Multiple conjugate functions and multiplicative Lipschitz classes

Ferenc Móricz¹

¹ Bolyai Institute University of Szeged Aradi vértanúk tere 1 6720 Szeged, Hungary

Colloquium Mathematicum, Tome 115 (2009) no. 1, pp. 21-32

Résumé

We extend the classical theorems of I. I. Privalov and A. Zygmund from single to multiple conjugate functions in terms of the multiplicative modulus of continuity. A remarkable corollary is that if a function $f$ belongs to the multiplicative Lipschitz class $\mathop{\rm Lip}(\alpha_1, \ldots, \alpha_N)$ for some $0\alpha_1, \ldots, \alpha_N1$ and its marginal functions satisfy $f(\cdot, x_2, \ldots, x_N) \in \mathop{\rm Lip} \beta_1, \ldots, f(x_1, \ldots, x_{N-1}, \cdot)\in \mathop{\rm Lip} \beta_N$ for some $0\beta_1, \ldots, \beta_N 1$ uniformly in the indicated variables $x_{l}$, $1\le l \le N$, then $\widetilde f^{(\eta_1, \ldots, \eta_N)} \in \mathop{\rm Lip} (\alpha_1, \ldots, \alpha_N)$ for each choice of $(\eta_1, \ldots, \eta_N)$ with $\eta_{l} = 0$ or $1$ for $1\le l \le N$.

Détail

DOI : 10.4064/cm115-1-3

Keywords: extend classical theorems privalov zygmund single multiple conjugate functions terms multiplicative modulus continuity remarkable corollary function belongs multiplicative lipschitz class mathop lip alpha ldots alpha alpha ldots alpha its marginal functions satisfy cdot ldots mathop lip beta ldots ldots n cdot mathop lip beta beta ldots beta uniformly indicated variables widetilde eta ldots eta mathop lip alpha ldots alpha each choice eta ldots eta eta