Caffarelli Kohn Nirenberg Inequalities Research Articles

Hardy–Sobolev–Rellich, Hardy–Littlewood–Sobolev and Caffarelli–Kohn–Nirenberg Inequalities on General Lie Groups

In this paper, we establish a number of geometrical inequalities such as Hardy, Sobolev, Rellich, Hardy–Littlewood–Sobolev, Caffarelli–Kohn–Nirenberg, Gagliardo-Nirenberg inequalities and their critical versions for an ample class of sub-elliptic differential operators on general connected Lie groups, which include both unimodular and non-unimodular cases in compact and noncompact settings. We also obtain the corresponding uncertainty type principles.

Functional Inequalities on Symmetric Spaces of Noncompact Type and Applications

The aim of this paper is to begin a systematic study of functional inequalities on symmetric spaces of noncompact type of higher rank. Our first main goal of this study is to establish the Stein–Weiss inequality, also known as a weighted Hardy–Littlewood–Sobolev inequality, for the Riesz potential on symmetric spaces of noncompact type. This is achieved by performing delicate estimates of ground spherical function with the use of polyhedral distance on symmetric spaces and by combining the integral Hardy inequality developed by Ruzhansky and Verma with the sharp Bessel-Green-Riesz kernel estimates on symmetric spaces of noncompact type obtained by Anker and Ji. As a consequence of the Stein–Weiss inequality, we deduce Hardy–Sobolev, Hardy–Littlewood–Sobolev, Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities on symmetric spaces of noncompact type. The second main purpose of this paper is to show the applications of aforementioned inequalities for studying nonlinear PDEs on symmetric spaces. Specifically, we show that the Gagliardo-Nirenberg inequality can be used to establish small data global existence results for the semilinear wave equations with damping and mass terms for the Laplace–Beltrami operator on symmetric spaces.

Improved Caffarelli–Kohn–Nirenberg Inequalities and Uncertainty Principle

Improved Caffarelli–Kohn–Nirenberg Inequalities and Uncertainty Principle

The Caffarelli–Kohn–Nirenberg inequalities for radial functions

The Caffarelli–Kohn–Nirenberg inequalities for radial functions

Improved Caffarelli–Kohn–Nirenberg inequalities in unit ball and sharp constants in dimension three

We show that the Caffarelli–Kohn–Nirenberg inequalities in unit ball Bn can be improved by subtraction of Hardy term. In three dimension and 0≤a<12, we show that the sharp constant coincides with that in R3. This is an analogous result to that of the sharp constant in the n−12-th order Hardy-Sobolev-Maz’ya inequality in the unit ball of dimension n when n is odd. As an application, we obtain a sharp Sobolev inequality on hyperbolic Caffarelli–Kohn–Nirenberg space introduced by L. Dupaigne, I. Gentil and S. Zugmeyer.

Caffarelli–Kohn–Nirenberg inequalities for curl-free vector fields and second order derivatives

Caffarelli–Kohn–Nirenberg inequalities for curl-free vector fields and second order derivatives

Reverse Stein–Weiss, Hardy–Littlewood–Sobolev, Hardy, Sobolev and Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups

Abstract In this note, we prove the reverse Stein–Weiss inequality on general homogeneous Lie groups. The results obtained extend previously known inequalities. Special properties of homogeneous norms and the reverse integral Hardy inequality play key roles in our proofs. Also, we prove reverse Hardy, Hardy–Littlewood–Sobolev, L p {L^{p}} -Sobolev and L p {L^{p}} -Caffarelli–Kohn–Nirenberg inequalities on homogeneous Lie groups.

Rellich, Gagliardo—Nirenberg, Trudinger and Caffarelli—Kohn—Nirenberg inequalities for Dunkl operators and applications

In this paper we introduce several extended versions of the classical Caffarelli—Kohn—Nirenberg inequalities. Moreover, we obtain weighted higher order Rellich, weighted Gagliardo—Nirenberg, Caffarelli—Kohn—Nirenberg, Trudinger inequalities and the uncertainty principle for Dunkl operators. Furthermore, we give an application of the Gagliardo—Nirenberg inequality to the Cauchy problem for the nonlinear damped wave equations for the Dunkl Laplacian.

Symmetry for positive critical points of Caffarelli–Kohn–Nirenberg inequalities

We consider positive critical points of Caffarelli–Kohn–Nirenberg inequalities and prove a Liouville type result which allows us to give a complete classification of the solutions in a certain range of parameters, providing a symmetry result for positive solutions. The governing operator is a weighted p-Laplace operator, which we consider for a general p∈(1,d). For p=2, the symmetry breaking region for extremals of Caffarelli–Kohn–Nirenberg inequalities was completely characterized in Dolbeault et al. (2016). Our results extend this result to a general p and are optimal in some cases.

Functional Inequalities: Nonlinear Flows and Entropy Methods as a Tool for Obtaining Sharp and Constructive Results

Interpolation inequalities play an essential role in analysis with fundamental consequences in mathematical physics, nonlinear partial differential equations (PDEs), Markov processes, etc., and have a wide range of applications in various other areas of Science. Research interests have evolved over the years: while mathematicians were originally focussed on abstract properties (for instance appropriate notions of functional spaces for the existence of weak solutions in PDEs), more qualitative questions (for instance, bifurcation diagrams, multiplicity of the solutions in PDEs and their qualitative behaviour) progressively emerged. The use of entropy methods in nonlinear PDEs is a typical example: in some cases, the optimal constant in the inequality can be interpreted as an optimal rate of decay of an entropy for an associated evolution equation. Much more has been learned by adopting this point of view. This paper aims at illustrating some of these recent aspect of entropy-entropy production inequalities, with applications to stability in Gagliardo–Nirenberg–Sobolev inequalities and symmetry results in Caffarelli–Kohn–Nirenberg inequalities. Entropy methods provide a framework which relates nonlinear regimes with their linearized counterparts. This framework allows to prove optimality results, symmetry results and stability estimates. Some emphasis will be put on the hidden structure which explain such properties. Related open problems will be listed.

Sharp Caffarelli–Kohn–Nirenberg inequalities on Riemannian manifolds: the influence of curvature

We first establish a family of sharp Caffarelli–Kohn–Nirenberg type inequalities (shortly, sharp CKN inequalities) on the Euclidean spaces and then extend them to the setting of Cartan–Hadamard manifolds with the same best constant. The quantitative version of these inequalities also is proved by adding a non-negative remainder term in terms of the sectional curvature of manifolds. We next prove several rigidity results for complete Riemannian manifolds supporting the Caffarelli–Kohn–Nirenberg type inequalities with the same sharp constant as in the Euclidean space of the same dimension. Our results illustrate the influence of curvature to the sharp CKN inequalities on the Riemannian manifolds. They extend recent results of Kristály (J. Math. Pures Appl. 119 (2018), 326–346) to a larger class of the sharp CKN inequalities.

Sharp Rellich–Sobolev inequalities and weighted Adams inequalities involving Hardy terms for bi-Laplacian

Motivated by a recent work of Dong, Lam and Lu concerning the sharp weighted Trudinger–Moser and Caffarelli–Kohn–Nirenberg inequalities, we derive a sharp Rellich–Sobolev inequality on Rn with n≥5 and weighted Adams inequalities involving Hardy terms on R4. The procedure is based on a quasi-conformal mapping type transform and decomposition into spherical harmonics since the symmetrization arguments do not work in dealing with these inequalities.

Sharp [formula omitted]-Caffarelli–Kohn–Nirenberg inequalities for Grushin vector fields

In this paper sharp L2-Caffarelli–Kohn–Nirenberg (CKN) inequalities on Euclidean space equipped with Grushin vector fields are established. By use of a Kelvin-like transform, and Grushin cylindrical coordinates, it is shown here that these inequalities are sharp for all possible parameter values for these inequalities.

Maximizers for Fractional Caffarelli–Kohn–Nirenberg and Trudinger–Moser Inequalities on the Fractional Sobolev Spaces

This paper is mainly concerned with the existence of extremals for the fractional Caffarelli–Kohn–Nirenberg and Trudinger–Moser inequalities. We first establish the existence of extremals of the fractional Caffarelli–Kohn–Nirenberg inequalities in Theorem 1.1. It is also proved that the extremals of this inequality are the ground-state solutions of some fractional p-Laplacian equation. Then, we use the method of considering the level-sets of functions under consideration first developed by Lam and Lu (Adv Math 231(6):3259–3287, 2012; J Differ Equ 255(3), 298–325, 2013) and combining with a new compactness argument and the fractional rearrangement inequalities to establish the existence of extremals for the fractional Trudinger–Moser inequalities with the Dirichlet norm (see Theorems 1.2, 1.3, 1.6 and 1.7). Radial symmetry of extremals and concentration-compactness principle for the fractional Trudinger–Moser inequalities are also established in Theorems 1.4, 1.8 and 1.9. Finally, we investigate some relationship between the best constants of the fractional Trudinger–Moser and Caffarelli–Kohn–Nirenberg inequalities in the asymptotic sense (see Theorem 1.11).

Fractional logarithmic inequalities and blow-up results with logarithmic nonlinearity on homogeneous groups

In this paper, we prove the fractional logarithmic Sobolev inequality on homogeneous groups. Also, we establish fractional logarithmic Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups. In addition, we show blow-up results for the fractional heat equation with logarithmic nonlinearity on homogeneous groups.

Sharp Caffarelli–Kohn–Nirenberg-Type Inequalities on Carnot Groups

Abstract The main purpose of this paper is to establish several general Caffarelli–Kohn–Nirenberg (CKN) inequalities on Carnot groups G (also known as stratified groups). These CKN inequalities are sharp for certain parameter values. In case G is an Iwasawa group, it is shown here that the L 2 {L^{2}} -CKN inequalities are sharp for all parameter values except one exceptional case. To show this, generalized Kelvin transforms K σ {K_{\sigma}} are introduced and shown to be isometries for certain weighted Sobolev spaces. An interesting transformation formula for the sub-Laplacian with respect to K σ {K_{\sigma}} is also derived. Lastly, these techniques are shown to be valid for establishing CKN-type inequalities with monomial and horizontal norm weights.

Classification of radial solutions to equations related to Caffarelli–Kohn–Nirenberg inequalities

This article studies the qualitative and quantitative properties of radial solutions to an elliptic equation related to the Euler–Lagrange equations for certain sharp Caffarelli–Kohn–Nirenberg inequalities. Namely, we examine the equation $$\begin{aligned} -\hbox {div} (|x|^{a} D u ) = |x|^{b}u^{p}, ~ u > 0,\, \text{ in } \mathbb {R}^N, \end{aligned}$$where \(p > 1\), \(N \ge 2\), \(N-2+a \ge 0\) and \(b > - N\). The main results establish the properties of radially symmetric solutions including existence, uniqueness, and classification results as well as results on the asymptotic and intersecting behaviour of such solutions.

Sharp weighted Trudinger–Moser–Adams inequalities on the whole space and the existence of their extremals

Though there have been extensive works on the existence of maximizers for sharp first order Trudinger–Moser inequalities, much less is known for that of the maximizers for higher order Adams’ inequalities. In this paper, we mainly study the existence of extremals for sharp weighted Trudinger–Moser–Adams type inequalities with the Dirichlet and Sobolev norms (also known as the critical and subcritical Trudinger–Moser–Adams inequalities), see Theorems 1.1, 1.2, 1.3, 1.5, 1.7, 1.9 and 1.11. First, we employ the method based on level-sets of functions under consideration and Fourier transform to establish stronger weighted Trudinger–Moser–Adams type inequalities with the Dirichlet norm in $$W^{2,\frac{n}{2}}(\mathbb {R}^n)$$ and $$W^{m,2}(\mathbb {R}^{2m})$$ respectively. While the first order sharp weighted Trudinger–Moser inequality and its existence of extremal functions was established by Dong and the second author using a quasi-conformal type transform (Dong and Lu in Calc Var Partial Differ Equ 55:55–88, 2016), such a transform does not work for the Adams inequality involving higher order derivatives. Since the absence of the Polya–Szego inequality and the failure of change of variable method for higher order derivatives for weighted inequalities, we will need several compact embedding results (Lemmas 2.1, 3.1 and 5.2). Through the compact embedding and scaling invariance of the subcritical Adams inequality, we investigate the attainability of best constants. Second, we employ the method developed by Lam et al. (Adv Math 352:1253–1298, 2019) which uses the relationship between the supremums of the critical and subcritical inequalities (see also Lam in Proc Amer Math Soc 145:4885–4892, 2017) to establish the existence of extremals for weighted Adams’ inequalities with the Sobolev norm. Third, using the Fourier rearrangement inequality established by Lenzmann and Sok (A sharp rearrangement principle in Fourier space and symmetry results for PDEs with arbitrary order, arXiv:1805.06294v1 ), we can reduce our problem to the radial case and then establish the existence of the extremal functions for the non-weighted Adams inequalities. As an application, we derive new results on high-order critical Caffarelli–Kohn–Nirenberg interpolation inequalities whose parameters extend those proved by Lin (Commun Partial Differ Equ 11:1515–1538, 1986) (see Theorems 1.13 and 1.14). Furthermore, we also establish the relationship between the best constants of the weighted Trudinger–Moser–Adams type inequalities and the Caffarelli–Kohn–Nirenberg inequalities in the asymptotic sense (see Theorems 1.13 and 1.14).

Weighted inequalities for the fractional Laplacian and the existence of extremals

In this paper, we obtain improved versions of Stein–Weiss and Caffarelli–Kohn–Nirenberg inequalities, involving Besov norms of negative smoothness. As an application of the former, we derive the existence of extremals of the Stein–Weiss inequality in certain cases, some of which are not contained in the celebrated theorem of Lieb [Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. of Math. (2) 118(2) (1983) 101–116].

Quantitative a priori estimates for fast diffusion equations with Caffarelli–Kohn–Nirenberg weights. Harnack inequalities and Hölder continuity

We study a priori estimates for a class of non-negative local weak solution to the weighted fast diffusion equation ut=|x|γ∇⋅(|x|−β∇um), with 0<m<1 posed on cylinders of (0,T)×RN. The weights |x|γ and |x|−β, with γ<N and γ−2<β≤γ(N−2)/N can be both degenerate and singular and need not belong to the class A2, a typical assumption for this kind of problems. This range of parameters is optimal for the validity of a class of Caffarelli–Kohn–Nirenberg inequalities, which play the role of the standard Sobolev inequalities in this more complicated weighted setting.The weights that we consider are not translation invariant and this causes a number of extra difficulties and a variety of scenarios: for instance, the scaling properties of the equation change when considering the problem around the origin or far from it. We therefore prove quantitative – with computable constants – upper and lower estimates for local weak solutions, focussing our attention where a change of geometry appears. Such estimates fairly combine into forms of Harnack inequalities of forward, backward and elliptic type. As a consequence, we obtain Hölder continuity of the solutions, with a quantitative (even if non-optimal) exponent. Our results apply to a quite large variety of solutions and problems. The proof of the positivity estimates requires a new method and represents the main technical novelty of this paper.Our techniques are flexible and can be adapted to more general settings, for instance to a wider class of weights or to similar problems posed on Riemannian manifolds, possibly with unbounded curvature. In the linear case, m=1, we also prove quantitative estimates, recovering known results in some cases and extending such results to a wider class of weights.