Related papers: The finite Hilbert transform acting on $L^\infty$

We study a reproducing kernel Hilbert space of functions defined on the positive integers and associated to the binomial coefficients. We introduce two transforms, which allow us to develop a related harmonic analysis in this Hilbert space.…

We show that the Hilbert transform does not map $L^1(M_{\Phi}w)$ to $L^{1,\infty}(w)$ for every Young function $\Phi$ growing more slowly than $t\log\log ({\rm e}^{\rm e}+t)$. Our proof is based on a construction of M.C. Reguera and C.…

We present and study a novel class of one-dimensional Hilbert space eigenfunction transforms that diagonalize analytic difference operators encoding the (reduced) two-particle relativistic hyperbolic Calogero-Moser dynamics. The scattering…

There is a canonical unitary transformation from $L^2(\R)$ onto the Fock space $F^2$, called the Bargmann transform. We study the action of the Bargmann transform on several classical integral operators on $L^2(\R)$, including the…

In this note, we define a bounded variant on the Hilbert projective metric on an infinite dimensional space $E$ and study the contraction properties of the projective maps associated with positive linear operators on $E$. More precisely, we…

The main result: for every sequence $\{\omega_m\}_{m=1}^\infty$ of positive numbers ($\omega_m>0)$ there exists an isometric embedding $F:[0,1]\to L_1[0,1]$ which is nowhere differentiable, but for each $t\in [0,1]$ the image $F_t$ is…

We characterize the weights for the Stieltjes transform and the Calder\'on operator to be bounded on the weighted variable Lebesgue spaces $L_w^{p(\cdot)}(0,\infty)$, assuming that the exponent function $p(\cdot)$ is log-H\"older continuous…

We prove Hilbert transform identities involving conformal maps via the use of Rellich identity and the solution of the Neumann problem in a graph Lipschitz domain in the plane. We obtain as consequences new $L^2$-weighted estimates for the…

We prove the weak type (1,1) estimate for maximal function of the truncated rough Hilbert transform considered in [9] and [10]

We prove that the analogue for the Hilbert transform of a classical weighted inequality by Fefferman and Stein for the Hardy Littlewood maximal operator does not hold. This is a sequel to paper arXiv:1008.3943 by the first author, which…

For any natural number $k$, consider the $k$-linear Hilbert transform $$ H_k( f_1,\dots,f_k )(x) := \operatorname{p.v.} \int_{\bf R} f_1(x+t) \dots f_k(x+kt)\ \frac{dt}{t}$$ for test functions $f_1,\dots,f_k: {\bf R} \to {\bf C}$. It is…

Let f(t,X) be an irreducible polynomial over the field of rational functions k(t), where k is a number field. Let O be the ring of integers of k. Hilbert's irreducibility theorem gives infinitely many integral specializations of t to values…

Let $L=-\Delta +|x|^2$ be the Hermite operator on $\mathbb{R}^n$, and $T$ be a Calder\'on-Zygmund type operator that is modelled on certain singular integrals related to $L$. We establish necessary and sufficient conditions for $T$ to be…

We show that if the Hilbert transform with values in a Banach space is $L^p$ bounded, then so is the dyadic Hilbert transform, with a linear relation of the norms.

An approximation result for the bilinear Hilbert transform is proved and used for the inversion of the bilinear Hilbert transform. Also, p-Lebesgue points $(p\geq 1)$ are analyzed.

In this work we define a Fourier transform for each $f\in L^{p(\cdot)}(\mathbb{R})$, for a large class of exponent functions $p(\cdot)$, as the distributional derivative of a H\"older continuous function. A norm is defined in the space of…

Let $T$ be a bounded linear operator on a Hilbert space. Then the Aluthge transform $\Delta T$ and the sequence $(\Delta^nT)$ of Aluthge iterates of $T$ are defined by \begin{align*} \Delta…

Let $p\in (1,\infty)$. In this paper, for any given measurable function $u:\ \mathbb{R}\rightarrow \mathbb{R}$ and a generalized plane curve $\gamma$ satisfying some conditions, the $L^p(\mathbb{R}^2)$ boundedness of the Hilbert transform…

We prove $L^p$, $p\in (1,\infty)$ estimates on the Hilbert transform along a one variable vector field acting on functions with frequency support in an annulus. Estimates when $p>2$ were proved by Lacey and Li in \cite{LL1}. This paper also…

The ordered eigenvalues define a Lipschitz map on the real vector space of Hermitian $d \times d$ matrices. We prove that this map acts continuously, but not uniformly continuously, by superposition on the Sobolev spaces $W^{1,q}$, for all…