We present the best constant and the existence of extremal functions for an Improved Hardy-Sobolev inequality. We prove that, under a proper transformation, this inequality is equivalent to the Sobolev inequality in $\mathbb{R}^N$. We also discuss the connection of the related functional spaces and as a result we obtain some Caffarelli - Kohn - Nirenberg inequalities. Our starting point is the existence of a minimizer for the Bliss' inequality and the indirect dependence of the Hardy inequality at the origin.
In this paper we deal with nonnegative distributional supersolutions for a class of linear elliptic equations involving inverse-square potentials and logarithmic weights. We prove sharp nonexistence results.
In this paper we study nonexistence of non-negative distributional supersolutions for a class of semilinear elliptic equations involving inverse-square potentials.
It is known that classical Hardy and Sobolev inequalities hold when the exponent $p$ and the dimension $N$ satisfy $p < N < \infty$. In this note, we consider two limits of Hardy and Sobolev inequalities as $p \nearrow N$ and $N \nearrow \infty$ in some sense.
The purpose of this text is twofold. We present a review of the existing stability results for Sobolev, Hardy-Littlewood-Sobolev (HLS) and related inequalities. We also contribute to the topic with some observations on constructive stability estimates for (HLS).
We obtain the sharp constant for the Hardy-Sobolev inequality involving the distance to the origin. This inequality is equivalent to a limiting Caffarelli-Kohn-Nirenberg inequality. In three dimensions, in certain cases the sharp constant coincides with the best Sobolev constant.
We study anti-symmetric solutions about the hyperplane $\{x_n=0\}$ to the following fractional Hardy-Hénon system $$ \left\{\begin{aligned} &(-Δ)^{s_1}u(x)=|x|^αv^p(x),\ \ x\in\mathbb{R}_+^n, \\&(-Δ)^{s_2}v(x)=|x|^βu^q(x),\ \ x\in\mathbb{R}_+^n, \\&u(x)\geq 0,\ \ v(x)\geq 0,\ \ x\in\mathbb{R}_+^n, \end{aligned}\right. $$ where $0<s_1,s_2<1$, $n>2\max\{s_1,s_2\}$. Nonexistence of anti-symmetric solutions are obtained in some appropriate domains of $(p,q)$ under some corresponding assumptions of $α,β$ via the methods of moving spheres and moving planes. Particularly, for the case $s_1=s_2$, one …
In this article, we derive the existence of positive solutions of a semi-linear, non-local elliptic PDE, involving a singular perturbation of the fractional laplacian, coming from the fractional Hardy-Sobolev-Maz'ya inequality, derived in this paper. We also derive symmetry properties and a precise asymptotic behaviour of solutions.
This paper is a second one following our work [CLZ13] in series, considering sharp Hardy- Littlewood-Sobolev inequalities on groups of Heisenberg type. The first important breakthrough was made by Frank and Lieb in [FL12]. In this paper, analogous results are obtained for octonionic Heisenberg group.
This work focuses on an improved fractional Sobolev inequality with a remainder term involving the Hardy-Littlewood-Sobolev inequality which has been proved recently. By extending a recent result on the standard Laplacian to the fractional case, we offer a new, simpler proof and provide new estimates on the best constant involved. Using endpoint differentiation, we also obtain an improved version of a Moser-Trudinger-Onofri type inequality on the sphere. As an immediate consequence, we derive an improved version of the Onofri inequality on the Euclidean space using the stereographic projection…
In this paper, we obtain necessary conditions and sufficient conditions on the initial data for the local-in-time solvability of the Cauchy problem \[ \partial_t u +(-Δ)^\fracθ{2} u=|x|^{-γ} u^p ,\quad x\in{\bf R}^N, t>0, \qquad u(0)=μ\quad \mbox{in} \quad {\bf R}^N, \] where $N\ge 1$, $0<θ\le2$, $p>1$, $γ>0$ and $μ$ is a nonnegative Radon measure on ${\bf R}^N$. Using these conditions, we attempt to identify the optimal strength of the singularity of $μ$ for the existence of solutions to this problem.
In this paper we study weighted Hardy-Sobolev spaces of vector valued functions analytic on double-napped cones of the complex plane. We introduce these spaces as a tool for complex scaling of linear ordinary differential equations with dilation analytic unbounded operator coefficients. As examples we consider boundary value problems in cylindrical domains and domains with quasicylindrical ends.
We provide sufficient conditions for boundary Hardy inequality to hold in bounded Lipschitz domains, complement of a point (the so-called point Hardy inequality), domain above the graph of a Lipschitz function, the complement of a bounded Lipschitz domain in fractional Orlicz-Sobolev setting. As a consequence, we get sufficient conditions for regional fractional Orlicz Poincaré inequality in bounded Lipschitz domains. Necessary conditions for fractional Orlicz Hardy and regional fractional Orlicz Poincaré inequalities are also given for bounded Lipschitz domains. Various sufficient conditions …
We show that the sharp constant in the Hardy-Littlewood-Sobolev inequality can be derived using the method that we employed earlier for a similar inequality on the Heisenberg group. The merit of this proof is that it does not rely on rearrangement inequalities; it is the first one to do so for the whole parameter range.
Involutivity is a well known necessary condition for integrability of smooth tangent distributions. We show that this condition is still necessary for integrability with Sobolev surfaces. We specialize our study to the left invariant horizontal distribution of the first Heisenberg group $\H^1$. Here we answer a question raised in a paper by Z.M.Balogh, R.Hoefer-Isenegger, J.T.Tyson.
We consider the optimizers $u$ in the Hardy-Sobolev inequality for the space $\dot{W}^{s,p}({\mathbb R}^N)$ with order of differentiability $s\in ]0,1[$. After proving existence through concentration-compactness, we derive the pointwise asymptotic $u(x)\simeq |x|^{-\frac{N-ps}{p-1}}$ for large $|x|$ and the summability estimate $u\in \dot{W}^{s,γ}({\mathbb R}^N)$ for all $γ>\frac{N(p-1)}{N-s}$. These estimates are optimal in the limit $s\to 1^-$, in which case optimizers are explicitly known.
In this note we prove the reverse Stein-Weiss inequality on general homogeneous Lie groups. The obtained results extend previously known inequalities. Special properties of homogeneous norms and the reverse integral Hardy inequality play key roles in our proofs. Also, we show reverse Hardy, Hardy-Littlewood-Sobolev, Lp-Sobolev and Lp-Caffarelli-Kohn-Nirenberg inequalities on homogeneous groups.
We give a new proof of certain cases of the sharp HLS inequality. Instead of symmetric decreasing rearrangement it uses the reflection positivity of inversions in spheres. In doing this we extend a characterization of the minimizing functions due to Li and Zhu.
In this paper we consider composition operators on Hardy-Sobolev spaces in connections with $BMO$-quasiconformal mappings. Using the duality of Hardy spaces and $BMO$-spaces we prove that $BMO$-quasiconformal mappings generate bounded composition operators from Hardy-Sobolev spaces to Sobolev spaces.
We prove a Hardy-Sobolev-Maz'ya inequality for arbitrary domains Ω\subset\R^N with a constant depending only on the dimension N\geq 3. In particular, for convex domains this settles a conjecture by Filippas, Maz'ya and Tertikas. As an application we derive Hardy-Lieb-Thirring inequalities for eigenvalues of Schrödinger operators on domains.
In the euclidean space, Sobolev and Hardy-Littlewood-Sobolev inequalities can be related by duality. In this paper, we investigate how to relate these inequalities using the flow of a fast diffusion equation in dimension $d\ge3$. The main consequence is an improvement of Sobolev's inequality when $d\ge5$, which involves the various terms of the dual Hardy-Littlewood-Sobolev inequality. In dimension $d=2$, Onofri's inequality plays the role of Sobolev's inequality and can also be related to its dual inequality, the logarithmic Hardy-Littlewood-Sobolev inequality, by a super-fast diffusion equat…
In this paper we study existence and nonexistence of nonnegative distributional solutions for a class of semilinear fractional elliptic equations involving the Hardy potential.
This is the first part of our research on certain sharp Hardy-Sobolev inequalities and the related elliptic equations. In this part we shall establish some sharp weighted Hardy-Sobolev inequalities whose weights are distance functions to the boundary.
In this paper, we establish several new anisotropic Hardy-Sobolev inequalities in mixed Lebesgue spaces and mixed Lorentz spaces, which covers many known corresponding results. As an application, this type of inequalities allows us to generalize some regularity criteria of the 3D axisymmetric Navier-Stokes equations.
We give a simple proof of the $λ= d-2$ cases of the sharp Hardy-Littlewood-Sobolev inequality for $d\geq 3$, and the sharp Logarithmic Hardy-Littlewood-Sobolev inequality for $d=2$ via a monotone flow governed by the fast diffusion equation.
On a compact Riemannian manifold, we prove the existence of multiple solutions for an elliptic equation with critical Sobolev growth and critical Hardy potential.
We investigate the existence of extremals for Hardy-Sobolev inequalities involving the Dirichlet fractional Laplacian of order s, 0<s<1, on half-spaces.
This paper is devoted to improvements of Sobolev and Onofri inequalities. The additional terms involve the dual counterparts, i.e. Hardy-Littlewood-Sobolev type inequalities. The Onofri inequality is achieved as a limit case of Sobolev type inequalities. Then we focus our attention on the constants in our improved Sobolev inequalities, that can be estimated by completion of the square methods. Our estimates rely on nonlinear flows and spectral problems based on a linearization around optimal Aubin-Talenti functions.
In this paper, we give a full classification of the nonexistence of positive weak solutions to the semilinear elliptic inequality involving the fractional Hardy potential in punctured and in exterior domains. Our methods are self-contained and new. The main ideas and key ingredients will be discussed in the next section after Theorem 1.1 for punctured domains, and after Theorem 1.4 for exterior domains. We will also explain why all the previous methods and techniques do not apply to our general setting. Let us give here a foretaste of our line of attack: Based on the imbalance between the Hard…
We give a sufficient condition for limiting Sobolev and Hardy inequalities to hold on stratified homogeneous groups. In the Euclidean case, this condition reduces to the known cancelling necessary and sufficient condition. We obtain in particular endpoint Korn--Sobolev and Korn--Hardy inequalities on stratified homogeneous groups.
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.
In this paper we establish several Hardy and Hardy-Sobolev type inequalities with homogeneous weights on the first orthant $\displaystyle \mathbb{R}_{*}^n:=\{(x_1, \ldots, x_n):x_1>0, \ldots, x_n>0 \}$. We then use some of them to produce Hardy type inequalities with remainder terms. Furthermore, we obtain some interpolation inequalities and Maz'ya type inequalities with remainder terms with the help of Maz'ya inequality and Sobolev inequality of Cabré and Ros-Orton on the upper half space $\mathbb{R}_{+}^n:=\{x=(x_1, \ldots, x_n)|\, x_n>0 \}$.
Linear interpolation inequalities that combine Hardy's inequality with sharp Sobolev embedding are obtained using classical arguments of Hardy and Littlewood (Bliss lemma). Such results are equivalent to Caffarelli-Kohn-Nirenberg inequalities with sharp constants. A one-dimensional convolution inequality for the exponential density is derived as an application of these methods.
A Stein-Weiss type inequality on Carnot groups is established by proving the boundedness of an integral operator and the Hardy-Littlewood-Sobolev type inequality on Carnot groups is also derived.
This paper studies the existence of extremal problems for the Hardy-Littlewood-Sobolev inequalities on compact manifolds without boundary via Concentration-Compactness principle.
We prove the attainability of the best constant in the fractional Hardy--Sobolev inequality with boundary singularity for the Spectral Dirichlet Laplacian. The main assumption is the average concavity of the boundary at the origin.
We prove the coincidence of the Sobolev and Hardy constants relative to the "Dirichlet" and "Navier" fractional Laplacians of any real order $m\in(0,\frac{n}{2})$ over bounded domains in $\mathbb R^n$.
Corrigenda to "$L^p$ estimates and asymptotic behavior for finite energy solutions of extremals to Hardy-Sobolev inequalities", Trans. Amer. Math. Soc. 363 (2011), no. 1, 37--62.
We prove an improved version of Poincaré-Hardy inequality in suitable subspaces of the Sobolev space on the hyperbolic space via Bessel pairs. As a consequence, we obtain a new Hardy type inequality with an improved constant (than the usual Hardy constant). Furthermore, we derive a new kind of improved Caffarelli-Kohn-Nirenberg inequality on the hyperbolic space.
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…
We study linear and non-linear equations related to the fractional Hardy--Sobolev inequality. We prove nondegeneracy of ground state solutions to the basic equation and investigate existence and qualitative properties, including symmetry of solutions to some perturbed equations.
We consider the second best constant in the Hardy-Sobolev inequality on a Riemannian manifold. More precisely, we are interested with the existence of extremal functions for this inequality. This problem was tackled by Djadli-Druet [5] for Sobolev inequalities. Here, we establish the corresponding result for the singular case. In addition, we perform a blow-up analysis of solutions Hardy-Sobolev equations of minimizing type. This yields informations on the value of the second best constant in the related Riemannian functional inequality.
