Double Hilbert transform along real-analytic surfaces in Rd+2

We give a necessary and sufficient condition for the doubleHilbert transform on ^d+2 to be bounded on L^p, 1 < p <. This generalizes a result of Carbery, Wainger and Wright [DukeMath. J. 101 (2000) 499–513] for d = 1.     

Fourier transform versus Hilbert transform

Certain relations between the Fourier transform of a function and the Hilbert transform of its derivative are revealed. They concern the integrability/non-integrability of both transforms. Certain applications are discussed.     

On the Hilbert transform

Для любой функцииf∈L ^p (|x|^α dx, гд е p >1, аа — любое действительное числ о, найдены необходимые и достаточные услови я для того, чтобы ее преобразова ние ГильбертаHf $$Hf(x) = \frac{1}{\pi }\mathop \smallint \limits_{ - \infty }^\infty (x - t)^{ - 1} f(t)dt, x \in ( - \infty ,\infty )$$ , также принадлежалоL ^p (|x|)^α dx.L ^p (|x|)^α dx — класс измеримых фу нкцийf определенных на (- ∞, ∞) и удовлетворяющих условию $$\mathop \smallint \limits_{ - \infty }^\infty |f(x)|^p |x|^\alpha dx< \infty$$ . Ниже сформулированы наиболее интересные случаи: Теорема 1. Теорема 2.      

Hilbert transform of

There are two general ways to evaluate the Hilbert transform of a function of real variable . We can extend to a harmonic function in the upper half plane by the Poisson integral formula. Non-tangential limit of its harmonic conjugate exists almost everywhere and is defined to be the Hilbert transform of . There is also a singular integral formula for the Hilbert transform of . It is fairly difficult to directly evaluate the Hilbert transform of . In this paper we give an explicit formula for the Hilbert transform of , where is a function in the Cartwright class.

     

The Hilbert transform of a measure

Let e be a homogeneous subset of ℝ in the sense of Carleson. Let μ be a finite positive measure on ℝ and H _μ(x) its Hilbert transform. We prove that if lim _t→∞ t|e∩{x ‖H _μ(x)| > t}| = 0, then μ _s (e) = 0, where μ_s is the singular part of μ.     

A characterization of the Hilbert transform

In this note the Hilbert transform is characterized in terms of function algebras with respect to pointwise multiplication.

     

一般Hilbert二重级数定理的注记

本文证明，对任意正整数这里，是使上式成立的与r有关的最大值；5／16＝0．3806471＋．由此改进了一般Hilbert二重级数定理．     

Distributional estimates for the bilinear Hilbert transform

We obtain size estimates for the distribution function of the bilinear Hilbert transform acting on a pair of characteristic functions of sets of finite measure, that yield exponential decay at infinity and blowup near zero to the power −2/3 (modulo some logarithmic factors). These results yield all known L^p bounds for the bilinear Hilbert transform and provide new restricted weak type endpoint estimates on L^p1 × L^p2 when either 1/p₁ + 1/p₂ = 3/2 or one of p₁, p₂ is equal to 1. As a consequence of this work we also obtain that the square root of the bilinear Hilbert transform of two characteristic functions is exponentially integrable over any compact set.     

The bilinear Hilbert transform acting on Hermite and Laguerre functions

We obtain several formulas for the action of the bilinear Hilbert transform on pairs of Hermite and Laguerre functions. The results can be given as linear combinations of products of Hermite or Laguerre functions. We show also that for the generalized bilinear Hilbert transforms the results cannot be expressed in such a simple way.     

A Remark on the Arcsine Distribution and the Hilbert transform

Journal of Fourier Analysis and Applications - It is known that if $$(p_n)_{n in mathbb {N}}$$ is a sequence of orthogonal polynomials in $$L^2([-1,1], w(x)dx)$$ , then the roots are distributed...     

Maximal theorems for the directional Hilbert transform on the plane

For a Schwartz function on the plane and a non-zero define the Hilbert transform of in the direction to be

p.v.

Let be a Schwartz function with frequency support in the annulus , and . We prove that the maximal operator maps into weak , and into for . The estimate is sharp. The method of proof is based upon techniques related to the pointwise convergence of Fourier series. Indeed, our main theorem implies this result on Fourier series.

     

Double Operator Integrals in a Hilbert Space

Double operator integrals are a convenient tool in many problems arising in the theory of self-adjoint operators, especially in the perturbation theory. They allow to give a precise meaning to operations with functions of two ordered operator-valued non-commuting arguments. In a different language, the theory of double operator integrals turns into the problem of scalarvalued multipliers for operator-valued kernels of integral operators.The paper gives a short survey of the main ideas, technical tools and results of the theory. Proofs are given only in the rare occasions, usually they are replaced by references to the original papers. Various applications are discussed.     

The finite Hilbert transform and weighted Sobolev spaces

The boundedness of the finite Hilbert transform operator on certain weighted L_p spaces is well known. We extend this result to give the boundedness of that operator on certain weighted Sobolev spaces. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Characterizing the Hilbert transform by the Bedrosian theorem

It is proved that a bounded linear translation invariant operator on L²(Rd) satisfies the Bedrosian theorem if and only if it is a linear combination of the compositions of the partial Hilbert transforms and the identity operator. This observation justifies a definition of multidimensional analytic signals in the papers [T. Bulow, G. Sommer, Hypercomplex signals—a novel extension of the analytic signal to the multidimensional case, IEEE Trans. Signal Process. 49 (2001) 2844-2852] and [S.L. Hahn, Multidimensional complex signals with single-orthant spectra, Proc. IEEE 80 (1992) 1287-1300].