arXiv subjects

Huoxiong Wu

Publications and source records attributed to Huoxiong Wu.

At least 19 recordsLinked to original sources

arXiv2026

BMO-Type characterizations and endpoint estimates for commutators of intrinsic square functions on Orlicz-Hardy spaces

The commutators of intrinsic square functions associated with the intrinsic Littlewood--Paley $g$-function, the intrinsic $g_λ^{*}$-function and the intrinsic Lusin area function can be used to characterize BMO-type function spaces through their boundedness properties. In this paper, we show that, for $b\in {\rm BMO}(\mathbb{R}^{n})$, the boundedness of each of these commutators from the Musielak--Orlicz Hardy space $H^φ(\mathbb{R}^{n})$ to $L^φ(\mathbb{R}^{n})$ is equivalent to $b\in {\rm BMO}_φ(\mathbb{R}^{n})$ (a nontrivial subspace of $\rm{BMO}(\mathbb{R}^{n})$), under suitable assumptions on the growth function $φ$. This extends the weighted characterization of Han and Wu [Proc. Amer. Math. Soc. 152(1) (2024), 281--293] to the Musielak--Orlicz setting. In addition, under suitable assumptions on a growth function $Φ$, we prove that, for $b\in {\rm BMO}_Φ(\mathbb{R}^{n})$, these commutators are bounded from the $b$-adapted Orlicz--Hardy space $H_{b}^Φ(\mathbb{R}^{n})$ to $L^{1}(\mathbb{R}^{n})$ and from Orlicz Hardy space $H^Φ(\mathbb{R}^{n})$ to $L^{1,\infty}(\mathbb{R}^{n})$.

math.CA↗

arXiv2026

On the relationship between block spaces and Orlicz spaces

Let $1 -1$. Let $(X,d,μ)$ be an $s$-Ahlfors-regular quasi-metric measure space. Suppose that $B^{0,v}_q(X)$ is the block space which consists of all functions that admit a decomposition into $q$-blocks supported on balls. In this paper, we study the relationship between the block space $B^{0,v}_q(X)$ and the Orlicz-type space $L(\log^+\!\!L)^{1+v}(X)$. More precisely, we show that the block space $B_q^{0,v}(X)$ is a proper subspace of the Orlicz space $L(\log^+\!\!L)^{1+v}(X)$ for any fixed $1 -1$. Namely, $$B_q^{0,v}(X)\subsetneq L(\log^+\!\!L)^{1+v}(X),$$ which gives a confirmed answer to a longstanding open problem concerning the relationship between block spaces and Orlicz-type spaces on the unit sphere $\mathbb S^{n-1}$. We further show that $L(\log^+\!\!L)^{1+v}(X)$ is the smallest Orlicz-type space containing $B^{0,v}_{q}(X)$. We also introduce a generalized block space $\mathscr B_q^{0,v}(X)$ that depends only on the measure structure and show that this space is equivalent to the Orlicz space $L(\log^+\!\!L)^{1+v}(X)$ when $μ(X)<\infty$. Finally, we consider two special cases that further clarify the roles of the parameter $q$ and the logarithmic weight.

math.CA↗

arXiv2026

The derivative of the fractional discrete Laplacian is an exotic Riesz potential

Let $Δ_{N}$ be the multidimensional discrete Laplacian on $\mathbb{Z}^N$ ($N\ge1$). In this note, we prove that, when $N=1$, the right hand derivative of $(-Δ_1)^s$ at $0$ is an exotic discrete Riesz potential (namely, the endpoint case: the order is 0) in Stein-Wainger sense (J. Anal. Math. 2000), and when $N\ge 2$, the corresponding derivative is also an exotic discrete Riesz potential with an additional corrector. A similar conclusion for the left hand derivative case is also considered. All results obtained in this note extend the logarithmic Laplacian of Chen-Weth (Comm. PDEs. 2019) to the discrete setting.

math.AP↗

arXiv2026

Characterizations of a class of Musielak--Orlicz BMO spaces via commutators of Riesz potential operators

The fractional integral operators $I_α$ can be used to characterize the Musielak--Orlicz Hardy spaces. This paper shows that for $b\in \rm BMO(\mathbb R^n)$, the commutators $[b,I_α]$ generated by fractional integral operators $I_α$ with $b$ are bounded from the Musielak--Orlicz Hardy spaces $H^{φ_1}(\mathbb R^n)$ to the Musielak--Orlicz spaces $L^{φ_2}(\mathbb R^n)$ (where $1<u<\infty$ and $φ_1$, $φ_2$ are growth functions) if and only if $b\in \mathcal {BMO}_{φ_1,u}(\mathbb R^n)$, which are a class of non-trivial subspaces of $\rm BMO(\mathbb R^n)$. Additionally, we obtain the boundedness of the commutator $[b,I_α]$ from $H^{φ_1}(\mathbb R^n)$ to $H^{φ_2}(\mathbb R^n)$. The corresponding results are also provided for commutators of fractional integrals associated with general homogeneous kernels.

math.CA↗

arXiv2025

Strichartz estimates involving orthonormal systems at the critical summability exponent

The primary objective of this paper is to investigate the orthonormal Strichartz estimates at the critical summability exponent for the Schrödinger operator $e^{itΔ}$ with initial data from the homogeneous Sobolev space $\dot{H}^s (\mathbb{R}^n)$. We prove new global strong-type orthonormal Strichartz estimates in the interior of $ODCA$ at the optimal summability exponent $α=q$, thereby substantially supplymenting the work of Bez-Hong-Lee-Nakamura-Sawano \cite{Bez-Hong-Lee-Nakamura-Sawano}. Our approach is based on restricted weak-type orthonormal estimates, real interpolation argument and the advantageous condition $q<p$ in the interior of $ODCA$.

math.AP↗

arXiv2025

Orthonormal Strichartz estimates for Dunkl-Schrödinger equation of initial data with Sobolev regularity

Let $Δ_κ$ be the Dunkl-Laplacian on $\mathbb{R}^n$. The main aim of this paper is to investigate the orthonormal Strichartz estimates for the Schrödinger equation with initial data from the homogeneous Dunkl-Sobolev space $\dot{H}_κ^s (\mathbb{R}^n)$. Our approach is based on restricted weak-type orthonormal estimates, frequency-localized estimates for the Dunkl-Schrödinger propagator $e^{itΔ_κ}$, and a series of successive real and complex interpolation techniques.

math.FA↗

arXiv2025

Matrix weighted inequalities for fractional type integrals associated to operators with new classes of weights

Let $e^{-tL}$ be a analytic semigroup generated by $-L$, where $L$ is a non-negative self-adjoint operator on $L^2(\mathbb{R}^d)$. Assume that the kernels of $e^{-tL}$, denoted by $p_t(x,y)$, only satisfy the upper bound: for all $N>0$, there are constants $c,C>0$ such that \begin{align}\label{upper bound} |p_t(x,y)|\leq\frac{C}{t^{d/2}}e^{-\frac{|x-y|^2}{ct}}\Big(1+\frac{\sqrt{t}}{ρ(x)}+ \frac{\sqrt{t}}{ρ(y)}\Big)^{-N} \end{align} holds for all $x,y\in\mathbb{R}^d$ and $t>0$. We first establish the quantitative matrix weighted inequalities for fractional type integrals associated to $L$ with new classes of matrix weights, which are nontrivial extension of the results established by Li, Rahm and Wick [23]. Next, we give new two-weight bump conditions with Young functions satisfying wider conditions for fractional type integrals associated to $L$, which cover the result obtained by Cruz-Uribe, Isralowitz and Moen [6]. We point out that the new classes of matrix weights and bump conditions are larger and weaker than the classical ones given in [17] and [6], respectively. As applications, our results can be applied to settings of magnetic Schrödinger operator, Laguerre operators, etc.

math.CA↗

arXiv2025

The uniform quantitive weighted boundedness of fractional Marcinkiewicz integral and its commutator

Suppose that $Ω\in L^{\infty}(\mathbb{S} ^{n-1})$ is homogeneous of degree zero with mean value zero. Then we consider a fractional type Marcinkiewicz integral operator $$μ_{Ω,β}f(x) = \left ( \int_{0}^{\infty } \left | \int_{\left | x-y \right |\le t }^{} \frac{Ω(x-y)}{\left | x-y \right |^{n-1-β} } f(y)dy \right | ^{2}\frac{dt}{t^3} \right )^{\frac{1}{2} },\quad 0<β<n.$$ Our main contribution is the quantitive weighted result of the classical Marcinkiewicz integral $μ_Ω$ proved by Hu and Qu [Math. Ineq. appl., 22(2019), 885-899] can be recovered from the quantitative weighted estimates of $μ_{Ω,β}$ in this paper when $β\to 0^+$. As inference, we also gives the uniform quantitive weighted bounds for the corresponding fractional commutators of $μ_{Ω,β}$ when $β\rightarrow 0^+$.

math.CA↗

arXiv2025

Weighted variational inequalities for heat semigroups associated with Schrödinger operators related to critical radius functions

Let $\mathcal{L}$ be a Schrödinger operator and $\mathcal{V}_\varrho(e^{-t\mathcal{L}})$ be the variation operator of heat semigroup associated to $\mathcal{L}$ with $\varrho>2$. In this paper, we first obtain the quantitative weighted $L^p$ bounds for $\mathcal{V}_\varrho(e^{-t\mathcal{L}})$ with a class of weights related to critical radius functions, which contains the classical Muckenhoupt weights as a proper subset. Next, a new bump condition, which is weaker than the classical bump condition, is given for two-weight inequality of $\mathcal{V}_\varrho(e^{-t\mathcal{L}})$, and the weighted mixed weak type inequality corresponding to Sawyer's conjecture for $\mathcal{V}_\varrho(e^{-t\mathcal{L}})$ are obtained. Furthermore, the quantitative restricted weak type $(p,p)$ bounds for $\mathcal{V}_\varrho(e^{-t\mathcal{L}})$ are also given with a new class of weights $A_{p}^{ρ,θ,\mathcal{R}}$, which is larger than the classical $A_{p}^{\mathcal{R}}$ weights. Meanwhile, several characterizations of $A_{p,q,α}^{ρ,θ,\mathcal{R}}$ in terms of restricted weak type estimates of maximal operators are established.

math.CA↗

arXiv2024

Orthonormal Strichartz inequalities and their applications on abstract measure spaces

The main objective of this paper is to extend certain fundamental inequalities from a single function to a family of orthonormal systems. In the first part of the paper, we consider a non-negative, self-adjoint operator $L$ on $L^2(X,μ)$, where $(X,μ)$ is a measure space. Under the assumption that the kernel $K_{it}(x,y)$ of the Schrödinger propagator $e^{itL}$ satisfies a uniform $L^\infty$-decay estimate of the form \begin{equation*} \sup_{x,y\in X}|K_{it}(x,y)|\lesssim |t|^{-\frac{n}{2}},\,|t|<T_0, \text{ for some }n\geq1, \end{equation*} where $T_0\in(0,+\infty]$, we establish Strichartz estimates for the Schrödinger propagator $e^{itL}$ and using a duality principle argument by Frank-Sabin \cite{FS}, we extend it for a system of infinitely many fermions on $L^2(X)$. We also obtain orthonormal Strichartz estimates for a class of dispersive semigroup $U(t)=e^{itϕ(L)}ψ(\sqrt{L}),$ where $ϕ: \mathbb{R}^+\rightarrow \mathbb{R}$ is a smooth function and $ψ\in C_c^\infty([\frac{1}{2},2])$. As an application of these orthonormal versions of Strichartz estimates, we prove the well-posedness for the Hartree equation in the Schatten spaces. In the next part of the paper, we obtain some new orthonormal Strichartz estimates, which extend prior work of Kenig-Ponce-Vega \cite{Kenig-Ponce-Vega} for single functions. Using those orthonormal versions of Kenig-Ponce-Vega result, we prove the orthonormal restriction theorem for the Fourier transform on some particular noncompact hypersurface of the form $S=\{(ξ, ϕ(ξ): ξ\in \mathbb{R})\}$, where $ϕ$ satisfies certain growth condition.

math.FA↗

arXiv2023

Decay estimates for a class of semigroups related to self-adjoint operators on metric measure spaces

Assume that $(X,d,μ)$ is a metric space endowed with a non-negative Borel measure $μ$ satisfying the doubling condition and the additional condition that $μ(B(x,r))\gtrsim r^n$ for any $x\in X, \,r>0$ and some $n\geq1$. Let $L$ be a non-negative self-adjoint operator on $L^2(X,μ)$. We assume that $e^{-tL}$ satisfies a Gaussian upper bound and the Schrödinger operator $e^{itL}$ satisfies an $L^1\to L^\infty$ decay estimate of the form \begin{equation*} \|e^{itL}\|_{L^1\to L^\infty} \lesssim |t|^{-\frac{n}{2}}. \end{equation*} Then for a general class of dispersive semigroup $e^{itϕ(L)}$, where $ϕ: \mathbb{R}^+ \to \mathbb{R}$ is smooth, we establish a similar $L^1\to L^\infty$ decay estimate by a suitable subordination formula connecting it with the Schrödinger operator $e^{itL}$. As applications, we derive new Strichartz estimates for several dispersive equations related to Hermite operators, twisted Laplacians and Laguerre operators.

math.AP↗

arXiv2021

Well-posedness of Navier-Stokes equations established by the decaying speed of single norm

The decaying speed of a single norm more truly reflects the intrinsic harmonic analysis structure of the solution of the classical incompressible Navier-Stokes equations. No previous work has been able to establish the well-posedness under the decaying speed of a single norm with respect to time, and the previous solution space is contained in the intersection of two spaces defined by different norms. In this paper, for some separable initial space $X$, we find some new solution space which is not the subspace of $L^{\infty}(X)$. We use parametric Meyer wavelets to establish the well-posedness via the decaying speed of a single norm only, without integral norm to $t$.

math.AP↗

arXiv2021

On the Bounds of Weak $(1,1)$ Norm of Hardy-Littlewood Maximal Operator with $L\log L({\mathbb S^{n-1}})$ Kernels

Let $Ω\in L^1{({\mathbb S^{n-1}})}$, be a function of homogeneous of degree zero, and $M_Ω$ be the Hardy-Littlewood maximal operator associated with $Ω$ defined by $M_Ω(f)(x) = \sup_{r>0}\frac1{r^n}\int_{|x-y| λ\}| = n^{-1}\|Ω\|_{L^1({\mathbb S^{n-1}})}\|f\|_{L^1({\mathbb R^n})}.$$ This removes the smoothness restrictions on the kernel $Ω$, such as Dini-type conditions, in previous results. To prove our result, we present a new upper bound of $\|M_Ω\|_{L^1\to L^{1,\infty}}$, which essentially improves the upper bound $C(\|Ω\|_{L\log L({\mathbb S^{n-1}})}+1)$ given by Christ and Rubio de Francia. As a consequence, the upper and lower bounds of $\|M_Ω\|_{L^1\to L^{1,\infty}}$ are obtained for $Ω\in L\log L {({\mathbb S^{n-1}})}$.

math.CA↗

arXiv2021

A remark on ill-posedness

Norm inflation implies certain discontinuous dependence of the solution on the initial value. The well-posedness of the mild solution means the existence and uniqueness of the fixed points of the corresponding integral equation. For ${\rm BMO}^{-1}$, Auscher-Dubois-Tchamitchian proved that Koch-Tataru's solution is stable. In this paper, we construct a non-Gauss flow function to show that, for classic Navier-Stokes equations, wellposedness and norm inflation may have no conflict and stability may have meaning different to $L^{\infty}(({\rm BMO}^{-1})^{n})$.

math.AP↗

arXiv2021

Bump conditions and two-weight inequalities for commutators of fractional integrals

This paper gives new two-weight bump conditions for the sparse operators related to iterated commutators of fractional integrals. As applications, the two-weight bounds for iterated commutators of fractional integrals under more general bump conditions are obtained. Meanwhile, the necessity of two-weight bump conditions as well as the converse of Bloom type estimates for iterated commutators of fractional integrals are also given.

math.CA↗

arXiv2021

Limiting weak-type behaviors for singular integrals with rough $L\log L(\mathbb{S}^n)$ kernels

Let $Ω$ be a function of homogeneous of degree zero and vanish on the unit sphere $\mathbb {S}^n$. In this paper, we investigate the limiting weak-type behavior for singular integral operator $T_Ω$ associated with rough kernel $Ω$. We show that, if $Ω\in L\log L(\mathbb S^{n})$, then $\lim_{λ\to0^+}λ|\{x\in\mathbb{R}^n:|T_Ω(f)(x)|>λ\}| = n^{-1}\|Ω\|_{L^1(\mathbb {S}^n)}\|f\|_{L^1(\mathbb{R}^n)},\quad0\le f\in L^1(\mathbb{R}^n).$ Moreover,$(n^{-1}\|Ω\|_{L^1(\mathbb{S}^{n-1})}$ is a lower bound of weak-type norm of $T_Ω$ when $Ω\in L\log L(\mathbb{S}^{n-1})$. Corresponding results for rough bilinear singular integral operators defined in the form $T_{\vecΩ}(f_1,f_2) = T_{Ω_1}(f_1)\cdot T_{Ω_2}(f_2)$ have also been established.

math.CA↗

arXiv2021

A note on extrapolation of compactness

This note is devoted to the study of Hytönen's extrapolation theorem of compactness on weighted Lebesgue spaces. Two criteria of compactness of linear operators in the two-weight setting are obtained. As applications, we obtain two-weight compactness of commutators of Calderón--Zygmund operators, fractional integrals and bilinear Calderón--Zygmund operators.

math.AP↗

arXiv2021

Sparse dominations and weighted variation inequalities for singular integrals and commutators

This paper gives the pointwise sparse dominations for variation operators of singular integrals and commutators with kernels satisfying the $L^r$-Hörmander conditions. As applications, we obtain the strong type quantitative weighted bounds for such variation operators as well as the weak-type quantitative weighted bounds for the variation operators of singular integrals and the quantitative weighted weak-type endpoint estimates for variation operators of commutators, which are completely new even in the unweighted case. In addition, we also obtain the local exponential decay estimates for such variation operators.

math.CA↗