Angelo Alvino, Adele Ferone, Anna Mercaldo, Futoshi Takahashi, Roberta Volpicelli
We prove an improved version of the trace-Hardy inequality, so-called Kato's inequality, on the half-space in Finsler context. The resulting inequality extends the former one obtained by \cite{AFV} in Euclidean context. Also we discuss the validity of the same type of inequalities on open cones.
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In this paper we present a unified simple approach to anisotropic Hardy inequalities in various settings. We consider Hardy inequalities which involve a Finsler distance from a point or from the boundary of a domain. The sharpness and the non-attainability of the constants in the inequalities are also proved.
Series expansion of weighted Finsler-Kato-Hardy inequalities
本文研究涉及一般 Finsler 度量的加权各向异性 Hardy 不等式与迹 Hardy 不等式，通过统一方法建立尖锐插值，并在有界域中添加余项实现无限级数改进，将欧几里得背景下的结果推广至 Finsler 框架及锥情形。
We establish a bipolar Hardy inequality on complete, not necessarily reversible Finsler manifolds. We show that our result strongly depends on the geometry of the Finsler structure, namely on the reversibility constant $r_F$ and the uniformity constant $l_F$. Our result represents a Finslerian counterpart of the Euclidean multipolar Hardy inequality due to Cazacu and Zuazua (2013) and the Riemannian case considered by Faraci, Farkas and Kristály (2018).
In this paper we derive both local and global geometric inequalities on general Riemannnian and Finsler manifolds and prove generalized Caffarelli-Kohn-Nirenberg type and Hardy type inequalities on Finsler manifolds, illuminating curvatures of both Riemannian and Finsler manifolds influence geometric inequalities.
We establish Hardy inequalities involving a weight function on complete, not necessarily reversible Finsler manifolds. We prove that the superharmonicity of the weight function provides a sufficient condition to obtain Hardy inequalities. Namely, if $ρ$ is a nonnegative function and $-\boldsymbolΔ ρ\geq 0$ in weak sense, where $\boldsymbolΔ$ is the Finsler-Laplace operator defined by $ \boldsymbolΔ ρ= \mathrm{div}(\boldsymbol{\nabla} ρ)$, then we obtain the generalization of some Riemannian Hardy inequalities given in D'Ambrosio and Dipierro (Ann. Inst. H. Poincaré, 2013). By extending the res…
In this paper we deal with a class of inequalities which interpolate the Kato's inequality and the Hardy's inequality in the half space. Starting from the classical Hardy's inequality in the half space $\rnpiu =\R^{n-1}\times(0,\infty)$, we show that, if we replace the optimal constant $\frac{(n-2)^2}{4}$ with a smaller one $\frac{(β-2)^2}{4}$, $2\le β<n$, then we can add an extra trace-term equals to that one that appears in the Kato's inequality. The constant in the trace remainder term is optimal and it tends to zero when $β$ goes to $n$, while it is equal to the optimal constant in the Kat…
Optimal Hardy-weights for the Finsler $p$-Dirichlet integral with a potential
本文研究了Finsler p-Dirichlet积分带势函数时的最优Hardy权重构造问题，提出了在特定条件下的数学方法和理论贡献。
Let $0<σ<n/2$ and $H=(-Δ)^σ+V(x)$ be Schrödinger type operators on $\mathbb R^n$ with a class of scaling-critical potentials $V(x)$, which include the Hardy potential $a|x|^{-2σ}$ with a sharp coupling constant $a\ge -C_{σ,n}$ ($C_{σ,n}$ is the best constant of Hardy's inequality of order $σ$). In the present paper we consider several sharp global estimates for the resolvent and the solution to the time-dependent Schrödinger equation associated with $H$. In the case of the subcritical coupling constant $a>-C_{σ,n}$, we first prove {\it uniform resolvent estimates} of Kato--Yajima type for all …
Sharp Hardy and spectral gap inequalities on special irreversible Finsler manifolds
本文研究特殊不可逆Finsler流形上Hardy和谱间隙不等式的sharpness，通过构造具体例子和条件证明其有效性。
Finsler $p$-Laplace equation with a potential: Maz'ya-type characterization and attainments of the Hardy constant
本文研究带位势的Finsler $p$-Laplace方程，通过建立Bregman距离双边估计得到Hardy权重的Maz'ya型刻画，并给出Hardy常数可达的充分条件。
Combining the sharp isoperimetric inequality established by Z. Balogh and A. Kristály [Math. Ann., in press, doi:10.1007/s00208-022-02380-1] with an anisotropic symmetrization argument, we establish sharp Morrey-Sobolev inequalities on $n$-dimensional Finsler manifolds having nonnegative $n$-Ricci curvature. A byproduct of this method is a Hardy-Sobolev-type inequality in the same geometric setting. As applications, by using variational arguments, we guarantee the existence/multiplicity of solutions for certain eigenvalue problems and elliptic PDEs involving the Finsler-Laplace operator. Our r…
On the sharp multi-bubble stability for fractional Hardy-Sobolev equations -- A quantitative approach in low dimensions
本文在低维情形下研究了分数 Hardy-Sobolev 方程的多气泡稳定性，通过定量方法证明了能量接近气泡叠加的函数的刚性，并构造反例证明线性速率最优。
This paper is devoted to investigate an interpolation inequality between the Brezis-Vázquez and Poincaré inequalities (shortly, BPV inequality) on nonnegatively curved spaces. As a model case, we first prove that the BPV inequality holds on any Minkowski space, by fully characterizing the existence and shape of its extremals. We then prove that if a complete Finsler manifold with nonnegative Ricci curvature supports the BPV inequality, then its flag curvature is identically zero. In particular, we deduce that a Berwald space of nonnegative Ricci curvature supports the BPV inequality if and onl…
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