If H denotes the classical Hilbert transform and Hu(x) = v(x), then the functions u(x) and v(x) are the values on the real axis of a pair of conjugate functions, harmonic in the upper half-plane. This note gives a generalization of the above concepts in which the Laplace equation Deltau = 0 is replaced by the Yukawa equation Deltau = mu(2)u and in which the Cauchy-Riemann equations have a corre...

x0 Introduction In this paper, we prove suucient conditions on pairs of weights (u; v) (scalar, matrix or operator valued) so that the Hilbert transform Hf(x) = p: v: Z f(y) x ? y dy; is bounded from L 2 (u) to L 2 (v). When u = v are scalar, the classical results were given in HMW] and CF]. Earlier, HS] gave a characterization of these weights by complex methods which has been generalized by C...

We obtain estimates for the distribution of values of functions Íll the weighted BMOq, spaces, BMO�(R), that let us find equivalent norms. It is also obtained that a suitable redefinition of the HilbeÍt transform is a bounded operator from these spaces into themselves. This is achieved for a certain cIass of weights w.

The aim of the present paper is to characterize the classes of weights which ensure the validity of one-weighted strong, weak or extra-weak type estimates in Orlicz classes for the integral operator H0f(x) = 2 π ∫ ∞ 0 yf(y) x2 − y2 dy, x ∈ (0,∞).

Riesz fractional derivatives of a function, Dα xf(x) (also called Riesz potentials), are defined as fractional powers of the Laplacian. Asymptotic expansions for large x are computed for the Riesz fractional derivatives of the Airy function of the first kind, Ai(x), and the Scorer function, Gi(x). Reduction formulas are provided that allow one to express Riesz potentials of products of Airy fun...

Some inequalities for the Hilbert transform of the product of two functions are given.

Let X be a recurrent Lévy process with no negative jumps and n the measure of its excursions away from 0. Using Lamperti’s connection [13] that links X to a continuous state branching process, we determine the joint distribution under n of the variables C T = ∫ T 0 1{Xs>0}X −1 s ds and C− T = ∫ T 0 1{Xs<0}|Xs|ds, where T denotes the duration of the excursion. This provides a new insight on an i...