arXiv subjects

The Anh Bui

Publications and source records attributed to The Anh Bui.

At least 19 recordsLinked to original sources

arXiv2026

Fractional Leibniz rules for the Dunkl Laplacian in Besov and Triebel--Lizorkin spaces

Let $L$ be the Dunkl Laplacian on the Euclidean space $\mathbb{R}^N$ associated with a normalized root system $R$ and a multiplicity function $k(ν)\geq 0$, $ν\in R$. We establish a Leibniz-type rule for the fractional powers of $L$ on Besov and Triebel--Lizorkin spaces in the Dunkl setting. Our approach exploits the interplay between spectral multipliers and the Dunkl transform, together with the support properties of the distributions associated with Dunkl translations. These results extend the corresponding Leibniz-type estimates previously established on $L^p$ spaces to the broader setting of Besov and Triebel--Lizorkin spaces.

math.AP↗

arXiv2026

Equivalence of Sobolev norms in Lebesgue spaces for Hardy operators in a half-space

We consider Hardy operators, i.e., homogeneous Schrödinger operators consisting of the ordinary or fractional Laplacian in a half-space plus a potential, which only depends on the appropriate power of the distance to the boundary of the half-space. We compare the scales of homogeneous $L^p$-Sobolev spaces generated by these Hardy operators with and without potential with each other. To that end, we prove and use new square function estimates for operators with slowly decaying heat kernels. Our results hold for all admissible coupling constants in the local case and for repulsive potentials in the fractional case, and extend those obtained recently in $L^2$. They also cover attractive potentials in the fractional case, once expected heat kernel estimates are available.

math.AP↗

arXiv2026

Dunkl paraproducts and fractional Leibniz rules for the Dunkl Laplacian

We establish fractional Leibniz rules for the Dunkl Laplacian $Δ_k$ of the form $$\|(-Δ_k)^s(fg)\|_{L^p(dμ_k)} \lesssim \|(-Δ_k)^s f\|_{L^{p_1}(dμ_k)} \|g\|_{L^{p_2}(dμ_k)} + \|f\|_{L^{p_1}(dμ_k)} \|(-Δ_k)^s g\|_{L^{p_2}(dμ_k)}.$$ Our approach relies on adapting the classical paraproduct decomposition to the Dunkl setting. In the process, we develop several new auxiliary results. Specifically, we show that for a Schwartz function $f$, the function $(-Δ_k)^s f$ satisfies a pointwise decay estimate; we establish a version of almost orthogonality estimates adapted to the Dunkl framework; and we investigate the boundedness of Dunkl paraproduct operators on the Lebesgue spaces.

math.FA↗

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

Bilinear and Fractional Leibniz Rules Beyond Euclidean Spaces: Weighted Besov and Triebel--Lizorkin Estimates

We establish fractional Leibniz rules in weighted settings for nonnegative self-adjoint operators on spaces of homogeneous type. Using a unified method that avoids Fourier transforms, we prove bilinear estimates for spectral multiplier on weighted Hardy, Besov and Triebel-Lizorkin spaces. Our approach is flexible and applies beyond the Euclidean setting-covering, for instance, nilpotent Lie groups, Grushin operators, and Hermite expansions-thus extending classical Kato-Ponce inequalities. The framework also yields new weighted bilinear estimates including fractional Leibniz rules for Hermite, Laguerre, and Bessel operator, with applications to scattering formulas and related PDE models.

math.CA↗

arXiv2025

Hardy spaces, Campanato spaces and higher order Riesz transforms associated with Bessel operators

Let $ν= (ν_1, \ldots, ν_n) \in (-1/2, \infty)^n$, with $n \ge 1$, and let $Δ_ν$ be the multivariate Bessel operator defined by \[ Δ_ν = -\sum_{j=1}^n\left( \frac{\partial^2}{\partial x_j^2} - \frac{ν_j^2 - 1/4}{x_j^2} \right). \] In this paper, we develop the theory of Hardy spaces and BMO-type spaces associated with the Bessel operator $Δ_ν$. We then study the higher-order Riesz transforms associated with $Δ_ν$. First, we show that these transforms are Calderón-Zygmund operators. We further prove that they are bounded on the Hardy spaces and BMO-type spaces associated with $Δ_ν$.

math.CA↗

arXiv2025

Weighted norm inequalities of higher-order Riesz transforms associated with Laguerre expansions

Let $ν=(ν_1,\ldots,ν_n)\in (-1,\vc)^n$, $n\ge 1$, and let $\mathcal{L}_ν$ be a self-adjoint extension of the differential operator \[ L_ν:= \sum_{i=1}^n \left[-\frac{\partial^2}{\partial x_i^2} + x_i^2 + \frac{1}{x_i^2}(ν_i^2 - \frac{1}{4})\right] \] on $C_c^\infty(\mathbb{R}_+^n)$ as the natural domain. The $j$-th partial derivative associated with $L_ν$ is given by \[ δ_{ν_j} = \frac{\partial}{\partial x_j} + x_j-\frac{1}{x_j}\Big(ν_j + \f{1}{2}\Big), \ \ \ \ j=1,\ldots, n. \] In this paper, we investigate the weighted estimates of the higher-order Riesz transforms $δ_ν^k\mathcal L^{-|k|/2}_ν, k\in \mathbb N^n$, where $δ_ν^k=δ_{ν_n}^{k_n}\ldots δ_{ν_1}^{k_1}$. This completes the description of the boundedness of the higher-order Riesz transforms with the full range $ν\in (-1,\vc)^n$.

math.CA↗

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

Harmonic analysis in Dunkl settings

Let $L$ be the Dunkl Laplacian on the Euclidean space $\mathbb R^N$ associated with a normalized root $R$ and a multiplicity function $k(ν)\ge 0, ν\in R$. In this paper, we first prove that the Besov and Triebel-Lizorkin spaces associated with the Dunkl Laplacian $L$ are identical to the Besov and Triebel-Lizorkin spaces defined in the space of homogeneous type $(\mathbb R^N, \|\cdot\|, dw)$, where $dw({\rm x})=\prod_{ν\in R}\langle ν,{\rm x}\rangle^{k(ν)}d{\rm x}$. Next, consider the Dunkl transform denoted by $\mathcal{F}$. We introduce the multiplier operator $T_m$, defined as $T_mf = \mathcal{F}^{-1}(m\mathcal{F}f)$, where $m$ is a bounded function defined on $\mathbb{R}^N$. Our second aim is to prove multiplier theorems, including the Hörmander multiplier theorem, for $T_m$ on the Besov and Tribel-Lizorkin spaces in the space of homogeneous type $(\mathbb R^N, \|\cdot\|, dw)$. Importantly, our findings present novel results, even in the specific case of the Hardy 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

Riesz transforms, Hardy spaces and Campanato spaces associated with Laguerre expansions

Let $ν\in [-1/2,\infty)^n$, $n\ge 1$, and let $\mathcal{L}_ν$ be a self-adjoint extension of the differential operator \[ L_ν:= \sum_{i=1}^n \left[-\frac{\partial^2}{\partial x_i^2} + x_i^2 + \frac{1}{x_i^2}(ν_i^2 - \frac{1}{4})\right] \] on $C_c^\infty(\mathbb{R}_+^n)$ as the natural domain. In this paper, we first prove that the Riesz transform associated with $\mathcal L_ν$ is a Calderón-Zygmund operator, answering the open problem in [JFA, 244 (2007), 399-443]. In addition, we develop the theory of Hardy spaces and Campanato spaces associated with $\mathcal{L}_ν$. As applications, we prove that the Riesz transform related to $\mathcal{L}_ν$ is bounded on these Hardy spaces and Campanato spaces, completing the description of the boundedness of the Riesz transform in the Laguerre expansion setting.

math.CA↗

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 norm inequalities of some singular integrals associated with Laguerre expansions

Let $ν=(ν_1,\ldots,ν_n)\in (-1,\infty)^n$, $n\ge 1$, and let $\mathcal{L}_ν$ be a self-adjoint extension of the differential operator \[ L_ν:= \sum_{i=1}^n \left[-\frac{\partial^2}{\partial x_i^2} + x_i^2 + \frac{1}{x_i^2}(ν_i^2 - \frac{1}{4})\right] \] on $C_c^\infty(\mathbb{R}_+^n)$ as the natural domain. In this paper, we investigate the weighted estimates of singular integrals in the Laguerre setting including the maximal function, the Riesz transform and the square functions associated to the Laguerre operator $\mathcal L_ν$. In the special case of the Riesz transform, the paper completes the description of the Riesz transform for the full range of $ν\in (-1,\infty)^n$ which significant improves the result in [J. Funct. Anal. 244 (2007), 399--443] for $ν_i\ge -1/2, ν_i\notin (-1/2,1/2)$ for $i=1,\ldots, n$.

math.CA↗

arXiv2024

New Sparse Domination and Weighted Estimates for Fractional Operators Beyond Calderón-Zygmund Theory

Let $L$ be a closed, densely defined operator on $L^2(\mathbb{R}^n)$ satisfying suitable $L^p-L^q$ off-diagonal estimates of order $κ> 0$. This paper aims to investigate the two-weight estimate and the Bloom weighted estimate for the fractional operator $L^{-α/κ}$ with $0 < α< n$ through the method of sparse domination. Our assumptions on the operators are minimal, and our result applies to a wide range of differential operators. As a byproduct, we also establish a new sparse domination criterion for a general class of fractional operators, including the classical fractional integral.

math.CA↗

arXiv2024

Existence of solutions semilinear parabolic equations with singular initial data in the Heisenberg group

In this paper we obtain necessary conditions and sufficient conditions on the initial data for the solvability of fractional semilinear heat equations with power nonlinearities in the Heisenberg group $\mathbb{H}^N$. Using these conditions, we can prove that $1+2/Q$ separates the ranges of exponents of nonlinearities for the global-in-time solvability of the Cauchy problem (so-called the Fujita-exponent), where $Q=2N+2$ is the homogeneous dimension of $\mathbb{H}^N$, and identify the optimal strength of the singularity of the initial data for the local-in-time solvability. Furthermore, our conditions lead sharp estimates of the life span of solutions with nonnegative initial data having a polynomial decay at the space infinity.

math.AP↗

arXiv2023

Calderón-Zygmund operators and endpoint spaces for Hermite expansions

Let $L=-Δ+|x|^2$ be the Hermite operator on $\mathbb{R}^n$, and $T$ be a Calderón-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↗

arXiv2023

Sharp estimates for imaginary powers of Bessel operators

Let $L f(x):=-\frac{d^2}{dx^2}f(x)-\frac{ r}{x}\frac{d}{dx}f(x),\quad x>0, r>0$ be the Bessel operator on $((0,\infty), |\cdot|, x^rdx)$. In this paper, we prove the sharp weak type $(1,1)$ estimate for the imaginary power $L^{iα}, α\in \mathbb R$, of the Bessel operator.

math.CA↗

arXiv2022

Calderón-Zygmund operators on local hardy spaces

We give necessary and sufficient conditions for inhomogeneous Calderón-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↗