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Fu Ken Ly

Publications and source records attributed to Fu Ken Ly.

9 recordsLinked to original sources
arXiv2023

Calder\'on-Zygmund operators and endpoint spaces for Hermite expansions

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 bounded on various function spaces including the Hardy spaces and the Lipschitz spaces associated to $L$. We then apply our results to study the boundedness of the Riesz transforms and pseudo-multipliers associated to $L$.

math.CA↗
arXiv2022

On the $L^2$ boundedness of pseudo-multipliers for Hermite expansions

We give various conditions for Hermite pseudo-multipliers to be bounded on $L^2(\mathbb{R}^n)$. As a by-product we also give results on $L^p(\mathbb{R}^n)$, as well as new results for pseudo-multipliers for the Gaussian measure setting. One of our key tools is a new integration by-parts formula for Hermite expansions.

math.CA↗
arXiv2022

Calder\'on-Zygmund operators on local hardy spaces

We give necessary and sufficient conditions for inhomogeneous Calder\'on-Zgymund operators to be bounded on the local hardy spaces $h^p(\mathbb{R}^n)$. We then give applications to local and truncated Riesz transforms, as well as pseudo-differential operators defined by amplitudes.

math.CA↗
arXiv2021

Pseudo-multipliers and smooth molecules on Hermite Besov and Hermite Triebel--Lizorkin spaces

We obtain new molecular decompositions and molecular synthesis estimates for Hermite Besov and Hermite Triebel--Lizorkin spaces and use such tools to prove boundedness properties of Hermite pseudo-multipliers on those spaces. The notion of molecule we develop leads to boundedness of pseudo-multipliers associated to symbols of Hörmander-type adapted to the Hermite setting on spaces with non-positive smoothness; in particular, we obtain continuity results for such operators on Lebesgue and Hermite local Hardy spaces. As a byproduct of our results on boundedness properties of pseudo-multipliers, we show that Hermite Besov spaces and Hermite Triebel--Lizorkin spaces are closed under non-linearities.

math.CA↗
arXiv2018

Maximal function characterizations for Hardy spaces on spaces of homogeneous type with finite measure and applications

We prove nontangential and radial maximal function characterizations for Hardy spaces associated to a non-negative self-adjoint operator satisfying Gaussian estimates on a space of homogeneous type with finite measure. This not only addresses an open point in the literature, but also gives a complete answer to the question posed by Coifman and Weiss in the case of finite measure. We then apply our results to give maximal function characterizations for Hardy spaces associated to second order elliptic operators with Neumann and Dirichlet boundary conditions, Schrödinger operators with Dirichlet boundary conditions, and Fourier--Bessel operators.

math.CA↗
arXiv2017

Maximal function characterizations for new local Hardy type spaces on spaces of homogeneous type

Let $X$ be a space of homogeneous type and let $\mathfrak{L}$ be a nonnegative self-adjoint operator on $L^2(X)$ enjoying Gaussian estimates. The main aim of this paper is twofold. Firstly, we prove the (local) nontangential and radial maximal function charaterization for the local Hardy spaces associated to $\mathfrak{L}$. This deduces the maximal function charaterization for local Hardy spaces in the sense of Coifman and Weiss provided that $\mathfrak{L}$ satisfies certain extra conditions. Secondly, we introduce the local Hardy space associated to the critical function $ρ$ which is motivated by the theory of Hardy spaces related to Schrödinger operators and includes the local Hardy spaces of Coifman and Weiss as a special case. Then we prove that these local Hardy spaces can be characterized by the the local nontangential and radial maximal function charaterization related to $\mathfrak{L}$ and $ρ$ and the global maximal function charaterizations associated to `perturbations' of $\mathfrak{L}$. As applications, we apply our theory to obtain a number of new results on maximal characterizations for the local Hardy type spaces in various settings ranging from Shrödinger operators on manifolds to Shrödinger operators on connected and simply connected nilpotent Lie groups.

math.FA↗
arXiv2016

Weighted estimates for powers and Smoothing estimates of Schrödinger operators with inverse-square potentials

Let $\mathcal{L}_a$ be a Schrödinger operator with inverse square potential $a|x|^{-2}$ on $\mathbb{R}^d, d\geq 3$. The main aim of this paper is to prove weighted estimates for fractional powers of $\mathcal{L}_a$. The proof is based on weighted Hardy inequalities and weighted inequalities for square functions associated to $\mathcal{L}_a$. As an application, we obtain smoothing estimates regarding the propagator $e^{it\mathcal{L}_a}$.

math.AP↗
arXiv2015

$T1$ criterions for generalised Calderón--Zygmund type operators on Hardy and BMO spaces associated to Schrödinger operators and applications

Suppose $L=-Δ+V$ is a Schrödinger operator on $\mathbb{R}^n$ with a potential $V$ belonging to certain reverse Hölder class $RH_σ$ with $σ\geq n/2$. The main aim of this paper is to provide necessary and sufficient conditions in terms of $T1$ criteria for a generalised Calderón--Zygmund type operator with respect to $L$ to be bounded on Hardy spaces $H^p_L(\mathbb{R}^n)$ and on BMO type spaces BMO$_L^α(\mathbb{R}^n)$ associated with $L$. As applications, we prove the boundedness for several singular integral operators associated to $L$. Our approach is flexible enough to prove the boundedness of the Riesz transforms related to $L$ with $n/2 \leq σ<n$ which were investigated in \cite{MSTZ} under the stronger condition $σ\geq n$. Thus our results not only recover existing results in \cite{MSTZ} but also contains new results in literature.

math.AP↗