부산대학교 도서관

Let $\Psi$ be a convex function, and let $f$ be a real-valued function on [0, 1]. Let a modulus of continuity associated to $\Psi$ be given as $$Q_\Psi (\delta,f) = \inf\Bigg\{\lambda: \frac{1}{\delta}\iint_{|x - y| \leqslant\delta}\Psi\Big(\frac{|f(x) - f(y)|}{\lambda}\Big) dx dy \leqslant \Psi(1)\Bigg\}$$. It is shown that $\int^1_0Q_\Psi(\delta, f) d (- \Psi^{-1} (c/\delta)) < \infty$ guarantees the essential continuity of $f$, and, in fact, a uniform Lipschitz estimate is given. In the case that $\Psi(u) = \exp u^2$ the above condition reduces to $$\int^1_0 Q_{\exp u^2} (\delta, f) \frac{d\delta}{\delta\sqrt{\log(c/\delta)}} < \infty$$. This exponential square condition is satisfied almost surely by the random Fourier series $f_t(x) = \sum^\infty_{n = 1}a_nR_n(t)e^{inx}$, where ${R_n}$ is the Rademacher system, as long as $$\int^1_0\sqrt{a^2_n\sin^2(n\delta/2)} \frac{d\delta}{\delta\sqrt{\log(1/\delta)}} < \infty$$. Hence, the random essential continuity of $f_t(x)$ is guaranteed by each of the above conditions.