For minimizers u ∈W 1,p(x)(Ω) of quasiconvex integral functionals of the type F [u] := ∫ Ω f(x,Du(x)) dx with p(x) growth in the class K := {u ∈ W 1,p(x)(Ω) : u ≥ ψ}, where ψ ∈ W 1,p(x)(Ω) is a given obstacle function, we show estimates of Calderón-Zygmund type, i.e. |Dψ|p(·) ∈ L =⇒ |Du|p(·) ∈ L , for any q > 1, provided that the modulus of continuity ω of the exponent function p satisfies the ...

Let ω : [0,+∞[→ [0,+∞[ be a modulus of continuity. Consider the integral functional

We show that there exist solutions of drift-diffusion equations in two dimensions with divergence-free super-critical drifts, that become discontinuous in finite time. We consider classical as well as fractional diffusion. However, in the case of classical diffusion and time-independent drifts we prove that solutions satisfy a modulus of continuity depending only on the local L norm of the drif...

In the product space H × R, we obtain uniform a priori C horizontal length estimates, uniform a priori C boundary gradient estimates, as well as uniform modulus of continuity, for a class of horizontal minimal equations. In two independent variables, we derive a certain uniform global a priori C estimates and we infer an existence result.

The purpose of this note is to prove a central limit theorem for the L -modulus of continuity of the Brownian local time obtained in [2], using techniques of stochastic analysis. The main ingredients of the proof are an asymptotic version of Knight’s theorem and the Clark-Ocone formula for the L-modulus of the Brownian local time.

We present a self-contained and modern survey of some existing quasi-sure results via the connection to the Brownian sheet. Among other things, we prove that quasi-every continuous function: (i) satisfies the local law of the iterated logarithm; (ii) has Lévy’s modulus of continuity for Brownian motion; (iii) is nowhere differentiable; and (iv) has a nontrivial quadratic variation. We also pres...

Given a diffeomorphism of the interval, consider the uniform norm of the derivative of its n-th iteration. We get a sequence of real numbers called the growth sequence. Its asymptotic behavior is an invariant which naturally appears both in smooth dynamics and in geometry of the diffeomorphisms groups. We find sharp estimates for the growth sequence of a given diffeomorphism in terms of the mod...

Abstract. We study inequalities between general integral moduli of continuity of a function and the tail integral of its Fourier transform. We obtain, in particular, a refinement of a result due to D. B. H. Cline [2] (Theorem 1.1 below). We note that our approach does not use a regularly varying comparison function as in [2]. A corollary of Theorem 1.1 deals with the equivalence of the two-side...

In this paper two kinds of Kantorovich-type q-Bernstein-Stancu operators are introduced, and the statistical approximation properties of these operators are investigated. Furthermore, by means of modulus of continuity, the rates of statistical convergence of these operators are also studied. MSC: 41A10; 41A25; 41A36

A uniqueness theorem is proven for the problem of the recovery of a complex valued compactly supported 2-D function from the modulus of its Fourier transform. An application to the phase problem in optics is discussed.