arXiv subjects

Xuan Thinh Duong

Publications and source records attributed to Xuan Thinh Duong.

At least 19 recordsLinked to original sources

arXiv2026

Endpoint estimates for commutators of singular integral operators associated with admissible functions

Let $(\mathcal X, d, μ)$ be a space of homogeneous type and let $ρ$ be an admissible function on $\mathcal X$. In this paper, we introduce a new class of singular integral operators associated with $ρ$, including a wide range of operators arising in harmonic analysis. For such an operator $T$ and a function $b$ belonging to suitable localized $\mathrm{BMO}$ spaces associated with $ρ$, which are strictly larger than the classical space $\mathrm{BMO}(\mathcal X)$, we establish the boundedness of the commutator $[b, T]$ from the Hardy space $H^1_ρ(\mathcal X)$ into $L^{1,\infty}(\mathcal X)$, $L^1(\mathcal X)$, and $H^1_ρ(\mathcal X)$. Moreover, the boundedness of $[b,T]$ on $H^1_ρ(\mathcal X)$ is characterized by necessary and sufficient conditions. We then apply this to investigate the boundedness of commutators of singular integrals in various settings, including Schrödinger operators on stratified Lie groups and Laguerre operators of convolution type. Our results are new even for Schrödinger operators on $\mathbb R^n$.

math.CA↗

arXiv2026

Endpoint Estimates for Commutators of the Dirichlet Riesz Transform on the Half-Space

Let $\mathfrak R=\nabla(-Δ_D)^{-1/2}$ be the Riesz transform associated with the Dirichlet Laplacian on the upper half-space $\mathbb R^n_+$, and let $H^1_r(\mathbb R^n_+)$ denote the Hardy space of restrictions. We establish an endpoint theory for commutators with $\mathfrak R$ on $H^1_r(\mathbb R^n_+)$. For every $b\in \mathrm{BMO}(\mathbb R^n_+)$, the commutator $[b,\mathfrak R]$ extends boundedly from $H^1_r(\mathbb R^n_+)$ to $L^{1,\infty}(\mathbb R^n_+)$. We then prove the sharp strong-endpoint characterization \[ [b,\mathfrak R]:H^1_r(\mathbb R^n_+)\to H^1_r(\mathbb R^n_+) \quad\Longleftrightarrow\quad b\in \mathrm{BMO}^{\log}(\mathbb R^n_+). \] Moreover, the $\mathrm{BMO}^{\log}$ norm is quantitatively equivalent to the sum of the usual $\mathrm{BMO}$ norm and the $H^1_r$ operator norms of the component commutators $[b,\mathfrak R_j]$. The logarithmic condition records the distance to the boundary and is therefore intrinsic to the Dirichlet geometry.

math.CA↗

arXiv2026

Inequalities for radial solutions to the wave equation on the hyperbolic space

Let $\mathbb H^n$ be the $n$-dimensional hyperbolic space, and $Δ_{\mathbb H^n}$ be the Laplace--Beltrami operator on $\mathbb H^n$. Let $u=u(t,\cdot)$ be the solution to the wave equation $\partial^2_{t}u-\big(Δ_{\mathbb H^n}+\frac{1}{4}(n-1)^2\operatorname{Id}\big)u=0$ with initial conditions $u(0,x)=f(x)$ and $\partial_tu(0,x)=0$. In this article we show that for a radial function $f$, $2n/(n-1)\leq p<\infty$, $γ<1/p$ and $T\geq 0$, there exists a constant $C_p$ independent of $f$ and $T$ such that $$ \left(\int_T^{T+1}\|u(t,\cdot)\|^p_p\mathrm d t\right)^{\frac1p}\leq C_{p,γ}\,\mathrm e^{(n-1)\left(\frac12-\frac1p\right)T}\left\|(-Δ_{\mathbb H^n})^{\frac{n-1}2\left(\frac12-\frac1p\right)-\frac\gamma2}f\right\|_p. $$ We also show that the exponential part $\mathrm e^{(n-1)(1/2-1/p)T}$ of the bound is optimal.

math.AP↗

arXiv2026

Equivalence of Sobolev norms for Kolmogorov operators with scaling-critical drift

We consider the ordinary or fractional Laplacian plus a homogeneous, scaling-critical drift term. This operator is non-symmetric but homogeneous, and generates scales of $L^p$-Sobolev spaces which we compare with the ordinary homogeneous Sobolev spaces. Unlike in previous studies concerning Hardy operators, i.e., ordinary or fractional Laplacians plus scaling-critical scalar perturbations, handling the drift term requires an additional, possibly technical, restriction on the range of comparable Sobolev spaces, which is related to the unavailability of gradient bounds for the associated semigroup.

math.AP↗

arXiv2025

The Cauchy--Szegö Projection for domains in $\mathbb C^n$ with minimal smoothness: weighted theory

Let $D\subset\mathbb C^n$ be a bounded, strongly pseudoconvex domain whose boundary $bD$ satisfies the minimal regularity condition of class $C^2$. A 2017 result of Lanzani \& Stein states that the Cauchy--Szegö projection $S_ω$ defined with respect to a bounded, positive continuous multiple $ω$ of induced Lebesgue measure, {maps $L^p(bD, ω)$ to $L^p(bD, ω)$ continuously} for any $1<p<\infty$. Here we show that $S_ω$ satisfies explicit quantitative bounds in $L^p(bD, Ω)$, for any $1<p<\infty$ and for any $Ω$ in the maximal class of \textit{$A_p$}-measures, that is for $Ω_p = ψ_pσ$ where $ψ_p$ is a Muckenhoupt $A_p$-weight and $σ$ is the induced Lebesgue measure (with $ω$'s as above being a sub-class). Earlier results rely upon an asymptotic expansion and subsequent pointwise estimates of the Cauchy--Szegö kernel, but these are unavailable in our setting of minimal regularity {of $bD$}; at the same time, more recent techniques that allow to handle domains with minimal regularity (Lanzani--Stein 2017) are not applicable to $A_p$-measures. It turns out that the method of {quantitative} extrapolation is an appropriate replacement for the missing tools. To finish, we identify a class of holomorphic Hardy spaces defined with respect to $A_p$-measures for which a meaningful notion of Cauchy--Szegö projection can be defined when $p=2$.

math.CV↗

arXiv2025

The commutator of the Cauchy--Szegő Projection for domains in $\mathbb C^n$ with minimal smoothness: weighted regularity

Let $D\subset\mathbb C^n$ be a bounded, strongly pseudoconvex domain whose boundary $bD$ satisfies the minimal regularity condition of class $C^2$, and let $S_ω$ denote the Cauchy--Szegő projection defined with respect to (any) positive continuous multiple $ω$ of induced Lebesgue measure for the boundary of $D$. We characterize compactness and boundedness (the latter with explicit bounds) of the commutator $[b, S_ω]$ in the Lebesgue space $L^p(bD, Ω_p)$ where $Ω_p$ is any measure in the Muckenhoupt class $A_p(bD)$, $1<p<\infty$. We next fix $p =2$ and we let $S_{Ω_2}$ denote the Cauchy--Szegő projection defined with respect to (any) measure $Ω_2 \in A_2(bD)$, which is the largest class of reference measures for which a meaningful notion of Cauchy-Leray measure may be defined. We characterize boundedness and compactness in $L^2(bD, Ω_2)$ of the commutator $\displaystyle{[b,S_{Ω_2}]}$.

math.CV↗

arXiv2025

Hardy spaces and Campanato spaces associated with Laguerre expansions and higher order Riesz transforms

Let $\mathcal{L}_ν$ be the Laguerre differential operator which is the self-adjoint extension of the differential operator \[ L_ν:= \sum_{i=1}^n \left[-\frac{\partial^2}{\partial x_i^2} + x_i^2 + \frac{1}{x_i^2} \left(ν_i^2 - \frac{1}{4} \right) \right] \] initially defined on $C_c^\infty(\mathbb{R}_+^n)$ as its natural domain, where $ν\in [-1/2,\infty)^n$, $n \geq 1$. In this paper, we first develop the theory of Hardy spaces $H^p_{\mathcal{L}_ν}$ associated with $\mathcal{L}_ν$ for the full range $p \in (0,1]$. Then we investigate the corresponding BMO-type spaces and establish that they coincide with the dual spaces of $H^p_{\mathcal{L}_ν}$. Finally, we show boundedness of higher-order Riesz transforms on Lebesgue spaces, as well as on our new Hardy and BMO-type spaces.

math.CA↗

arXiv2025

On subordinated semigroups and Hardy spaces associated to fractional powers of operators

Let $L$ be a positive self-adjoint operator on $L^2(X)$, where $X$ is a $σ$-finite metric measure space. When $α\in (0,1)$, the subordinated semigroup $\{\exp(-tL^α):t \in \mathbb{R}^+\}$ can be defined on $L^2(X)$ and extended to $L^p(X)$. We prove various results about the semigroup $\{\exp(-tL^α):t \in \mathbb{R}^+\}$, under different assumptions on $L$. These include the weak type $(1,1)$ boundedness of the maximal operator $f \mapsto \sup _{t\in \mathbb{R}^+}\exp(-tL^α)f$ and characterisations of Hardy spaces associated to the operator $L$ by the area integral and vertical square function.

math.FA↗

arXiv2024

Hardy spaces, Besov spaces and Triebel--Lizorkin spaces associated with a discrete Laplacian and applications

Consider the discrete Laplacian $Δ_d$ defined on the set of integers $\mathbb Z$ by \[ Δ_d f(n) = -f(n+1) + 2f(n) -f(n-1), \ \ \ \ n\in \mathbb Z, \] where $f$ is a function defined on $\mathbb Z$. In this paper, we define Hardy spaces, Besov spaces and Triebel--Lizorkin spaces associated with $Δ_d$ and then show that these function spaces coincide with the classical function spaces defined on $\mathbb Z$. As applications, we prove the boundedness of the spectral multipliers and the Riesz transforms associated with $Δ_d$ on these function spaces.

math.CA↗

arXiv2024

Weighted Estimates of Singular Integrals and Commutators in the Zygmund Dilation Setting

The main purpose of this paper is to establish weighted estimates for singular integrals associated with Zygmund dilations via a discrete Littlewood--Paley theory, and then apply it to obtain the upper bound of the norm of commutators of such singular integrals with a function in the little bmo space associated with Zygmund dilations. Examples of such singular integrals associated with Zygmund dilations include a class of singular integrals studied by Ricci--Stein and Fefferman--Pipher, as well as a singular integral along a particular surface studied by Nagel--Wainger. We show that the lower bound of the norm of this commutator is not true for any singular integral in the class considered in Ricci--Stein and Fefferman--Pipher, but does in fact hold for the specific singular integral studied in Nagel--Wainger. In particular this implies that the family of singular integrals studied in these papers is not sufficiently general to contain the operator of Nagel--Wainger, which we show is of significance in this theory.

math.CA↗

arXiv2024

Global-in-time maximal regularity for the Cauchy problem of the heat equation in BMO and applications

In this article, we establish global-in-time maximal regularity for the Cauchy problem of the classical heat equation $\partial_t u(x,t)-Δu(x,t)=f(x,t)$ with $u(x,0)=0$ in a certain $\rm BMO$ setting, which improves the local-in-time result initially proposed by Ogawa and Shimizu in \cite{OS, OS2}. In further developing our method originally formulated for the heat equation, we obtain analogous global ${\rm BMO}$-maximal regularity associated to the Schrödinger operator $\mathcal L=-Δ+V$, where the nonnegative potential $V$ belongs to the reverse Hölder class ${\rm RH}_q$ for some $q> n/2$. This extension includes several inhomogeneous estimates as ingredients, such as Carleson-type estimates for the external forces. Our new methodology is to exploit elaborate heat kernel estimates, along with matched space-time decomposition on the involving integral-type structure of maximal operators, as well as some global techniques such as those from de Simon's work and Schur's lemma. One crucial trick is to utilize the mean oscillation therein to contribute a higher and necessary decay order for global-in-time estimates.

math.AP↗

arXiv2023

The Garnett-Jones Theorem on BMO spaces associated with operators and applications

Let $X$ be a metric space with doubling measure, and $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel satisfies the Gaussian upper bound. Let $f$ be in the space $ {\rm BMO}_L(X)$ associated with the operator $L$ and we define its distance from the subspace $L^{\infty}(X)$ under the $ {\rm BMO}_L(X)$ norm as follows: $$ {\rm dist} (f, L^{\infty}):= \inf_{g\in L^{\infty}} \|f -g\|_{{\rm BMO}_L(X)}. $$ In this paper we prove that ${\rm dist} (f, L^{\infty})$ is equivalent to the infimum of the constant $\varepsilon$ in the John-Nirenberg inequality for the space ${\rm BMO}_L(X)$: $$ \sup_B { μ\big(\{ x\in B: |f(x)-e^{-{r_B^2}L}f(x)|>λ\}\big) \over μ(B)} \leq e^{-λ/\varepsilon}\ \ \ \ {\rm for\ large\ } λ. $$ This extends the well-known result of Garnett and Jones \cite{GJ1} for the classical ${\rm BMO}$ space (introduced by John and Nirenberg). As an application, we show that a ${\rm BMO}_L(X)$ function with compact support can be decomposed as the summation of an $L^\infty$-function and the integral of the heat kernel (associated with $L$) against a finite Carleson measure on $X\times[0,\infty)$. The key new technique is a geometric construction involving the semigroup $e^{-tL}$. We also resort to several fundamental tools including the stopping time argument and the random dyadic lattice.

math.CA↗

arXiv2023

The Riesz Transform and Fractional Integral Operators in the Bessel Setting

Fix $λ>0$. Consider the Bessel operator $\triangle_λ:=-\frac{d^2}{dx^2}-\frac{2λ}{x} \frac d{dx}$ on $\mathbb{R_+}$, where $\mathbb{R_+}:=(0,\infty)$ and $dm_λ:=x^{2λ}dx$ with $dx$ the Lebesgue measure. We provide a deeper study of the Bessel Riesz transform and fractional integral operator via the related Besov and Triebel--Lizorkin spaces associated with $\triangle_λ$. Moreover, we investigate some possible characterization of the commutator of fractional integral operator, which was missing in the literature of the Bessel setting.

math.CA↗

arXiv2022

The Two Weight Inequality for Poisson Semigroup on Manifold with Ends

Let $M = \mathbb R^m \sharp \mathcal R^n$ be a non-doubling manifold with two ends $\mathbb R^m \sharp \mathcal R^n$, $m > n \ge 3$. Let $Δ$ be the Laplace--Beltrami operator which is non-negative self-adjoint on $L^2(M)$. Then $Δ$ and its square root $\sqrtΔ$ generate the semigroups $e^{-tΔ}$ and $e^{-t\sqrtΔ}$ on $L^2(M)$, respectively. We give testing conditions for the two weight inequality for the Poisson semigroup $e^{-t\sqrtΔ}$ to hold in this setting. In particular, we prove that for a measure $μ$ on $M_{+}:=M\times (0,\infty)$ and $σ$ on $M$: $$ \|\mathsf{P}_σ(f)\|_{L^2(M_{+};μ)} \lesssim \|f\|_{L^2(M;σ)}, $$ with $\mathsf{P}_σ(f)(x,t):= \int_M \mathsf{P}_t(x,y)f(y) \,dσ(y)$ if and only if testing conditions hold for the Poisson semigroup and its adjoint. Further, the norm of the operator is shown to be equivalent to the best constant in these testing conditions.

math.CA↗

arXiv2022

The Schrödinger equation in $L^p$ spaces for operators with heat kernel satisfying Poisson type bounds

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. In this paper, we study sharp endpoint $L^p$-Sobolev estimates for the solution of the initial value problem for the Schrödinger equation, $i \partial_t u + L u=0 $ and show that for all $f\in L^p(X), 1<p<\infty,$ \begin{eqnarray*} \left\| e^{itL} (I+L)^{-{σn}} f\right\|_{p} \leq C(1+|t|)^{σn} \|f\|_{p}, \ \ \ t\in{\mathbb R}, \ \ \ σ\geq \big|{1\over 2}-{1\over p}\big|, \end{eqnarray*} where the semigroup $e^{-tL}$ generated by $L$ satisfies a Poisson type upper bound. This extends the previous result in \cite{CDLY1} in which the semigroup $e^{-tL}$ generated by $L$ satisfies the exponential decay.

math.AP↗

arXiv2021

Fundamental properties of Cauchy--Szegő projection on quaternionic Siegel upper half space and applications

We investigate the Cauchy--Szegő projection for quaternionic Siegel upper half space to obtain the pointwise (higher order) regularity estimates for Cauchy--Szegő kernel and prove that the Cauchy--Szegő kernel is non-zero everywhere, which further yields a non-degenerated pointwise lower bound. As applications, we prove the uniform boundedness of Cauchy--Szegő projection on every atom on the quaternionic Heisenberg group, which is used to give an atomic decomposition of regular Hardy space $ H^p$ on quaternionic Siegel upper half space for $2/3<p\leq1$. Moreover, we establish the characterisation of singular values of the commutator of Cauchy--Szegő projection based on the kernel estimates and on the recent new approach by Fan--Lacey--Li. The quaternionic structure (lack of commutativity) is encoded in the symmetry groups of regular functions and the associated partial differential equations.

math.CV↗

arXiv2021

Almost everywhere convergence of spectral sums for self-adjoint operators

Let $L$ be a non-negative self-adjoint operator acting on the space $L^2(X)$, where $X$ is a metric measure space. Let ${ L}=\int_0^{\infty} λdE_{ L}(λ)$ be the spectral resolution of ${ L}$ and $S_R({ L})f=\int_0^R dE_{ L}(λ) f$ denote the spherical partial sums in terms of the resolution of ${ L}$. In this article we give a sufficient condition on $L$ such that $$ \lim_{R\rightarrow \infty} S_R({ L})f(x) =f(x),\ \ {\rm a.e.} $$ for any $f$ such that ${\rm log } (2+L) f\in L^2(X)$. These results are applicable to large classes of operators including Dirichlet operators on smooth bounded domains, the Hermite operator and Schrödinger operators with inverse square potentials.

math.CA↗

arXiv2021

Sharp endpoint estimates for Schrödinger groups on Hardy spaces

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. Suppose that the heat kernel of $L$ satisfies the Davies-Gaffney estimates of order $m\geq 2$. Let $H^1_L(X)$ be the Hardy space associated with $L.$ In this paper we show sharp endpoint estimate for the Schrödinger group $e^{itL}$ associated with $L$ such that \begin{eqnarray*} \left\| (I+L)^{-{n/2}}e^{itL} f\right\|_{ L^1(X)} + \left\| (I+L)^{-{n/2}}e^{itL} f\right\|_{ H^1_L(X)} \leq C(1+|t|)^{n/2}\|f\|_{H^1_L(X)}, \ \ \ t\in{\mathbb R} \end{eqnarray*} for some constant $C=C(n, m)>0$ independent of $t$. By a duality and interpolation argument, it gives a new proof of a recent result of \cite{CDLY} for { sharp} endpoint $L^p$-Sobolev bound for $e^{itL}$: $$ \left\| (I+L)^{-s }e^{itL} f\right\|_{ L^p(X)} \leq C (1+|t|)^{s} \|f\|_{ L^p(X)}, \ \ \ t\in{\mathbb R}, \ \ \ s\geq n\big|{1\over 2}-{1\over p}\big| $$ for every $1<p<\infty$ when the heat kernel of $L$ satisfies a Gaussian upper bound, which extends the classical results due to Miyachi ) for the Laplacian on the Euclidean space ${\mathbb R}^n$.

math.AP↗