Related papers: $L^p(\mathbb{R}^2)$-boundedness of Hilbert Transfo…

Using some resolution of singularities and oscillatory integral methods in conjunction with appropriate damping and interpolation techniques, L^p boundedness theorems for p > 2 are obtained for maximal operators over a wide range of…

Let $H^n\cong \Bbb R^{2n}\ltimes \Bbb R$ be the Heisenberg group and let $\mu_t$ be the normalized surface measure for the sphere of radius $t$ in $\Bbb R^{2n}$. Consider the maximal function defined by $Mf=\sup_{t>0} |f*\mu_t|$. We prove…

Let $M$ be a closed complex submanifold in ${\mathbb C}^N$ with the complete K\"ahler metric induced by the Euclidean metric. Several finiteness theorems on the $L^p$ Bergman space of holomorphic sections of a given Hermitian line bundle…

We consider a conjecture attributed to Muckenhoupt and Wheeden which suggests a positive relationship between the continuity of the Hardy-Littlewood maximal operator and the Hilbert transform in the weighted setting. Although continuity of…

By using an $H^{\infty}$ joint functional calculus for strongly commuting operators, we derive a scheme to deduce the $L^p$ boundedness of certain $d$-dimensional Riesz transforms from the $L^p$ boundedness of appropriate one-dimensional…

We study the lacunary analogue of the $\delta$-discretised spherical maximal operators introduced by Hickman and Jan\v{c}ar, for $\delta \in (0, 1/2)$, and establish the boundedness on $L^p$ for all $1 < p < \infty$, along with the endpoint…

Let $M$ be a von Neumann algebra with a faithful normal finite trace $t$, and $H^\infty$ be a finite, maximal, subdiagonal of $M$. Fundamental theorems on conjugate functions for weak* Dirichlet algebras are shown to be a bounded linear map…

We prove uniform uniform $L^{p}$ bounds for the family of bilinear Hilbert transforms $\mathrm{BHT}_{\beta} [f_1, f_2] (x) := \mathrm{p.v.} \int_{\mathbb{R}} f_1 (x - t) f_2 (x + \beta t) \frac{\mathrm{d} t}{t}$. We show that the operator…

In recent articles it was proved that when $\mu$ is a finite, radial measure in $\real^n$ with a bounded, radially decreasing density, the $L^p(\mu)$ norm of the associated maximal operator $M_\mu$ grows to infinity with the dimension for a…

This paper studies the $L^{p}$ boundedness of bilinear Fourier multipliers in the local $L^{2}$ range. We assume a H\"{o}rmander condition relative to a singular set that is a finite union of Lipschitz curves. The H\"{o}rmander condition is…

As a corollary to our main result we deduce sharp A_p$ inequalities for T being either the Hilbert transform in dimension d=1, the Beurling transform in dimension d=2, or a Riesz transform in any dimension d\ge 2. For T_{\ast} the maximal…

We prove that the Hardy--Littlewood maximal operator $M$ is bounded on the variable Lebesgue space $L^{p(\cdot)}(X,d,\mu)$, with $1<p_-\le p_+<\infty$, over an unbounded space of homogeneous type $(X,d,\mu)$ with a Borel-semiregular measure…

For any $p\in[1,\infty]$, we prove that the set of simple functions taking at most $k$ different values is proximinal in $L^p$ for all $k\geq 1$. We introduce the class of uniformly approximable subsets of $L^p$, which is larger than the…

We study the $L^{p},$ $1\leqslant p\leqslant \infty,$ boundedness for Riesz transforms of the form $V^{a}(-\frac{1}{2}\Delta+V)^{-a},$ where $a>0$ and $V$ is a non-negative potential. We prove that $V^{a}(-\frac{1}{2}\Delta+V)^{-a}$ is…

This article focuses on $L^p$ estimates for objects associated to elliptic operators in divergence form: its semigroup, the gradient of the semigroup, functional calculus, square functions and Riesz transforms. We introduce four critical…

In graphs and Riemannian manifolds where the kernel of the diffusion semigroup satisfies pointwise sub-Gaussian estimates, we study the range of parameters \( p \in (1, \infty) \) and \( \gamma \in [0, 1] \) for which the quantities \(…

We study the Littlewood-Paley-Stein functions associated with Hodge-de Rham and Schr{\"o}dinger operators on Riemannian manifolds. Under conditions on the Ricci curvature we prove their boundedness on L p for p in some interval (p 1 , 2]…

In this paper, we establish the $L^{p}(\mathbb{R}^{d})$-boundedness of the variation operator and the $\delta$-jump operator for generalized spherical means, and we also show the necessary conditions for the…

Let $L$ be a one-to-one operator of type $\omega$ in $L^2(\mathbb{R}^n)$, with $\omega\in[0,\,\pi/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. Let $p(\cdot):\ \mathbb{R}^n\to(0,\,1]$…

For $p>p_0=\frac{2\lambda}{2\lambda+1}$ with $\lambda>0$, the Hardy space $H_{\lambda}^p(\mathbb{R}_+^2)$ associated with the Dunkl transform $\mathcal{F}_\lambda$ and the Dunkl operator $D$ on the real line $\mathbb{R}$, where…