Sharp Inequalities Research Articles

Improved higher-order Sobolev inequalities on CR sphere

We improve higher-order CR Sobolev inequalities on S2n+1 under the vanishing of higher order moments of the volume element. As an application, we give a new and direct proof of the classification of minimizers of the CR invariant higher-order Sobolev inequalities. In the same spirit, we prove almost sharp Sobolev inequalities for GJMS operators to general CR manifolds, and obtain the existence of minimizers in C2k(N) of higher-order CR Yamabe-type problems when Yk(N)<Yk(Hn).

Sharper bounds for the numerical radius

In this paper, we discuss and present new sharp inequalities for the numerical radii of Hilbert space operators. In particular, if A and B are bounded linear operators on a Hilbert space, we present new upper bounds for . The main tool to obtain our results is using block matrix techniques. Among many interesting results, and as an application of the new inequalities, we obtain the following bound for the numerical radius of an operator T, where , , and denote the numerical radius, the usual operator norm, and the absolute value operator, respectively. Other difference inequalities will be presented too.

On the local systolic optimality of Zoll contact forms

We prove a normal form for contact forms close to a Zoll one and deduce that Zoll contact forms on any closed manifold are local maximizers of the systolic ratio. Corollaries of this result are: (1) sharp local systolic inequalities for Riemannian and Finsler metrics close to Zoll ones, (2) the perturbative case of a conjecture of Viterbo on the symplectic capacity of convex bodies, (3) a generalization of Gromov’s non-squeezing theorem in the intermediate dimensions for symplectomorphisms that are close to linear ones.

A twist in sharp Sobolev inequalities with lower order remainder terms

Abstract Let ( M , g ) {(M,g)} be a smooth compact Riemannian manifold of dimension n ≥ 3 {n\geq 3} . Let also A be a smooth symmetrical positive ( 0 , 2 ) {(0,2)} -tensor field in M. By the Sobolev embedding theorem, we can write that there exist K , B &gt; 0 {K,B&gt;0} such that for any u ∈ H 1 ⁢ ( M ) {u\in H^{1}(M)} , (0.1) ∥ u ∥ L 2 ⋆ 2 ≤ K ⁢ ∥ ∇ A ⁡ u ∥ L 2 2 + B ⁢ ∥ u ∥ L 1 2 \|u\|_{L^{2^{\star}}}^{2}\leq K\|\nabla_{A}u\|_{L^{2}}^{2}+B\|u\|_{L^{1}}^{2} where H 1 ⁢ ( M ) {H^{1}(M)} is the standard Sobolev space of functions in L 2 {L^{2}} with one derivative in L 2 {L^{2}} , | ∇ A ⁡ u | 2 = A ⁢ ( ∇ ⁡ u , ∇ ⁡ u ) {|\nabla_{A}u|^{2}=A(\nabla u,\nabla u)} and 2 ⋆ {2^{\star}} is the critical Sobolev exponent for H 1 {H^{1}} . We compute in this paper the value of the best possible K in (0.1) and investigate the validity of the corresponding sharp inequality.

Sharp inequalities of Bienaymé–Chebyshev and Gauß type for possibly asymmetric intervals around the mean

Sharp upper bounds are proved for the probability that a standardized random variable takes on a value outside a possibly asymmetric interval around 0. Six classes of distributions for the random variable are considered, namely the general class of ‘distributions’, the class of ‘symmetric distributions’, of ‘concave distributions’, of ‘unimodal distributions’, of ‘unimodal distributions with coinciding mode and mean’, and of ‘symmetric unimodal distributions’. In this way, results by Gauß (Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores 5:1–58, 1823), Bienaymé (C R Hebd Séance Acad Sci Paris 37:309–24, 1853), Bienaymé (C R Hebd Séance Acad Sci Paris 37:309–24, 1853), Chebyshev (Journal de mathématiques pures et appliqués (2) 12:177–184, 1867), and Cantelli (Atti del Congresso Internazionale dei Matematici 6:47–59, 1928) are generalized. For some of the known inequalities, such as the Gauß inequality, an alternative proof is given.

Sharp Uncertainty Principle inequality for solenoidal fields

This paper solves the L2 version of Maz'ya's open problem [1, Section 3.9] on the sharp uncertainty principle inequality∫RN|∇u|2dx∫RN|u|2|x|2dx≥CN(∫RN|u|2dx)2 for solenoidal (namely divergence-free) vector fields u=u(x) on RN. The best value of the constant for N≥3 turns out to be CN=14(N2−4(N−3)+2)2 which exceeds the original value N2/4 for unconstrained fields. Moreover, we show the attainability of CN and specify the profiles of the extremal solenoidal fields: for N≥4, the extremals are proportional to a poloidal field that is axisymmetric and unique up to the axis of symmetry; for N=3, there exist extremal toroidal fields, in addition to extremal poloidal fields; for N=2, the extremal fields are all toroidal.

On the maximizing problem associated with critical Sobolev inequality under inhomogeneous constraints

In this paper, we study a maximizing problem associated with the critical Sobolev inequality under inhomogeneous constraints. The problem of this type was previously studied by Ishiwata and Wadade [On the effect of equivalent constrain on a maximizing problem associated with the Sobolev type embedding in . Math Ann. 2016;364:1043–1068; On the maximizing problem associated with Sobolev type embeddings under inhomogeneous constraints. Appl Anal. 2019;98(10):1916–1934], Ishiwata [On variational problems associated with Sobolev type inequalities and related topics. Available from: http://www.rism.it/doc/Ishiwata.pdf] and Nguyen [Maximizers for the variational problems associated with Sobolev type inequalities under constraints. Math Ann. 2018;372(1–2):229–255]. Our results give a complete picture of the effect of the constraints on the attainability and non-attainability of the problem. The sharp Sobolev inequality plays a crucial role in our argument. Our method also provides a new and simple proof for the recent results of Ishiwata and Wadade [On the maximizing problem associated with Sobolev type embeddings under inhomogeneous constraints. Appl Anal. 2019;98(10):1916–1934] concerning the sub-critical Sobolev-type inequalities (or Gagliardo–Nirenberg inequalities).

Estimating the probability that a given vector is in the convex hull of a random sample

For a d-dimensional random vector X, let p_{n, X}(theta ) be the probability that the convex hull of n independent copies of X contains a given point theta . We provide several sharp inequalities regarding p_{n, X}(theta ) and N_X(theta ) denoting the smallest n for which p_{n, X}(theta )ge 1/2. As a main result, we derive the totally general inequality 1/2 le alpha _X(theta )N_X(theta )le 3d + 1, where alpha _X(theta ) (a.k.a. the Tukey depth) is the minimum probability that X is in a fixed closed halfspace containing the point theta . We also show several applications of our general results: one is a moment-based bound on N_X(mathbb {E}!left[ Xright] ), which is an important quantity in randomized approaches to cubature construction or measure reduction problem. Another application is the determination of the canonical convex body included in a random convex polytope given by independent copies of X, where our combinatorial approach allows us to generalize existing results in random matrix community significantly.

Minimal surfaces and Schwarz lemma

We prove a sharp Schwarz lemma type inequality for the Weierstrass–Enneper parameterization of minimal disks. It states the following. If F:D→Σ is a conformal harmonic parameterization of a minimal disk Σ⊂R3, where D is the unit disk and |Σ|=πR2, then |Fx(z)|(1−|z|2)≤R. If for some z the previous inequality is equality, then the surface is an affine image of a disk, and F is linear up to a Möbius transformation of the unit disk.

Sharp systolic inequalities for rotationally symmetric 2-orbifolds

We show that suitably defined systolic ratios are globally bounded from above on the space of rotationally symmetric spindle orbifolds and that the upper bound is attained precisely at so-called Besse metrics, i.e. Riemannian orbifold metrics all of whose geodesics are closed. The first type of systolic ratios that we consider are defined in terms of closed geodesics that lift to contractible loops on certain covers of the unit sphere bundle. The second type of systolic ratios are defined in terms of the kth shortest closed geodesic, where the number k depends on the underlying orbifold. Our results generalize a corresponding result of Abbondandolo, Bramham, Hryniewicz and Salomão for spheres of revolution, even in the manifold case. Moreover, they complement recent results by Abbondandolo, Mazzucchelli and the first named author on local systolic inequalities for Besse Reeb flows on closed 3-manifolds.

Sharp inequalities related to the Adamovic-Mitrinovic, Cusa, Wilker and Huygens results

In this paper, we establish sharp inequalities for trigonometric functions. For example, we consider the Wilker inequality and prove that for 0 &lt; x &lt; ?/2 and n ? 1, 2 + (?n?1 j=2 dj+1x2j+ ?nx2n) x3 tan x &lt; (sin x/x)2 + tan x/x &lt; 2 + (?n?1 j=3 dj+1x2j+ Dnx2n) x3 tan x with the best possible constants ?n = dn and Dn = 2?6 ? 168?4 + 15120/945?4 (2/?) 2n ? ?n?1 j=2 dj+1 (2/?/)2n?2j , where dk = 22k+2 ((4k + 6) |B2k+2| + (?1)k+1)/(2k + 3)! and Bk are the Bernoulli numbers (k ? N0 := N? {0}). This improves and generalizes the results given by Mortici, Nenezic and Malesevic.

A sharp Gagliardo-Nirenberg inequality and its application to fractional problems with inhomogeneous nonlinearity

&lt;p style='text-indent:20px;'&gt;The purpose of this paper is threefold. Firstly, we establish a Gagliardo-Nirenberg inequality with optimal constant, which involves a fractional norm and an inhomogeneous nonlinearity. Secondly, as an application of this inequality, we study ground state standing waves to a nonlinear Schrödinger equation (NLS) with a mixed fractional Laplacians and a inhomogeneous nonlinearity, and consider a minimization problem which gives the existence of ground state solutions with prescribed mass. In particular, by making use of this Gagliardo-Nirenberg inequality and its optimal constant, we give a sufficient and necessary condition for the existence results. Finally, we develop local wellposedness theory for NLS with a mixed fractional Laplacians and a inhomogeneous nonlinearity. In the process, we prove Strichartz estimates in Lorentz spaces which may be of independent interest.&lt;/p&gt;

Sharp exponential type inequalities for the arc lemniscate sine function with applications

Sharp exponential type inequalities for the arc lemniscate sine function with applications

Sharp Observability Inequalities for Hyperbolic Systems with Potentials

This paper is devoted to a sharp internal/boundary observability inequality for a hyperbolic system with a zero order potential. For this purpose, we first establish a new Carleman estimate for hyperbolic operator in H1-norm. Based on this Carleman estimate and a modified auxiliary optimal control problem, we obtain Carleman estimate for hyperbolic operator in L2-norm. Then, by virtue of a modified energy estimate and a delicate treatment of the observation region, we obtain an internal observability estimate with the observability constant of the order exp (C‖q‖2/3L∞(Q;ℝN×N)), with q the potential involved in the system. We also address the same problem for boundary observation. Compared with the related results in the literature, the main contributions of this paper are the observability constant is sharper, the waiting time T is shorter and the internal (or boundary) observation domain is smaller.

Corrigendum and Addendum to : "SINGULAR weighted sharp trudinger-moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations" DOI: 10.3934/dcds.2021137

We correct the proof of Theorem 1.11 and Theorem 1.12 and we add a new result of sharpness of the Trudinger-Moser inequality for the doubly exponential case.

New sharp inequalities of Mitrinovic-Adamovic type

In this paper, new sharp Mitrinovic-Adamovic inequalities for circular functions are established.

Bernstein-type bounds for beta distribution

This work obtains sharp closed-form exponential concentration inequalities of Bernstein type for the ubiquitous beta distribution, improving upon sub-Gaussian and sub-gamma bounds previously studied in this context. The proof leverages a novel handy recursion of order 2 for central moments of the beta distribution, obtained from the hypergeometric representations of moments; this recursion is useful for obtaining explicit expressions for central moments and various tail approximations.

Sharp inequalities of Iyengar-Madhava Rao-Nanjundiah type including cos(x/√3 + ax^r)

Sharp inequalities of Iyengar-Madhava Rao-Nanjundiah type including cos(x/√3 + ax^r)

Sharp inequalities for the complete elliptic integrals of the first and second kinds

By studying the monotonicity properties of K(r), E(r) and some combinations of elementary functions and special functions, some new inequalities for the complete elliptic integrals of the first and second kinds are established. where K(r) =??/2 0 (1 ? r2 sin2?)?1/2d?, E(r)=??/2 0 (1 ? r2 sin2?)1/2d? denote the complete elliptic integrals of the first and second kind, r ? (0,1).

ON ONE INEQUALITY OF DIFFERENT METRICS FOR TRIGONOMETRIC POLYNOMIALS

We study the sharp inequality between the uniform norm and \(L^p(0,\pi/2)\)-norm of polynomials in the system \(\mathscr{C}=\{\cos (2k+1)x\}_{k=0}^\infty\) of cosines with odd harmonics. We investigate the limit behavior of the best constant in this inequality with respect to the order \(n\) of polynomials as \(n\to\infty\) and provide a characterization of the extremal polynomial in the inequality for a fixed order of polynomials.