Related papers: Single annulus $L^p$ estimates for Hilbert transfo…

Stein conjectured that the Hilbert transform in the direction of a vector field is bounded on, say, $L^2$ whenever $v$ is Lipschitz. We establish a wide range of $L^p$ estimates for this operator when $v$ is a measurable, non-vanishing,…

Let v be a planar Lipschitz vector field. We prove that the r-th variation-norm Hilbert transform along v, composed with a standard Littlewood-Paley projection operator P_k, is bounded from L^2 to L^{2, \infty}, and from L^p to itself for…

In this paper, for general plane curves $\gamma$ satisfying some suitable smoothness and curvature conditions, we obtain the single annulus $L^p(\mathbb{R}^2)$-boundedness of the Hilbert transforms $H^\infty_{U,\gamma}$ along the variable…

We prove bounds for the truncated directional Hilbert transform in $L^p(\mathbb{R}^2)$ for any $1<p<\infty$ under a combination of a Lipschitz assumption and a lacunarity assumption. It is known that a lacunarity assumption alone is not…

We prove the $L^p (p > 3/2)$ boundedness of the directional Hilbert transform in the plane relative to measurable vector fields which are constant on suitable Lipschitz curves.

We investigate the Hilbert transform and the maximal operator along a class of variable non-flat polynomial curves $(P(t),u(x)t)$ with measurable $u(x)$, and prove uniform $L^p$ estimates for $1<p<\infty$. In particular, via the change of…

In this paper, for $1<p<\infty$, we obtain the $L^p$-boundedness of the Hilbert transform $H^{\gamma}$ along a variable plane curve $(t,u(x_1, x_2)\gamma(t))$, where $u$ is a Lipschitz function with small Lipschitz norm, and $\gamma$ is a…

We prove the $L^p$ bound for the Hilbert transform along variable non-flat curves $(t,u(x)[t]^\alpha+v(x)[t]^\beta)$, where $\alpha$ and $\beta$ satisfy $\alpha\neq \beta,\ \alpha\neq 1,\ \beta\neq 1.$ Comparing with the associated theorem…

In this paper, we show that Hilbert transforms along some curves are bounded on $L^p({\mathbb R}^n;X)$ for some $1<p<\infty$ and some UMD spaces $X$. In particular, we prove that the Hilbert transform along some curves are completely…

Let $D$ be a nonnegative integer and ${\mathbf{\Theta}}\subset S^1$ be a lacunary set of directions of order $D$. We show that the $L^p$ norms, $1<p<\infty$, of the maximal directional Hilbert transform in the plane $$ H_{{\mathbf{\Theta}}}…

We prove a conjecture of Lacey and Li in the case that the vector field depends only on one variable. Specifically: let v be a vector field defined on the unit square such that v(x,y) = (1,u(x)) for some measurable u from [0,1] to [0,1].…

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…

M. Lacey and C. Thiele proved in [27] (Annals of Math. (1997)) and [28] (Annals of Math. (1999)) that the bilinear Hilbert transform maps $L^{p_1}\times L^{p_2}\rightarrow L^{p}$ boundedly when $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$ with…

We prove variable coefficient versions of L^p boundedness results on Hilbert transforms and maximal functions along convex curves in the plane.

Let $H_k$ be the one dimensional Hilbert transform computed in the direction $(1,2^k)$ in the plane. We show that the maximal operator $\sup_k |H_kf|$ maps $L^p$ of the plane into itself for $1<p<\infty$. The same result with the Hilbert…

We prove the $L^2$ boundedness of the directional Hilbert transform in the plane relative to measurable vector fields which are constant on suitable Lipschitz curves.

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.

Let $\mathcal{H}^{\mathbb{T}}$ denote the Hilbert transform on the circle. The paper contains the proofs of the sharp estimates \begin{equation*} \frac{1}{2\pi}|\{ \xi\in\mathbb{T} : \mathcal{H}^{\mathbb{T}}f(\xi) \geq 1 \}| \leq…

The $L^p$ ($1<p<\infty$) and weak-$L^1$ estimates for the variation for Calder\'on-Zygmund operators with smooth odd kernel on uniformly rectifiable measures are proven. The $L^2$ boundedness and the corona decomposition method are two key…

We prove sharp $L^p$ estimates for a singular transport equation by building what we call a \emph{cascading solution}; the equation studies the combined effect of multiplying by a bounded function and application of the Hilbert transform.…