Amjad Hussain, Naqash Sarfraz, Ferit Gurbuz
In this article, we establish sharp weak endpoint estimates for $p$-adic fractional Hardy operator. In addition, sharp weak bounds for p-adic Hardy operator on p-adic central Morrey space are also obtained.
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Bounds for $p$-adic Hardy-type Operators and Commutator On $p$-adic Variable Herz-Morrey Spaces
本文研究分数p-adic Hardy算子在变指数p-adic Herz-Morrey空间中的有界性，并进一步估计了其与p-adic变指数CBMO函数生成的交换子算子的有界性。
The present article focuses on the bounds of weighted multilinear $p$-adic Hardy operators on the product of Herz spaces and Morrey-Herz spaces. The corresponding norm on both the cases are also obtain.
Sharp bounds on higher-dimensional product spaces for Hardy-type operators on Heisenberg group
本文研究Heisenberg群上Hardy算子在更高维Lebesgue乘积空间上的锐界，获得锐界常数，并给出加权Hardy算子和加权Cesàro算子的有界性。
We give the optimal embedding relations between local Hardy space and $α$-modulation spaces, which extend the results for the embedding relations between local Hardy and modulation spaces obtained by Kobayashi, Miyachi and Tomita in [Studia Math. 192 (2009), 79-96].
The current status concerning Hardy-type inequalities with sharp constants is presented and described in a unified convexity way. In particular, it is then natural to replace the Lebesgue measure $dx$ with the Haar measure $dx/x.$ There are also derived some new two-sided Hardy-type inequalities for monotone functions, where not only the two constants are sharp but also where the involved function spaces are (more) optimal. As applications, a number of both well-known and new Hardy-type inequalities are pointed out. And, in turn, these results are used to derive some new sharp information conc…
This paper focuses on the operator norm of the truncated Hardy-Littlewood maximal operator $M^b_a$ and the strong truncated Hardy-Littlewood maximal operator $\tilde{M}^{\boldsymbol{b}}_{\boldsymbol{a}}$, respectively. We first present the $L^1$-norm of $M^b_a$, and then the $L^1$-norm of $\tilde{M}^{\boldsymbol{b}}_{\boldsymbol{a}}$ is given. Our study may have some enlightening significance for the research on sharp constant for the classical Hardy-Littlewood maximal inequality.
Four Jacobi settings are considered in the context of Hardy's inequality: the trigonometric polynomials and functions, and the corresponding symmetrized systems. In the polynomial cases sharp Hardy's inequality is proved for the type parameters $α,β\in(-1,\infty)^d$, whereas in the function systems for $α,β\in[-1/2,\infty)^d$. The ranges of these parameters are the widest in which the corresponding orthonormal bases are composed of bounded functions. Moreover, the sharp $L^1$-analogues of Hardy's inequality are obtained with the same restrictions on the parameters $α$ and $β$.
Sharp bound for m-linear $n$-dimensional Hardy-Littlewood-Polya operator in Morrey space on Heisenberg group
本文在Heisenberg群上的Morrey空间中，获得了m-线性n维Hardy-Littlewood-Pólya算子及Hilbert算子的尖锐界。
The restricted maximal operators of partial sums with respect to bounded Vilenkin systems are investigated. We derive the maximal subspace of positive numbers, for which this operator is bounded from the Hardy space $%H_{p}$ to the Lebesgue space $L_{p}$ for all $0<p\leq 1.$ We also prove that the result is sharp in a particular sense.
Sharp multi-dimensional Hardy's inequality for the Laguerre functions of Hermite type is proved for the type parameter $\al\in[-1/2,\infty)^d$. As a consequence we obtain the corresponding result for the generalized Hermite expansions. In particular, it validate that the known version of Hardy's inequality for the Hermite functions is sharp.
In this paper, Hardy type operator $H_β$ on $\bR^{n}$ and its adjoint operator $H_β^{*}$ are investigated. We use novel methods to obtain two main results. One is that we obtain the operators $H_β$ and $H_β^{*}$ being bounded from $L^{p}(|x|^α)$ to $L^{q,\infty}(|x|^γ)$, and the bounds of the operators $H_β$ and $H_β^{*}$ are sharp worked out. In particular, when $α=γ=0$, the norm of $H_β$ is equal to $1$. The other is that we study limiting weak-type behavior for the operator $H_β$ and its optimal form was obtained.
Mixed radial-angular bounds for Hardy-type operators on Heisenberg groups
该研究利用旋转方法，在Heisenberg群的径向-角度混合空间中建立Hardy算子、对偶算子及加权Hardy、Cesàro算子的精确界。
We provide an improvement of Calderón and Torchinsky's version of the Hörmander multiplier theorem on Hardy spaces $H^p$ ($0<p<\infty$), by replacing the Sobolev space $L_s^2(A_0)$ by the Lorentz-Sobolev space $L_s^{τ^{(s,p)} ,\min(1,p) }(A_0)$, where $τ^{(s,p)} =\frac{n}{s-(n/\min{(1,p)}-n)}$ and $A_0$ is the annulus $\{ξ\in \mathbb{R}^n: 1/2<|ξ|<2\}$. Our theorem also extends that of Grafakos and Slavíková to the range $0<p\le 1$. Our result is sharp in the sense that the preceding Lorentz-Sobolev space cannot be replaced by a smaller Lorentz-Sobolev space $L^{r,q}_s(A_0)$ with $r< τ^{(s,p)}…
In this paper, we investigate a class of fractional Hardy type operators $\mathscr{H}_{β_{1},\cdots,β_{m}}$ defined on higher-dimensional product spaces $\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}}$. We use novel methods to obtain two main results. One is that we obtain the operator $\mathscr{H}_{β_{1},\cdots,β_{m}}$ is bounded from $L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}},|x|^γ)$ to $L^{q}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}},|x|^α)$ and the bounds of the operator $\mathsc…
A method of proving Hardy's type inequality for orthogonal expansions is presented in a rather general setting. Then sharp multi-dimensional Hardy's inequality associated with the Laguerre functions of convolution type is proved for type index $\al\in[-1/2,\infty)^d$. The case of the standard Laguerre functions is also investigated. Moreover, the sharp analogues of Hardy's type inequality involving $L^1$ norms in place of $H^1$ norms are obtained in both settings.
We show that the sharp constant in the classical $n$-dimensional Hardy-Leray inequality can be improved for axisymmetric divergence-free fields, and find its optimal value. The same result is obtained for $n=2$ without the axisymmetry assumption.
We obtain sharp two-sided inequalities between $L^p-$norms $(1<p<\infty)$ of functions $Hf$ and $H^*f$, where $H$ is the Hardy operator, $H^*$ is its dual, and $f$ is a nonnegative measurable function on $(0,\infty).$ In an equivalent form, it gives sharp constants in the two-sided relations between $L^p$-norms of functions $H\f-\f$ and $\f$, where $\f$ is a nonnegative nonincreasing function on $(0,+\infty)$ with $\f(+\infty)=0.$ In particular, it provides an alternative proof of a result obtained by N. Kruglyak and E. Setterqvist (2008) for $p=2k (k\in \N)$ and by S. Boza and J. Soria (2011)…
We give a new proof of Hardy's uncertainty principle, up to the end-point case, which is only based on calculus. The method allows us to extend Hardy's uncertainty principle to Schrödinger equations with non-constant coefficients. We also deduce optimal Gaussian decay bounds for solutions to these Schrödinger equations.
Hardy's inequality on $H^p$ spaces, $p\in(0,1]$, in the context of orthogonal expansions is investigated for general basis on a subset of $\mathbb{R}^d$ with Lebesgue measure. The obtained result is applied to various Hermite, Laguerre, and Jacobi expansions. For that purpose some delicate estimates of the higher order derivatives for the underlying functions and of the associated heat kernels are proved. Moreover, sharpness of studied Hardy's inequalities is justified by a construction of an explicit counterexample, which is adjusted to all considered settings.
In this paper, by using the rotation method, we calculate that the sharp bound for $n$-dimensional Hardy operator $\mathcal{H}$ on mixed radial-angular spaces. Furthermore, we also obtain the sharp bound for $n$-dimensional fractional Hardy operator $\mathcal{H}_β$ from $L^p_{|x|}L_θ^{\bar{p}}({\Bbb R}^n)$ to $L^q_{|x|}L_θ^{\bar{q}}({\Bbb R}^n)$, where $0<β<n$, $1<p,q,\bar{p},\bar{q}<\infty$ and $1/p-1/q=β/n$. By using duality, the corresponding results for the dual operators $\mathcal{H}^*$ and $\mathcal{H}^*_β$ are also established. In addition, the sharp weak-type estimate for $\mathcal{H}$…
Sharp bounds for a class of integral operators in weighted-type spaces on Heisenberg group
本文利用文献[1]的结论与方法，给出海森堡群加权型空间中一类非负核积分算子的锐界，并推广得到Hardy算子、Hardy Littlewood-Pólya算子及Hilbert算子的锐界。
We give a short proof of a recently established Hardy-type inequality due to Keller, Pinchover, and Pogorzelski together with its optimality. Moreover, we identify the remainder term which makes it into an identity.
Sharp Hardy's Inequalities in Hilbert Spaces
本文研究了哈代不等式在连续和离散情形下的最优常数，确定了精确值及其收敛速率，并构造了极值函数与几乎极值序列。
We give sharp remainder terms of $L^{p}$ and weighted Hardy and Rellich inequalities on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. As consequences, we obtain analogues of the generalised classical Hardy and Rellich inequalities and the uncertainty principle on homogeneous groups. We also prove higher order inequalities of Hardy-Rellich type, all with sharp constants. A number of identities are derived including weighted and higher order types.
We study Hardy-type inequalities associated to the quadratic form of the shifted Laplacian $-Δ_{\mathbb H^N}-(N-1)^2/4$ on the hyperbolic space ${\mathbb H}^N$, $(N-1)^2/4$ being, as it is well-known, the bottom of the $L^2$-spectrum of $-Δ_{\mathbb H^N}$. We find the optimal constant in the resulting Poincaré-Hardy inequality, which includes a further remainder term which makes it sharp also locally. A related inequality under suitable curvature assumption on more general manifolds is also shown. Similarly, we prove Rellich-type inequalities associated with the shifted Laplacian, in which at …
We establish a sharp Adams-type inequality invoking a Hardy inequality for any even dimension. This leads to a non compact Sobolev embedding in some Orlicz space. We also give a description of the lack of compactness of this embedding in the spirit of \cite{Bahouri}.
A sharp logarithmic condition for the Hardy operator on $L^{1}(0,\infty)$ and $\ell^1$
本文研究了Hardy算子在 $L^{1}(0,\infty)$ 和 $\ell^1$ 上的有界性问题，引入修正后的Hardy算子并确定其可映射到可积函数空间的最大子空间，得出一个精确的对数可积性条件。
Sharp variational inequalities for the Hardy-Littlewood maximal operator on finite undirected graphs
本文研究了 Hardy-Littlewood 最大算子在完全图上的尖锐 p-变分不等式，回答了 Liu 和 Xue 的问题，并通过计算找到连接图上最多五个顶点的 1-变分不等式尖锐常数，提出路径图的猜想。
A general class of weighted multilinear Hardy-Cesàro operators that acts on the product of Lebesgue spaces and central Morrey spaces. Their sharp bounds are also obtained. In addition, we obtain sufficient and necessary conditions on weight functions so that the commutators of these weighted multilinear Hardy-Cesàro operators (with symbols in central BMO space) are bounded on the product of central Morrey spaces. These results extends known results on multilinear Hardy operators.
In this paper we characterize subsequences of Fejér means with respect to Vilenkin systems, which are bounded from the Hardy space $H_{p}$ to the Lebesgue space $L_{p},$ for all $0<p<1/2.$ The result is in a sense sharp.
We establish sharp Hardy-Adams inequalities on hyperbolic space $\mathbb{B}^{4}$ of dimension four. Namely, we will show that for any $α>0$ there exists a constant $C_α>0$ such that \[ \int_{\mathbb{B}^{4}}(e^{32π^{2} u^{2}}-1-32π^{2} u^{2})dV=16\int_{\mathbb{B}^{4}}\frac{e^{32π^{2} u^{2}}-1-32π^{2} u^{2}}{(1-|x|^{2})^{4}}dx\leq C_α. \] for any $u\in C^{\infty}_{0}(\mathbb{B}^{4})$ with \[ \int_{\mathbb{B}^{4}}\left(-Δ_{\mathbb{H}}-\frac{9}{4}\right)(-Δ_{\mathbb{H}}+α)u\cdot udV\leq1. \] As applications, we obtain a sharpened Adams inequality on hyperbolic space $\mathbb{B}^{4}$ and an inequal…
Sharp Hardy-Leray inequality for solenoidal fields
本文计算了 $\mathbb{R}^N$ 上无源向量场Hardy-Leray不等式的最佳常数，在无对称性假设下推导并简化了该常数，并证明其在积分有限的函数空间内无法达到。
In this paper, we got several sharp Hardy-Littlewood-Sobolev-type inequalities on quaternionic Heisenberg groups (a general form due to Folland and Stein [FS74]), using the symmetrization-free method in a paper of Frank and Lieb [FL12], where they considered the analogues on classical Heisenberg group. First, we give the sharp Hardy-Littlewood-Sobolev inequalities, both on quaternionic Heisenberg group and its equivalent on quaternionic sphere for exponent bigger than 4. The extremizer, as we guess, is almost uniquely constant function on sphere. Then their dual form, sharp conformally-invaria…
Sharp weak bounds for the Hardy-Littlewood-Pólya operator and the weak bounds for the multilinear integral operator
本文获得了n维Hardy-Littlewood-Pólya算子从加权Lebesgue空间到加权弱Lebesgue空间的算子范数，以及m线性n维积分算子的弱界。
This paper continues the program initiated in the works by the authors [60], [61] and [62] and by the authors with Li [51] and [52] to establish higher order Poincaré-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on real hyperbolic spaces using the method of Helgason-Fourier analysis on the hyperbolic spaces. The aim of this paper is to establish such inequalities on the Siegel domains and complex hyperbolic spaces. Firstly, we prove a factorization theorem for the operators on the complex hyperbolic space which is closely related to Geller' operator, as well as the CR inva…
By using, among other things, the Fourier analysis techniques on hyperbolic and symmetric spaces, we establish the Hardy-Sobolev-Maz'ya inequalities for higher order derivatives on half spaces. The proof relies on a Hardy-Littlewood-Sobolev inequality on hyperbolic spaces which is of its independent interest. We also give an alternative proof of Benguria, Frank and Loss' work concerning the sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half space. Finally, we show the sharp constant in the Hardy-Sobolev-Maz'ya inequality for bi-Laplacian in the upper half…
In this paper, we investigate the sharp Hardy-Littlewood-Sobolev inequalities on the Heisenberg group. On one hand, we apply the concentration compactness principle to prove the existence of the maximizers. While the approach here gives a different proof under the special cases discussed in a recent work of Frank and Lieb, we generalize the result to all admissible cases. On the other hand, we provide the upper bounds of sharp constants for these inequalities.
To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is shown that the sharp factor is meaningful for each finite interval and a classical sharp model is re-examined.
Though Adams and Hardy-Adams inequalities can be extended to general symmetric spaces of noncompact type fairly straightforwardly by following closely the systematic approach developed in our early works on real and complex hyperbolic spaces, higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities are more difficult to establish. The main purpose of this goal is to establish the Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane. A crucial part of our work is to establish appropriate factorization theorems on the…
The sharp reverse Hardy inequality in BMO for nonincreasing functions
本文针对正实数集上的Hardy算子，证明了非增局部可积函数的反向Hardy不等式中Korenovskii提出的因子e/4可去除，得到最优常数α₀，解决了相关猜想。
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 co…
In this paper we prove sharp weighted Hardy-type inequalities on Carnot groups with the homogeneous norm $N=u^{1/(2-Q)}$ associated to Folland's fundamental solution $u$ for the sub-Laplacian $Δ_{\mathbb{G}}$. We also prove uncertainty principle, Caffarelli-Kohn-Nirenberg and Rellich inequalities on Carnot groups.
Using the Fourier analysis techniques on hyperbolic spaces and Green's function estimates, we confirm in this paper the conjecture given by the same authors in [43]. Namely, we prove that the sharp constant in the $\frac{n-1}{2}$-th order Hardy-Sobolev-Maz'ya inequality in the upper half space of dimension $n$ coincides with the best $\frac{n-1}{2}$-th order Sobolev constant when $n$ is odd and $n\geq9$ (See Theorem 1.6). We will also establish a lower bound of the coefficient of the Hardy term for the $k-$th order Hardy-Sobolev-Maz'ya inequality in upper half space in the remaining cases of d…
Sharp gradient stability for a class of Hardy-Sobolev-Maz'ya inequalities
本文针对具有部分（更强）奇异权和非径向极值函数的一类Hardy-Sobolev-Maz'ya不等式，证明了尖锐梯度稳定性，这是首个关于非径向极值函数的稳定性结果。
On Hardy spaces associated with the twisted Laplacian and sharp estimates for the corresponding wave operator
本文刻画了扭曲 Laplace 算子对应的 Hardy 空间，并利用原子分解证明了相关波动算子从 Hardy 空间到 $L^p$ 的尖锐有界性条件。
We prove a one-dimensional Hardy inequality on the halfline with sharp constant, which improves the classical form of this inequality. As a consequence of this new inequality we can rederive known doubly weighted Hardy inequalities. Our motivation comes from the theory of Schrödinger operators and we explain the use of Hardy inequalities in that context.
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