Moduli of continuity for exponential Lipschitz classes

Abstract

Let Ψ \Psi be a convex function, and let f be a real-valued function on [0, 1]. Let a modulus of continuity associated to Ψ \Psi be given as \[ Q Ψ ( δ , f ) = inf { λ : 1 δ ∬ | x − y | ⩽ δ Ψ ( | f ( x ) − f ( y ) | λ ) d x d y ⩽ Ψ ( 1 ) } . {Q_\Psi }(\delta ,f) = \inf \left \{ {\lambda :\frac {1}{\delta }\iint \limits _{|x - y| \leqslant \delta } {\Psi \left ( {\frac {{|f(x) - f(y)|}}{\lambda }} \right )}\;dx\;dy\; \leqslant \Psi (1)} \right \}. \] It is shown that ∫ 0 1 Q Ψ ( δ , f ) d ( − Ψ − 1 ( c / δ ) ) > ∞ \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 Ψ ( u ) = exp u 2 \Psi (u) = \exp \;{u^2} the above condition reduces to \[ ∫ 0 1 Q exp u 2 ( δ , f ) d δ δ log ⁡ ( c / δ ) > ∞ . \int _0^1 {{Q_{\exp \;{u^2}}}\;(\delta ,f)\frac {{d\delta }}{{\delta \sqrt {\log (c/\delta )} }}\; > \infty .} \] This exponential square condition is satisfied almost surely by the random Fourier series f t ( x ) = Σ n = 1 ∞ a n R n ( t ) e i n x {f_t}(x) = \Sigma _{n = 1}^\infty {a_n}{R_n}(t){e^{inx}} , where { R n } \{ {R_n}\} is the Rademacher system, as long as \[ ∫ 0 1 a n 2 sin 2 ( n δ / 2 ) d δ δ log ⁡ ( 1 / δ ) > ∞ . \int _0^1 {\sqrt {a_n^2{{\sin }^2}(n\delta /2)} \frac {{d\delta }}{{\delta \sqrt {\log (1/\delta )} }}\; > \infty .} \] Hence, the random essential continuity of f t ( x ) {f_t}(x) is guaranteed by each of the above conditions.