Harmonic Hardy spaces on smooth domains

The role of the mean curvature in a Hardy-Sobolev trace inequality

The Hardy-Sobolev trace inequality can be obtained via Harmonic extensions on the half-space of the Stein and Weiss weighted Hardy-Littlewood-Sobolev inequality. In this paper we consider a bounded domain and study the influence of the boundary mean curvature in the Hardy-Sobolev trace inequality on the underlying domain. We prove existence of minimizers when the mean curvature is negative at the singular point of the Hardy potential.

Weighted Hardy inequality with higher dimensional singularity on the boundary

Let $Ω$ be a smooth bounded domain in $\mahbb R^N$ with $N\ge 3$ and let $Σ_k$ be a closed smooth submanifold of $δΩ$ of dimension $1\le k\le N-2$. In this paper we study the weighted Hardy inequality with weight function singular on $Σ_k$. In particular we provide sufficient and necessary conditions for existence of minimizers.

齐次变指数Herz-Morrey-Hardy空间上由Riesz位势生成的交换子的若干范数不等式

Some norm inequalities for commutators generated by the Riesz potentials on homogeneous variable exponent Herz-Morrey-Hardy spaces

本文受文献[3]启发，利用原子分解方法，在齐次变指数Herz-Morrey-Hardy空间上证明了由Riesz位势生成的交换子的有界性，丰富了调和分析中经典算子的不等式研究。

具有二次哈密顿量的薛定谔方程的Hardy不确定性原理

Hardy's Uncertainty principle for Schrödinger equations with quadratic Hamiltonians

本文将经典Hardy不确定性原理扩展至具有二次哈密顿量的薛定谔方程的所有传播子（metaplectic operators），提供分数阶傅里叶变换等具体实例，并提出与辛投影结构相关的特定方向高斯衰减条件。

The Smoothness of kernel in Hardy spaces

This paper provides a study of problems related to Hardy spaces left by G.\,Weiss in \cite{We}. First, We will prove that the Hardy spaces $H^p(\mathbb{R}^n)$ can be characterized by a fixed Lipschitz function.

Characterizations of product Hardy spaces in Bessel setting

In this paper, we work in the setting of Bessel operators and Bessel Laplace equations studied by Weinstein, Huber, and the harmonic function theory in this setting introduced by Muckenhoupt--Stein, especially the generalised Cauchy--Riemann equations and the conjugate harmonic functions. We provide the equivalent characterizations of product Hardy spaces associated with Bessel operators in terms of the Bessel Riesz transforms, non-tangential and radial maximal functions defined via Poisson and heat semigroups, based on the atomic decomposition, the extension of Merryfield's result which conne…

高阶div-curl型估计在局部可定位Hardy空间上对于椭圆线性微分算子的应用

Higher order div-curl type estimates for elliptic linear differential operators on localizable Hardy spaces

本文研究了在局部可定位Hardy空间上，针对椭圆线性微分算子的高阶div-curl型估计，并开发了新的光滑原子分解方法和Poincaré型不等式证明。

齐次变指数Herz-Morrey-Hardy空间上Riesz位势的若干不等式

Some Inequalities for Riesz Potential on Homogeneous Variable Exponent Herz-Morrey-Hardy Spaces

本文研究调和分析中Riesz位势在齐次变指数Herz-Morrey-Hardy空间上的有界性，并在一定条件下给出了相应不等式。

On the Hardy-Poincaré inequality with boundary singularities

Let $Ø$ be a smooth bounded domain in $\R^N$ with $N\ge 1$. In this paper we study the Hardy-Poincaré inequality with weight function singular at the boundary of $Ø$. In particular we provide sufficient and necessary conditions on the existence of minimizers.

A note on Hardy's inequalities with boundary singularities

Let $Ø$ be a smooth bounded domain in $\R^N$ with $N\ge 1$. In this paper we study the Hardy-Poincaré inequalities with weight function singular at the boundary of $Ø$. In particular we give sufficient conditions so that the best constant is achieved.

Local smoothing and Hardy spaces for Fourier integral operators

We show that the Hardy spaces for Fourier integral operators form natural spaces of initial data when applying $\ell^{p}$-decoupling inequalities to local smoothing for the wave equation. This yields new local smoothing estimates which, in a quantified manner, improve the bounds in the local smoothing conjecture on $\mathbb{R}^{n}$ for $p\geq 2(n+1)/(n-1)$, and complement them for $2<p<2(n+1)/(n-1)$. These estimates are invariant under application of Fourier integral operators, and they are essentially sharp.

Moderate solutions of semilinear elliptic equations with Hardy potential

Let $Ω$ be a bounded smooth domain in $\mathbb{R}^N$. We study positive solutions of equation (E) $-L_μu+ u^q = 0$ in $Ω$ where $L_μ=Δ+ \fracμ{δ^2}$, $0<μ$, $q>1$ and $δ(x)=\mathrm{dist}\,(x,\partialΩ)$. A positive solution of (E) is moderate if it is dominated by an $L_μ$-harmonic function. If $μ<C_H(Ω)$ (the Hardy constant for $Ω$) every positive $L_μ$- harmonic functions can be represented in terms of a finite measure on $\partialΩ$ via the Martin representation theorem. However the classical measure boundary trace of any such solution is zero. We introduce a notion of normalized boundary t…

通过超调和函数与 Hardy 不等式建立非光滑区域中 Poisson 方程和热方程的 Sobolev 空间理论

Sobolev space theory for Poisson's and the heat equations in non-smooth domains via superharmonic functions and Hardy's inequality

本文利用超调和函数、边界距离权函数与 Hardy 不等式，证明非光滑区域中带零 Dirichlet 条件的 Poisson 方程和热方程在加权 Sobolev 空间中唯一可解，并给出多类区域的应用。

Weighted Hardy inequality on Riemannian manifolds

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $N\geq 3$ and we let $Σ$ to be a closed submanifold of dimension $1 \leq k \leq N-2. $ In this paper we study existence and non-existence of minimizers of Hardy inequality with weight function singular on $Σ$ within the framework of Brezis-Marcus-Shafrir. In particular we provide necessary and sufficient conditions for existence of minimizers.

Hardy-Sobolev inequality with higher dimensional singularity

For $N\geq 4$, we let $Ω$ to be a smooth bounded domain of $\mathbb{R}^N$, $Γ$ a smooth closed submanifold of $Ω$ of dimension $k$ with $1\leq k \leq N-2$ and $h$ a continuous function defined on $Ω$. We denote by $ρ_Γ\left(\cdot\right):=\dist_g\left(\cdot, Γ\right)$ the distance function to $Γ$. For $σ\in (0,2)$, we study existence of positive solutions $u \in H^1_0\left(Ω\right)$ to the nonlinear equation $$ -Δu+h u=ρ_Γ^{-σ} u^{2^*(σ)-1} \qquad \textrm{in } Ω, $$ where $2^*(σ):=\frac{2(N-σ)}{N-2}$ is the critical Hardy-Sobolev exponent. In particular, we provide existence of solution under t…

流形上傅里叶积分算子的局部光滑性与Hardy空间

Local smoothing and Hardy spaces for Fourier integral operators on manifolds

本研究引入有界几何黎曼流形上傅里叶积分算子的Hardy空间，得到满足曲率条件的算子及紧流形波动方程的最优局部光滑估计，并将其应用于初值非$L^2$型Sobolev空间的非线性波动方程。

$L^1$ Hardy inequalities with weights

We prove sharp homogeneous improvements to $L^1$ weighted Hardy inequalities involving distance from the boundary. In the case of a smooth domain, we obtain lower and upper estimates for the best constant of the remainder term. These estimates are sharp in the sense that they coincide when the domain is a ball or an infinite strip. In the case of a ball, we also obtain further improvements.

A new proof of the Hardy-Rellich inequality in any dimension

The Hardy-Rellich inequality in the whole space with the best constant was firstly proved by Tertikas and Zographopoulos in Adv. Math. (2007) in higher dimensions $N\geq 5$. Then it was extended to lower dimensions $N\in \{3, 4\}$ by Beckner in Forum Math. (2008) and Ghoussoub-Moradifam in Math. Ann. (2011) by applying totally different techniques. In this note we refine the method implemented by Tertikas and Zographopoulos, based on spherical harmonics decomposition, to give an easy and compact proof of the optimal Hardy-Rellich inequality in any dimension $N\geq 3$. In addition, we provide m…

Sharp Hardy and Hardy--Sobolev inequalities with point singularities on the boundary

We study the Hardy inequality when the singularity is placed on the boundary of a bounded domain in $\mathbb{R}^n$ that satisfies both an interior and exterior ball condition at the singularity. We obtain the sharp Hardy constant $n^2/4$ in case the exterior ball is large enough and show the necessity of the large exterior ball condition. We improve Hardy inequality with the best constant by adding a sharp Sobolev term. We next produce criteria that lead to characterizing maximal potentials that improve Hardy inequality. Breaking the criteria one produces successive improvements with sharp con…

分数Hardy不等式与子流形上的奇异性

Fractional Hardy inequality with singularity on submanifold

本文研究了子流形上分数Hardy不等式，考虑了距离边界倒数作为权函数，并在临界情况下提供了最优对数修正。

Large solutions to semilinear elliptic equations with Hardy potential and exponential nonlinearity

On a bounded smooth domain we study solutions of a semilinear elliptic equation with an exponential nonlinearity and a Hardy potential depending on the distance to the boundary of the domain. We derive global a priori bounds of the Keller-Osserman type. Using a Phragmen-Lindelof alternative for generalized sub and super-harmonic functions we discuss existence, nonexistence and uniqueness of so-called large solutions, i.e., solutions which tend to infinity at the boundary. The approach develops the one used by the same authors for a problem with a power nonlinearity instead of the exponential n…

Riesz Transform Characterization of Hardy Spaces Associated with Ball Quasi-Banach Function Spaces

Let $X$ be a ball quasi-Banach function space satisfying some mild assumptions and $H_X(\mathbb{R}^n)$ the Hardy space associated with $X$. In this article, the authors introduce both the Hardy space $H_X(\mathbb{R}^{n+1}_+)$ of harmonic functions and the Hardy space $\mathbb{H}_X(\mathbb{R}^{n+1}_+)$ of harmonic vectors, associated with $X$, and then establish the isomorphisms among $H_X(\mathbb{R}^n)$, $H_{X,2}(\mathbb{R}^{n+1}_+)$, and $\mathbb{H}_{X,2}(\mathbb{R}^{n+1}_+)$, where $H_{X,2}(\mathbb{R}^{n+1}_+)$ and $\mathbb{H}_{X,2}(\mathbb{R}^{n+1}_+)$ are, respectively, certain subspaces o…

Hardy and Hardy-Sobolev inequalities on Riemannian manifolds

Let $ (M,g) $ be a smooth compact Riemannian manifold of dimension $ N \geq 3 $. Given $p_0 \in M$, $λ\in \mathcal{R}$ and $σ\in (0,2]$, we study existence and non existence of minimizers of the following quotient: \begin{equation}\label{Paper Equation} μ_{λ,σ}=\inf_{u \in H^1(M)\setminus \lbrace0\rbrace} \frac{\displaystyle\int_M |\nabla u|^2 dv_g -λ\int_M u^2 dv_g }{\biggl(\displaystyle\int_M ρ^{-σ} |u|^{2^*(σ)} dv_g\biggl)^{2/2^*(σ)}}, \end{equation} where $ρ(.):=dist(p_0,.)$ denoted the geodesic distance from $p \in M$ to $p_0$. In particular for $σ=2$, we provide sufficient and necessary …

Large solutions of semilinear equations with Hardy potential

We consider equations of the form $-L_μu +f(u)=0$ in a smooth domain $Ω$, where $L_μ=Δ+ μδ^{-2}$ and $δ(x)$ denotes the distance of the point $x$ to the boundary of the domain. The nonlinear term $f$ is positive, increasing and convex on $(0,\infty)$, satisfies the Keller-Osserman condition and some additional technical assumptions. The conditions are satisfied, in particular, by power and exponential nonlinearities. We discuss the question of existence and uniqueness of large solutions when $μ>0$.

一维边界分数阶 Hardy 不等式：临界情形

Boundary fractional Hardy's inequality in dimension one: The critical case

本文证明一维临界情形 $sp=1$ 的分数阶边界 Hardy 不等式，确立其最优性，并给出收敛到常值函数的光滑函数序列示例。

与部分调和振子相关的里斯变换和索伯列夫空间

Riesz transforms and Sobolev spaces associated to the partial harmonic oscillator

本文通过热核估计、符号演算和算子因式分解，建立部分调和振子相关的索伯列夫空间，证明里斯变换有界性及空间等价性，并在对应势空间得到三类修正不等式。

Inequalities of Hardy-Sobolev type in Carnot-Carathéodory spaces

We consider various types of Hardy-Sobolev inequalities on a Carnot-Carathéodory space $(\Om, d)$ associated to a system of smooth vector fields $X=\{X_1, X_2,...,X_m\}$ on $\RR^n$ satisfying the Hörmander's finite rank condition $rank Lie[X_1,...,X_m] \equiv n$. One of our main concerns is the trace inequality \int_{\Om}|ϕ(x)|^{p}V(x)dx\leq C\int_{\Om}|Xϕ|^{p}dx,\qquad ϕ\in C^{\infty}_{0}(\Om), where $V$ is a general weight, i.e., a nonnegative locally integrable function on $\Om$, and $1<p<+\infty$. Under sharp geometric assumptions on the domain $\Om\subset \Rn$ that can be measured equival…

Regularity for solutions of H-systems and n-harmonic maps with n/2 square integrable derivatives

We study the regularity of weak solutions for two elliptic systems involving the $n$-Laplacian and a critical nonlinearity in the right hand side: $H$-systems and $n$-harmonic maps into compact Riemannian manifolds. Under the assumptions that the solutions belong to $W^{n/2,2}$ in an even dimension $n$, we prove their continuty. The tools used in the proof involve Hardy spaces and BMO, and the Rivière--Uhlenbeck decomposition (with estimates in Morrey spaces). A prominent role is played by the Coifman--Rochberg--Weiss commutator theorem.

Weak solutions of semilinear elliptic equations with Leray-Hardy potential and measure data

We study existence and stability of solutions of (E 1) --$Δ$u + $μ$ |x| 2 u + g(u) = $ν$ in $Ω$, u = 0 on $\partial$$Ω$, where $Ω$ is a bounded, smooth domain of R N , N $\ge$ 2, containing the origin, $μ$ $\ge$ -- (N --2) 2 4 is a constant, g is a nondecreasing function satisfying some integral growth assumption and $ν$ is a Radon measure on $Ω$. We show that the situation differs according $ν$ is diffuse or concentrated at the origin. When g is a power we introduce a capacity framework to find necessary and sufficient condition for solvability.

Regularity estimates for Green operators of Dirichlet and Neumann problems on weighted Hardy spaces

In this paper we first study the generalized weighted Hardy spaces $H^p_{L,w}(X)$ for $0<p\le 1$ associated to nonnegative self-adjoint operators $L$ satisfying Gaussian upper bounds on the space of homogeneous type $X$ in both cases of finite and infinite measure. We show that the weighted Hardy spaces defined via maximal functions and atomic decompositions coincide. Then we prove weighted regularity estimates for the Green operators of the inhomogeneous Dirichlet and Neumann problems in suitable bounded or unbounded domains including bounded semiconvex domains, convex regions above a Lipschi…

Boundary singularities of semilinear elliptic equations with Leray-Hardy potential

We study existence and uniqueness of solutions of (E 1) --$Δ$u + $μ$ |x| ^{-2} u + g(u) = $ν$ in $Ω$, u = $λ$ on $\partial$$Ω$, where $Ω$ $\subset$ R N + is a bounded smooth domain such that 0 $\in$ $\partial$$Ω$, $μ$ $\ge$ -- N 2 4 is a constant, g a continuous nondecreasing function satisfying some integral growth condition and $ν$ and $λ$ two Radon measures respectively in $Ω$ and on $\partial$$Ω$. We show that the situation differs considerably according the measure is concentrated at 0 or not. When g is a power we introduce a capacity framework which provides necessary and sufficient cond…

One dimensional Weighted Hardy's Inequalities and application

In the present paper we shall improve one dimensional weighted Hardy inequalities with one-sided boundary condition by adding sharp remainders. As an application, we shall establish n dimensional weighted Hardy inequalities in a bounded smooth domain with weight functions being powers of the distance function d(x) to the boundary. Our results will be applicable to variational problems in a coming paper.

Schr{ö}dinger operators with Leray-Hardy potential singular on the boundary

We study the kernel function of the operator u $\rightarrow$ L $μ$ u = --$Δ$u + $μ$ |x| 2 u in a bounded smooth domain $Ω$ $\subset$ R N + such that 0 $\in$ $\partial$$Ω$, where $μ$ $\ge$ -- N 2 4 is a constant. We show the existence of a Poisson kernel vanishing at 0 and a singular kernel with a singularity at 0. We prove the existence and uniqueness of weak solutions of L $μ$ u = 0 in $Ω$ with boundary data $ν$ + k$δ$ 0 , where $ν$ is a Radon measure on $\partial$$Ω$ \ {0}, k $\in$ R and show that this boundary data corresponds in a unique way to the boundary trace of positive solution of L …

Hardy spaces of the conjugate Beltrami equation

We study Hardy spaces of solutions to the conjugate Beltrami equation with Lipschitz coefficient on Dini-smooth simply connected planar domains, in the range of exponents $1<\infty$. We analyse their boundary behaviour and certain density properties of their traces. We derive on the way an analog of the Fatou theorem for the Dirichlet and Neumann problems associated with the equation ${div}(σ\nabla u)=0$ with $L^p$-boundary data.

Weak type estimates on certain Hardy spaces for smooth cone type multipliers

Let $\varrho\in C^{\infty} ({\Bbb R}^d\setminus\{0\})$ be a non-radial homogeneous distance function satisfying $\varrho(tξ)=t\varrho(ξ)$. For $f\in\frak S ({\Bbb R}^{d+1})$ and $δ>0$, we consider convolution operator ${\Cal T}^δ$ associated with the smooth cone type multipliers defined by $$\hat {{\Cal T}^δ f}(ξ,τ)= (1-\frac{\varrho(ξ)}{|τ|} )^δ_+\hat f (ξ,τ), (ξ,τ)\in {\Bbb R}^d \times \Bbb R.$$ If the unit sphere $Σ_{\varrho}\fallingdotseq\{ξ\in {\Bbb R}^d : \varrho(ξ)=1\}$ is a convex hypersurface of finite type and $\varrho$ is not radial, then we prove that ${\Cal T}^{δ(p)}$ maps from $H…

Hermite expansions and Hardy's theorem

Assuming that both a function and its Fourier transform are dominated by a Gaussian of large variance, it is shown that the Hermite coefficients of the function decay exponentially. A sharp estimate for the rate of exponential decay is obtained in terms of the variance, and in the limiting case (when the variance becomes so small that the Gaussian is its own Fourier transform), Hardy's theorem on Fourier transform pairs is obtained. A quantitative result on the confinement of particle-like states of a quantum harmonic oscillator is obtained. A stronger form of the result is conjectured. Furthe…

Navier-Stokes方程在$\mathbb{T}^2$上具有Hardy空间梯度的病态解

Pathological solutions of Navier-Stokes equations on $\mathbb{T}^2$ with gradients in Hardy spaces

本文针对二维Navier-Stokes方程，在任意光滑初值下构造了多个梯度属于Hardy空间$\mathcal{H}^p$（$p\in(0,1)$）的非零解，证明$p=1$为涡量路径空间非唯一性与唯一性的阈值，并发展了周期域上的Hardy空间理论。

开集上适应于椭圆算子的 Hardy 空间

Hardy spaces adapted to elliptic operators on open sets

本文研究了开集上带混合边界条件的椭圆算子，利用适应平方函数定义 Hardy 空间并导出其原子刻画，推广了 Lipschitz 域上纯 Dirichlet/Neumann 边界条件的已知结果，建立了调和分析与底层几何的联系。

重整化变分原理与Hardy型不等式

Renormalized variational principles and Hardy-type inequalities

本文在Hardy不等式成立的有界区域上建立包含Hardy与Trudinger不等式的指数可积性结果，并用重整化泛函给出Liouville方程极大解的变分刻画及整体H1界。

具有加权Hardy类函数的Hermite展开

Hermite expansions of functions from the weighted Hardy class

本文研究了具有加权Hardy类函数的Hermite展开，通过分析Log型权重和指数型权重，证明了Hermite系数衰减估计及谐振子解的衰减率，并改进了亚临界Hardy不确定性原理。

Real-Variable Theory of Hardy Spaces Associated with Generalized Herz Spaces of Rafeiro and Samko

This book is devoted to exploring properties of generalized Herz spaces and establishing a complete real-variable theory of Hardy spaces associated with local and global generalized Herz spaces via a totally fresh perspective which means that the authors view these generalized Herz spaces as special cases of ball quasi-Banach function spaces. To be precise, in this book, the authors first study some basic properties of generalized Herz spaces and obtain boundedness and compactness characterizations of commutators on them. Then the authors introduce the associated Herz-Hardy spaces, local Herz-…