The main results in the paper are the weighted multipolar Hardy inequalities \begin{equation*} c\int_{\R^N}\sum_{i=1}^n\frac{u^2}{|x-a_i|^2}\,dμ\leq\int_{\R^N}|\nabla u |^2dμ+ K\int_{\R^N} u^2dμ, \end{equation*} in $\R^N$ for any $u$ in a suitable weighted Sobolev space, with $0<c\le c_{o,μ}$, $a_1,\dots,a_n\in \R^N$, $K$ constant. The weight functions $μ$ are of a quite general type.
The paper fits in the framework of the study of Kolmogorov operators \begin{equation*} Lu=Δu+\frac{\nabla μ}μ\cdot\nabla u, \end{equation*} perturbed by multipolar inverse square potentials, and of the related evolution problems.
The necessary and sufficient conditions for the existence of positive exponentially bounded in time solutions to the associated initial value problem are based on weighted Hardy inequalities. The optimality of the constant constant $c_{o,μ}$ allow us to state the nonexistence of positive solutions.
We follow the Cabré-Martel's approach. To this aim we state some properties of the operator $L$, of its corresponding $C_0$-semigroup and density results.
We prove a family of improved multipolar Poincaré-Hardy inequalities on Cartan-Hadamard manifolds. For suitable configurations of poles, these inequalities yield an improved multipolar Hardy inequality and an improved multipolar Poincaré inequality such that the critical unipolar singular mass is reached at any pole.
We prove multipolar Hardy inequalities on complete Riemannian manifolds, providing various curved counterparts of some Euclidean multipolar inequalities due to Cazacu and Zuazua [Improved multipolar Hardy inequalities, 2013]. We notice that our inequalities deeply depend on the curvature, providing (quantitative) information about the deflection from the flat case. By using these inequalities together with variational methods and group-theoretical arguments, we also establish non-existence, existence and multiplicity results for certain Schrödinger-type problems involving the Laplace-Beltrami …
We give necessary and sufficient conditions for the existence of weak solutions of a parabolic problem corresponding to the Kolmogorov operators perturbed by a multipolar inverse square potential with respect to the Gaussian probability measure which is the unique invariant measure for Ornstein-Uhlenbeck type operators. We state the optimality of the constant and, then, the nonexistence of positive exponentially bounded solutions to the parabolic problem.
\begin{abstract} In this paper we state the following weighted Hardy type inequality for any functions $φ$ in a weighted Sobolev space and for weight functions $μ$ of a quite general type \begin{equation*} c_{N,μ} \int_{\R^N}V\,φ^2μ(x)dx\le \int_{\R^N}|\nabla φ|^2μ(x)dx +C_μ\int_{\R^N}W φ^2μ(x)dx, \end{equation*} where $V$ is a multipolar potential and $W$ is a bounded function from above depending on $μ$. The method to get the result is based on the introduction of a suitable vector value function and on an integral identity that we state in the paper. We prove that the constant $c_{N,μ}$ in …
We proved some optimal Hardy inequalities in RNwhich is closely related to multipolar Schrödinger operators with mean-value type potentials, these sharp inequalities imply some multipolar type Heisenberg inequalities. We also obtained someimproved multipolar Hardy inequalities on bounded domains, moreover, we got the range of the best Hardy constant for a specific Hardy inequality.
In this paper we state the weighted Hardy inequality \begin{equation*} c\int_{{\mathbb R}^N}\sum_{i=1}^n \frac{φ^2 }{|x-a_i|^2}\, μ(x)dx\le \int_{{\mathbb R}^N} |\nablaφ|^2 \, μ(x)dx +k \int_{\mathbb{R}^N}φ^2 \, μ(x)dx \end{equation*} for any $ φ$ in a weighted Sobolev spaces, with $c\in]0,c_o[$ where $c_o=c_o(N,μ)$ is the optimal constant, $a_1,\dots,a_n\in \mathbb{R}^N$, $k$ is a constant depending on $μ$. We show the relation between $c$ and the closeness to the single pole. To this aim we analyze in detail the difficulties to be overcome to get the inequality.
In this paper, we consider the first order Hardy inequalities using simple equalities. This basic setting not only permits to derive quickly many well-known Hardy inequalities with optimal constants, but also supplies improved or new estimates in miscellaneous situations, such as multipolar potential, the exponential weight, hyperbolic space, Heisenberg group, the edge Laplacian, or the Grushin type operator.
In this paper, we show a weighted Hardy inequality in a limiting case for functions in weighted Sobolev spaces with respect to an invariant measure. We also prove that the constant in the left-hand side of the inequality is optimal. As applications, we establish the existence and nonexistence of positive exponentially bounded weak solutions to a parabolic problem involving the Ornstein-Uhlenbeck operator perturbed by a critical singular potential in two dimensional case, according to the size of the coefficient of the critical potential. These results can be considered as counterparts in the l…
We give general conditions to state the weighted Hardy inequality \[ c\int_{\mathbb{R}^N}\frac{φ^2} {|x|^2}dμ\leq\int_{\mathbb{R}^N}|\nabla φ|^2 dμ+C\int_{\mathbb{R}^N} φ^2dμ,\quad φ\in C_c^{\infty}(\mathbb{R}^N),\,c\leq c_{0,μ}, \] with respect to a probability measure $dμ$. Moreover, the optimality of the constant $c_{0,μ}$ is given. The inequality is related to the following Kolmogorov equation perturbed by a singular potential \[ Lu+Vu=\left(Δu+\frac{\nabla μ}μ\cdot \nabla u\right)+\frac{c}{|x|^2}u \] for which the existence of positive solutions to the corresponding parabolic problem can …
In this paper we study the optimization problem $$μ^\star(Ω):=\inf_{u\in \semi}\frac{\into |\n u|^2 \dx}{\into V u^2 \dx}$$ in a suitable functional space $\semi$. Here, $V$ is the multi-singular potential given by $$V:=\sum_{1\leq i<j\leq n} \frac{|a_i-a_j|^2}{|x-a_i|^2|x-a_j|^2}$$ and all the singular poles $a_1, \ldots, a_n$, $n\geq 2$, arise either in the interior or at the boundary of a smooth open domain $Ω\subset \rr^N$, with $N\geq 3$ or $N \geq 2$, respectively. For a bounded domain $Ω$ containing all the singularities in the interior, we prove that $μ^\star(Ω)>μ^\star(\rr^N)$ when $n…
In this work we prove sharp $L^p$ versions of multipolar Hardy inequalities in the case of a bipolar potential and $p\geq 2$, which were first developed in the case $p=2$ by Cazacu (CCM 2016) and Cazacu&Zuazua (Studies in phase space analysis with applications to PDEs, 2013). Our results are sharp and minimizers do exist in the energy space. New features appear when $p>2$ compared to the linear case $p=2$ at the level of criticality of the p-Laplacian $-Δ_p$ perturbed by a singular Hardy bipolar potential.
In his seminal 1934 paper on Brownian motion and the theory of gases Kolmogorov introduced a second order evolution equation which displays some challenging features. In the opening of his 1967 hypoellipticity paper Hörmander discussed a general class of degenerate Ornstein-Uhlenbeck operators that includes Kolmogorov's as a special case. In this note we combine semigroup theory with a nonlocal calculus for these hypoelliptic operators to establish new inequalities of Hardy-Littlewood-Sobolev type in the situation when the drift matrix has nonnegative trace. Our work has been influenced by ide…
\begin{abstract} We state the following weighted Hardy inequality \begin{equation*} c_{o, μ}\int_{{\R}^N}\frac{φ^2 }{|x|^2}\, dμ\le \int_{{\R}^N} |\nablaφ|^2 \, dμ+ K \int_{\R^N}φ^2 \, dμ\quad \forall\, φ\in H_μ^1 %\qquad c\le c_μ, \end{equation*} in the context of the study of the Kolmogorov operators \begin{equation*} Lu=Δu+\frac{\nabla μ}μ\cdot\nabla u \end{equation*} perturbed by inverse square potentials and of the related evolution problems. The function $μ$ in the drift term is a probability density on $\R^N$. We prove the optimality of the constant $c_{o, μ}$ and state existence and no…
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 article, we prove a weighted Hardy inequality for $1<p<+\infty$ and dimension $d\ge1$. If $p>d$, then we can deduce from our weighted Hardy inequality a Poincaré inequality. The proof of the weighted Hardy inequality is based on the method of vector fields firstly introduced by Mitidieri \cite{MR1769903}. By the same method, we show for $1<p<+\infty$ and $d\ge1$ that weighted Caffarelli-Kohn-Nirenberg inequalities hold true. As an application of our weighted Hardy inequality, we prove a non-existence result for a p-Kolmogorov parabolic equation.
We establish sharp upper and lower bounds on the heat kernel of the fractional Laplace operator perturbed by Hardy-type drift by transferring it to appropriate weighted space with singular weight.
Let $s_{n}(T)$ denote the $n$th approximation, isomorphism, Gelfand, Kolmogorov or Bernstein number of the Hardy-type integral operator $T$ given by $$ Tf(x)=v(x)\int_{a}^{x}u(t)f(t)dt,\,\,\,x\in(a,b)\,\,(-\infty<a<b<+\infty) $$ and mapping a Banach function space $E$ to itself. We investigate some geometrical properties of $E$ for which $$ C_{1}\int_{a}^{b}u(x)v(x)dx \leq\limsup\limits_{n\rightarrow\infty}ns_{n}(T) \leq \limsup\limits_{n\rightarrow\infty}ns_{n}(T)\leq C_{2}\int_{a}^{b}u(x)v(x)dx $$ under appropriate conditions on $u$ and $v.$ The constants $C_{1},C_{2}>0$ depend only on the s…
Université Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；Laboratoire de Mathématiques d’Orsay（奥赛数学实验室）；Australian National University-CNRS（澳大利亚国立大学-法国国家科学研究中心）；Westfälische Wilhelms-Universität Münster（明斯特西法利亚威廉大学）
