Moduli of continuity for exponential Lipschitz classes

Abstract:

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 \[ {Q_\Psi }(\delta ,f) = \inf \left \{ {\lambda :\frac {1}{\delta }\iint \limits _{|x - y| \leqslant \delta } {\Psi \left ( {\frac {{|f(x) - f(y)|}}{\lambda }} \right )}\;dx\;dy\; \leqslant \Psi (1)} \right \}.\] It is shown that $\smallint _0^1{Q_\Psi }(\delta ,f)\;d\;( - {\Psi ^{ - 1}}(c/\delta )) < \infty$ 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 \[ \int _0^1 {{Q_{\exp \;{u^2}}}\;(\delta ,f)\frac {{d\delta }}{{\delta \sqrt {\log (c/\delta )} }}\; < \infty .} \] This exponential square condition is satisfied almost surely by the random Fourier series ${f_t}(x) = \Sigma _{n = 1}^\infty {a_n}{R_n}(t){e^{inx}}$, where $\{ {R_n}\}$ is the Rademacher system, as long as \[ \int _0^1 {\sqrt {a_n^2{{\sin }^2}(n\delta /2)} \frac {{d\delta }}{{\delta \sqrt {\log (1/\delta )} }}\; < \infty .} \] Hence, the random essential continuity of ${f_t}(x)$ is guaranteed by each of the above conditions.