;. \] Let \(B\) denote the unit ball in \(\mathbb{R}^{n}\), let \(S\) be the boundary of \(B\), let \(d\sigma\) denote the normalized surface measure on \(S\), and let \(dB\) denote the volume measure on \(B\). For either a real valued or Clifford valued function \(f(x), \;x \in B\) and \(0 \leq r < 1\), define \[ M_{q}(r,f) = \bigg\{ \int_{S} | f(r\xi)| ^{q} d\sigma(\xi) \;, \;\;\;x = r\xi, r = | x| , \xi \in S, \;\;\;0 < q < \infty \bigg\} , \] and \(M_{\infty}(r,f) = \sup_{\xi \in S}{| f(r\xi)| }\). Further, define \[ \| f\| _{p,q,\alpha} = \bigg\{ \int_{B} M_{q}^{p}(r,f) (1 - r^{2})^{p\alpha - 1} dr \bigg\}^{\frac {1} {p}} \;\;\;0 < p < \infty \] and \(\| f\| _{\infty,q,\alpha} = \sup_{0<r<1}{(1 - r^{2})^{\alpha} M_{q}(r,f)}.\) Further define \[ Ef = \sum_{i=1}^{n} x_{i} \frac {\partial f} {\partial x_{i}} \;\;\;\text{and} \;\;\;E_{x}f = sf + Ef, \;\;s \geq 0 \;. \] The authors prove a number of Hardy-Littlewo…
Mathematics Subject Classification ID: 30G35
Mathematics Subject Classification ID: 30D45
zbMATH DE Number: 5011361
zbMATH Keywords: Hardy-Littlewood inequalities
zbMATH Keywords: Clifford analysis
author: Guangbin Ren
author: Uwe Kähler
reviewed by: Peter Lappan
MaRDI profile type: Publication
cites work: Q4308185
full work available at URL: https://doi.org/10.4171/zaa/1246
OpenAlex ID: W2008100696
description: scientific article; zbMATH DE number 5011361