Hardy-Littlewood inequalities and \(Q_p\)-spaces
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-Littlewood type inequalities dealing with the quantities mentioned above, and with some other quantities. A basic result shows that for \(0 < p, q \leq \infty, 0 < \alpha < \infty, \beta > 0\) and f is either a real--valued or Clifford--valued harmonic function, then \(\| f\| _{p,q,\alpha} \simeq | f(0)| + \| E_{s}^{\beta} f \| _{p,q,\alpha+k}\), where \(A \simeq B\) means that there exists a constant \(C > 0\) such that \(CA \leq B \leq A/C\). Using these results, the authors suggest a definition for \(Q_{p}\)--spaces in Clifford analysis by using the following \(Q_{p}\) norm: \[ \| f\| _{Q_{p}} = \sup_{a \in B} {\int_{B} \sum_{k=1}^{n} \bigg| \frac {\partial f} {\partial x_{k}}\bigg| ^{2} \big(1 - | \phi_{a}(x)| ^{2}\big)^{p} dB \;, f \in \text{ker} \;\partial}, \] where \(\phi_{a}(x) = (a - x) (1 - \bar{a}x)^{-1}, \;| a| < 1\), is a mapping of the unit disc of \(\mathbb{R}^{n}\) into itself. It is shown that other equivalent definitions are possible.
scientific article; zbMATH DE number 5011361