Adams Type Inequality Research Articles

Sharp Adams Type Inequalities for the Fractional Laplace–Beltrami Operator on Noncompact Symmetric Spaces

Sharp Adams Type Inequalities for the Fractional Laplace–Beltrami Operator on Noncompact Symmetric Spaces

Sharp Sobolev and Adams–Trudinger–Moser embeddings on weighted Sobolev spaces and their applications

Abstract We derive sharp Sobolev embeddings on a class of Sobolev spaces with potential weights without assuming any boundary conditions. Moreover, we consider the Adams-type inequalities for the borderline Sobolev embedding into the exponential class with a sharp constant. As applications, we prove that the associated elliptic equations with nonlinearities in both forms of polynomial and exponential growths admit nontrivial solutions.

On a weighted Adams type inequality and an application to a biharmonic equation

This paper deals with an improvement of a class of Adams type inequalities involving potentials and weights , which can decay to zero at infinity as , , and , , respectively. As an application of this result and by using minimax methods, we establish the existence of solutions for the following class of problems: where the nonlinear term can have critical exponential growth. Furthermore, when we prove that the solutions belong to the Sobolev space (bound state solutions).

Solving a Dirichlet problem on unbounded domains via a conformal transformation

In this paper, we solve the p-Dirichlet problem for Besov boundary data on unbounded uniform domains with bounded boundaries when the domain is equipped with a doubling measure satisfying a Poincaré inequality. This is accomplished by studying a class of transformations that have been recently shown to render the domain bounded while maintaining uniformity. These transformations conformally deform the metric and measure in a way that depends on the distance to the boundary of the domain and, for the measure, a parameter p. We show that the transformed measure is doubling and the transformed domain supports a Poincaré inequality. This allows us to transfer known results for bounded uniform domains to unbounded ones, including trace results and Adams-type inequalities, culminating in a solution to the Dirichlet problem for boundary data in a Besov class.

Sharp Adams type inequalities in Lorentz-Sobole space

<abstract><p>This article addresses several sharp weighted Adams type inequalities in Lorentz-Sobolev spaces by using symmetry, rearrangement and the Riesz representation formula. In particular, the sharpness of these inequalities were also obtained by constructing a proper test sequence.</p></abstract>

Estimate for concentration level of the Adams functional and extremals for Adams-type inequality

This paper is mainly concerned with the existence of extremals for the Adams inequality. We first establish an upper bound for the classical Adams functional along of all concentrated sequences in the higher order Sobolev space with homogeneous Navier boundary conditions WNm,nm(Ω), which in particular includes the classical Sobolev space W0m,nm(Ω), where Ω is a smooth bounded domain in Euclidean n-space. Secondly, based on the Concentration-compactness alternative due to Do Ó and Macedo, we prove the existence of extremals for the Adams inequality under Navier boundary conditions for second order derivatives at least for higher dimensions when Ω is an Euclidean ball.

Sharp Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on the Siegel domains and complex hyperbolic spaces

The aim of this paper is to establish higher order Poincaré-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on Siegel domains and complex hyperbolic spaces using the method of Helgason-Fourier analysis on the complex hyperbolic spaces. Firstly, we give a factorization theorem for the operators on the complex hyperbolic space which is closely related to Geller's operator, as well as the CR invariant differential operators on the Heisenberg group and CR sphere. (See Theorem 1.3.) Secondly, by using, among other things, the Kunze-Stein phenomenon on a closed linear group SU(1,n) and Helgason-Fourier analysis techniques on the complex hyperbolic spaces, we establish the Poincaré-Sobolev, Hardy-Sobolev-Maz'ya inequality on the Siegel domain Un and the unit ball BCn. (See Theorems 1.4, 1.5.) Finally, we establish the sharp Hardy-Adams inequalities and sharp Adams type inequalities on Sobolev spaces of any positive fractional order on the complex hyperbolic spaces. (See Theorems 1.9, 1.10 and 1.12.) The factorization theorem we prove on the complex hyperbolic spaces is considerably more difficult than in the case of real hyperbolic spaces and should be of its independent interest in the analysis on Heisenberg group and CR sphere and CR invariant differential operators therein.

A sharpened form of Adams-type inequalities on higher-order Sobolev spaces : a simple approach

AbstractIn this paper, we develop an extremely simple method to establish the sharpened Adams-type inequalities on higher-order Sobolev spaces $W^{m,\frac {n}{m}}(\mathbb {R}^n)$ in the entire space $\mathbb {R}^n$ , which can be stated as follows: Given $\Phi \left ( t\right ) =e^{t}-\underset {j=0}{\overset {n-2}{\sum }} \frac {t^{j}}{j!}$ and the Adams sharp constant $\beta _{n,m}$ . Then, $$ \begin{align*}\sup_{\|\nabla^mu\|_{\frac{n}{m}}^{\frac{n}{m}}+\|u\|_{\frac{n}{m}}^{\frac{n}{m}}\leq1}\int_{\mathbb{R}^n}\Phi\Big(\beta_{n,m} (1+\alpha \|u\|_{\frac{n}{m}}^{\frac{n}{m}} )^{\frac{m}{n-m}}|u|^{\frac{n}{n-m}}\Big)dx<\infty, \end{align*} $$ for any $0<\alpha <1$ . Furthermore, we construct a proper test function sequence to derive the sharpness of the exponent $\alpha $ of the above Adams inequalities. Namely, we will show that if $\alpha \ge 1$ , then the above supremum is infinite.Our argument avoids applying the complicated blow-up analysis often used in the literature to deal with such sharpened inequalities.

Sharp Green’s Function Estimates on Hadamard Manifolds and Adams Inequality

Abstract In this article, we establish estimates on Riesz-type kernels and prove the Adams-type inequality for $W^{k,p}(M)$ functions, where $(M,g)$ is an $n$-dimensional Hadamard manifold with sectional curvature bounded from below and above by a negative constant and $k$ is an integer satisfying $kp=n$.

Improved Adams-type inequalities and their extremals in dimension 2m

In this paper, we prove the existence of an extremal function for the Adams–Moser–Trudinger inequality on the Sobolev space [Formula: see text], where [Formula: see text] is any bounded, smooth, open subset of [Formula: see text], [Formula: see text]. Moreover, we extend this result to improved versions of Adams’ inequality of Adimurthi-Druet type. Our strategy is based on blow-up analysis for sequences of subcritical extremals and introduces several new techniques and constructions. The most important one is a new procedure for obtaining capacity-type estimates on annular regions.

Existence and nonexistence of extremals for critical Adams inequalities in [formula omitted] and Trudinger-Moser inequalities in [formula omitted

Though much progress has been made with respect to the existence of extremals of the critical first order Trudinger-Moser inequalities in W1,n(Rn) and higher order Adams inequalities on finite domain Ω⊂Rn, whether there exists an extremal function for the critical higher order Adams inequalities on the entire space Rn still remains open. The current paper represents the first attempt in this direction by considering the critical second order Adams inequality in the entire space R4. The classical blow-up procedure cannot apply to solving the existence of critical Adams type inequality because of the absence of the Pólya-Szegö type inequality. In this paper, we develop some new ideas and approaches based on a sharp Fourier rearrangement principle (see [31]), sharp constants of the higher-order Gagliardo-Nirenberg inequalities and optimal poly-harmonic truncations to study the existence and nonexistence of the maximizers for the Adams inequalities in R4 of the formS(α)=sup‖u‖H2=1⁡∫R4(exp⁡(32π2|u|2)−1−α|u|2)dx, where α∈(−∞,32π2). We establish the existence of the threshold α⁎, where α⁎≥(32π2)2B22 and B2≥124π2, such that S(α) is attained if 32π2−α<α⁎, and is not attained if 32π2−α>α⁎. This phenomenon has not been observed before even in the case of first order Trudinger-Moser inequality. Therefore, we also establish the existence and non-existence of an extremal function for the Trudinger-Moser inequality on R2. Furthermore, the symmetry of the extremal functions can also be deduced through the Fourier rearrangement principle.

An Adams type inequality of fractional integral operator on hypergroups

In this paper, an Adams type inequality of fractional integral operator will be presented on the generalized Morrey spaces over commutative hypergroups. This inequality needs an estimation of the fractional integral operator by the maximal operator. Besides, the almost decreasing of functions and Ahlfors condition of the measure are also involved.

Sharp Adams–Moser–Trudinger type inequalities in the hyperbolic space

The purpose of this paper is to establish some Adams–Moser–Trudinger inequalities, which are the borderline cases of the Sobolev embedding, in the hyperbolic space \mathbb H^n . First, we prove a sharp Adams’ inequality of order two with the exact growth condition in \mathbb H^n . Then we use it to derive a sharp Adams-type inequality and an Adachi–Tanakat-ype inequality. We also prove a sharp Adams-type inequality with Navier boundary condition on any bounded domain of \mathbb H^n , which generalizes the result of Tarsi to the setting of hyperbolic spaces. Finally, we establish a Lions-type lemma and an improved Adams-type inequality in the spirit of Lions in \mathbb H^n . Our proofs rely on the symmetrization method extended to hyperbolic spaces.

A Sharp Adams-Type Inequality for Weighted Sobolev Spaces

AbstractIn a classical work (Ann. Math.128, (1988) 385–398), D. R. Adams proved a sharp Trudinger–Moser inequality for higher-order derivatives. We derive a sharp Adams-type inequality and Sobolev-type inequalities associated with a class of weighted Sobolev spaces that is related to a Hardy-type inequality.

A simple proof of the Adams type inequalities in &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ {\mathbb R}^{2m} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;

We provide the simple proof of the Adams type inequalities in whole space $ {\mathbb R}^{2m} $. The main tools are the Fourier rearrangement technique introduced by Lenzmann and Sok [16], a Hardy–Rellich type inequality for radial functions, and the sharp Moser–Trudinger type inequalities in $ {\mathbb R}^2 $.

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).

Equivalence of critical and subcritical sharp Trudinger–Moser–Adams inequalities

Sharp Trudinger–Moser inequalities on the first order Sobolev spaces and their analogous Adams inequalities on high order Sobolev spaces play an important role in geometric analysis, partial differential equations and other branches of modern mathematics. Such geometric inequalities have been studied extensively by many authors in recent years and there is a vast literature. There are two types of such optimal inequalities: critical and subcritical sharp inequalities, both are with best constants. Critical sharp inequalities are under the restriction of the full Sobolev norms for the functions under consideration, while the subcritical inequalities are under the restriction of the partial Sobolev norms for the functions under consideration. There are subtle differences between these two type of inequalities. Surprisingly, we prove in this paper that these critical and subcritical Trudinger–Moser and Adams inequalities are actually equivalent. Moreover, we also establish the asymptotic behavior of the supremum for the subcritical Trudinger–Moser and Adams inequalities on the entire Euclidean spaces, and provide a precise relationship between the suprema for the critical and subcritical Trudinger–Moser and Adams inequalities. This relationship of supremum is useful in establishing the existence and nonexistence of extremal functions for the Trudinger–Moser inequalities. Since the critical Trudinger–Moser and Adams inequalities can be easier to prove than subcritical ones in some occasions, and more difficult to establish in other occasions, our results and the method suggest a new approach to both the critical and subcritical Trudinger–Moser and Adams type inequalities.

Application of an Adams type inequality to a two-chemical substances chemotaxis system

This paper deals with positive solutions of the fully parabolic system,{ut=Δu−χ∇⋅(u∇v)inΩ×(0,∞),τ1vt=Δv−v+winΩ×(0,∞),τ2wt=Δw−w+uinΩ×(0,∞), under homogeneous Neumann boundary conditions or mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded domain Ω⊂Rn (n≤4) with positive parameters τ1,τ2,χ>0 and nonnegative smooth initial data (u0,v0,w0).In the lower dimensional case (n≤3), it is proved that for all reasonable initial data solutions of the system exist globally in time and remain bounded.In the case n=4, it is shown that in the radially symmetric setting solutions to the Neumann boundary value problem of the system exist globally in time and remain bounded if ‖u0‖L1(Ω)<(8π)2/χ; as to the mixed boundary value problem, we will establish global existence and boundedness of solutions if ‖u0‖L1(Ω)<(8π)2/χ without radial symmetry.The key ingredients are a Lyapunov functional and an Adams type inequality. A Lyapunov functional of the above problems will be constructed and the constant (8π)2/χ is deduced from the critical constant in the Adams type inequality. This result is regarded as a generalization of the well-known 8π problem in the Keller–Segel system to higher dimensions.

Singular Adams inequality for biharmonic operator on Heisenberg Group and its applications

The goal of this paper is to establish singular Adams type inequality for biharmonic operator on Heisenberg group. As an application, we establish the existence of a solution to $$\Delta_{\mathbb{H}^n}^2 u=\frac{f(\xi,u)}{\rho(\xi)^a} \,\,\text{ in } \Omega,\,\, u|_{\partial\Omega}=0=\left.\frac{\partial u}{\partial \nu} \right|_{\partial\Omega},$$ where $${0\in \Omega \subseteq \mathbb{H}^4}$$ is a bounded domain, $$0 \leq a \leq Q,\,(Q=10).$$ The special feature of this problem is that it contains an exponential nonlinearity and singular potential.

Sharp Affine and Improved Moser–Trudinger–Adams Type Inequalities on Unbounded Domains in the Spirit of Lions

The purpose of this paper is threefold. First, we prove sharp singular affine Moser–Trudinger inequalities on both bounded and unbounded domains in \({\mathbb {R}}^{n}\). In particular, we will prove the following much sharper affine Moser–Trudinger inequality in the spirit of Lions (Rev Mat Iberoamericana 1(2):45–121, 1985) (see our Theorem 1.4): Let \(\alpha _{n}=n\left( \frac{n\pi ^{\frac{n}{2}}}{\Gamma (\frac{n}{2}+1)}\right) ^{\frac{1}{n-1}}\), \(0\le \beta 0\). Then there exists a constant \(C=C\left( n,\beta \right) >0\) such that for all \(0\le \alpha \le \left( 1-\frac{\beta }{n}\right) \alpha _{n}\) and \(u\in C_{0}^{\infty }\left( {\mathbb {R}}^{n}\right) \setminus \left\{ 0\right\} \) with the affine energy \(~{\mathcal {E}}_{n}\left( u\right) <1\), we have $$\begin{aligned} {\displaystyle \int \nolimits _{{\mathbb {R}}^{n}}} \frac{\phi _{n,1}\left( \frac{2^{\frac{1}{n-1}}\alpha }{\left( 1+{\mathcal {E}}_{n}\left( u\right) ^{n}\right) ^{\frac{1}{n-1}}}\left| u\right| ^{\frac{n}{n-1}}\right) }{\left| x\right| ^{\beta }}dx\le C\left( n,\beta \right) \frac{\left\| u\right\| _{n}^{n-\beta }}{\left| 1-{\mathcal {E}}_{n}\left( u\right) ^{n}\right| ^{1-\frac{\beta }{n}}}. \end{aligned}$$ Moreover, the constant \(\left( 1-\frac{\beta }{n}\right) \alpha _{n}\) is the best possible in the sense that there is no uniform constant \(C(n, \beta )\) independent of u in the above inequality when \(\alpha >\left( 1-\frac{\beta }{n}\right) \alpha _{n}\). Second, we establish the following improved Adams type inequality in the spirit of Lions (Theorem 1.8): Let \(0\le \beta 0\). Then there exists a constant \(C=C\left( m,\beta ,\tau \right) >0\) such that $$\begin{aligned} \underset{u\in W^{2,m}\left( {\mathbb {R}}^{2m}\right) , \int _{ {\mathbb {R}}^{2m}}\left| \Delta u\right| ^{m}+\tau \left| u\right| ^{m} \le 1}{\sup } {\displaystyle \int \nolimits _{{\mathbb {R}}^{2m}}} \frac{\phi _{2m,2}\left( \frac{2^{\frac{1}{m-1}}\alpha }{\left( 1+\left\| \Delta u\right\| _{m}^{m}\right) ^{\frac{1}{m-1}}}\left| u\right| ^{\frac{m}{m-1}}\right) }{\left| x\right| ^{\beta }}dx\le C\left( m,\beta ,\tau \right) , \end{aligned}$$ for all \(0\le \alpha \le \left( 1-\frac{\beta }{2m}\right) \beta (2m,2)\). When \(\alpha >\left( 1-\frac{\beta }{2m}\right) \beta (2m,2)\), the supremum is infinite. In the above, we use $$\begin{aligned} \phi _{p,q}(t)=e^{t}- {\displaystyle \sum \limits _{j=0}^{j_{\frac{p}{q}}-2}} \frac{t^{j}}{j!},\,\,\,j_{\frac{p}{q}}=\min \left\{ j\in {\mathbb {N}} :j\ge \frac{p}{q}\right\} \ge \frac{p}{q}. \end{aligned}$$ The main difficulties of proving the above results are that the symmetrization method does not work. Therefore, our main ideas are to develop a rearrangement-free argument in the spirit of Lam and Lu (J Differ Equ 255(3):298–325, 2013; Adv Math 231(6): 3259–3287, 2012), Lam et al. (Nonlinear Anal 95: 77–92, 2014) to establish such theorems. Third, as an application, we will study the existence of weak solutions to the biharmonic equation $$\begin{aligned} \left\{ \begin{array}{l} \Delta ^{2}u+V(x)u=f(x,u)\text { in }{\mathbb {R}}^{4}\\ u\in H^{2}\left( {\mathbb {R}}^{4}\right) ,~u\ge 0 \end{array} \right. , \end{aligned}$$ where the nonlinearity f has the critical exponential growth.