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Abstract

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 $0<\alpha_1, \ldots, \alpha_N<1$ 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 $1\le l \le N$.