Related papers: The Polynomial Carleson Operator

We prove a Tb Theorem that characterizes all Calderon-Zygmund operators that extend compactly on L^p(R^n), 1<p<\infty . The result, whose proof does not require the property of accretivity, can be used to prove compactness of the Double…

Consider the discrete maximal function acting on $\ell^2(\mathbb Z)$ functions \[ \mathcal{C}_{\Lambda} f( n ) := \sup_{ \lambda \in \Lambda} \left| \sum_{m \neq 0} f(n-m) \frac{e^{2 \pi i\lambda m^2}} {m} \right| \] where $\Lambda \subset…

This paper investigates the $L^p$ boundedness of wave operators for the Laplace operator with finite rank perturbations \begin{equation*} H=-\Delta+\sum\limits_{i=1}^N\langle\cdot\,, \varphi_i\rangle \varphi_i \qquad \mbox{on}\,\,\, \R^d.…

We provide elementary proofs that the 2-variation Carleson operator $V_2$ along with explicit bilinear multipliers adapted to $\{\xi_1 + \xi_2 = 0\}$ satisfy no $L^p$ estimates. Furthermore, we obtain $L^p \rightarrow L^p$ estimates when $2…

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…

This paper is devoted to establishing several types of $L^p$-boundedness of wave operators $W_\pm=W_\pm(H, \Delta^2)$ associated with the bi-Schr\"odinger operators $H=\Delta^{2}+V(x)$ on the line $\mathbb{R}$. Given suitable decay…

In this article we prove L^p estimates for a general maximal operator, which extend both the classical Coifman-Meyer and Carleson-Hunt theorems in harmonic analysis

In this paper, we study the bilinear cone multiplier operator in two dimensions. We establish $L^{p_1}\times L^{p_2}\to L^{p}$ boundedness for a regularized version of this operator over a broad range of exponents satisfying the H\"older…

In this work, we develop $L^p$ boundedness theory for pseudodifferential operators with rough (not even continuous in general) symbols in the $x$ variable. Moreover, the $B(L^p)$ operator norms are estimated explicitly in terms of scale…

We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$ when $H$ has a threshold eigenvalue. We adapt our recent results for $m\geq 1$ when…

We consider $\ell^r$ extensions of Calderon-Zygmund operators on weighted spaces $L^p(w)$ with $w$ an $A_p$ weight and $1 < p < \infty$. We give quantitative estimates of these operators' norm in terms of a given weight's $A_p$…

In this work we extend consideration of the polynomial Carleson operator to the setting of a Radon transform acting along the paraboloid in $\mathbb{R}^{n+1}$ for $n \geq 2$. Inspired by work of Stein and Wainger on the original polynomial…

In the paper arXiv:1708.02289 we have introduced new solvability methods for strongly elliptic second order systems in divergence form on a domains above a Lipschitz graph, satisfying $L^p$-boundary data for $p$ near $2$. The main novel…

We prove continuity and surjectivity of the trace map onto $L_p$, from a space of functions of locally bounded variation, defined by the Carleson functional. The extension map is constructed through a stopping time argument. This extends…

We prove various equivalent characterisations of the Hardy space $H^p_{\mathcal{L}}(\mathbb{C}^n)$ for $0<p<1$ associated with the twisted Laplacian $\mathcal{L}$ which generalises the result of [MPR81] for the case $p=1$. Using the atomic…

We develop a theory of Lp spaces based on outer measures rather than measures. This theory includes the classical Lp theory on measure spaces as special case. It also covers parts of potential theory and Carleson embedding theorems. The…

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…

We study maximal averages associated with singular measures on $\rr$. Our main result is a construction of singular Cantor-type measures supported on sets of Hausdorff dimension $1 - \epsilon$, $0 \leq \epsilon < {1/3}$ for which the…

In this paper we present in concise form recent results, with illustrative proofs, on solvability of the $L^p$ Dirichlet, Regularity and Neumann problems for scalar elliptic equations on Lipschitz domains with coefficients satisfying a…

In the first part of this study we consider the boundedness and compactness properties of Cauchy-Fantappie type operators on Poletsky-Stessin Hardy spaces $H^{p}_{u}(\mathbb{B}^{\textbf{p}})$ of complex ellipsoids. We show that boundedness…