Moment inequality and Hölder inequality for BSDEs

Under the Lipschitz and square integrable assumptions on the generator g of BSDEs, this paper proves that if g is positively homogeneous in （y, z） and is decreasing in y, then the Moment inequality for BSDEs with generator g holds in general, and if g is positively homogeneous and sub-additive in （y, z）, then the HSlder inequality and Minkowski inequality for BSDEs with generator g hold in general.     

Integral representations and the generalized Poincaré inequality on Carnot groups

We give some integral representations of the form f(x) = P(f)+K(?f) on two-step Carnot groups, where P(f) is a polynomial and K is an integral operator with a specific singularity. We then obtain the weak Poincaré inequality and coercive estimates as well as the generalized Poincaré inequality on the general Carnot groups.     

The <Emphasis Type="Italic">p</Emphasis>-Harmonic Capacity of an Asymptotically Flat 3-Manifold with Non-negative Scalar Curvature

As an inclusive \({(1,3)\ni p}\)—extension of Bray–Miao’s Theorem 1 and Corollary 1 (Invent Math 172:459–475, 2008) for p = 2, this note presents a sharp isoperimetric inequality for the p-harmonic capacity of a surface in the complete, smooth, asymptotically flat 3-manifold with non-negative scalar curvature, and then an optimal Riemannian Penrose type inequality linking the ADM/total mass and the p-harmonic capacity by means of the deficit of Willmore’s energy. Even in the Euclidean 3-space, the discovered result for \({p \not =2}\) is new and non-trivial.     

On vector variational-like inequalities and vector optimization problems with (<Emphasis Type="Italic">G</Emphasis>, <Emphasis Type="Italic">α</Emphasis>)-invexity

The aim of this paper is to study the relationship among Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and vector optimization problem involving (G, α)-invex functions. Furthermore, we establish equivalence among the solutions of weak formulations of Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and weak efficient solution of vector optimization problem under the assumption of (G, α)-invex functions. Examples are provided to elucidate our results.     

Remarks on curvature dimension conditions on graphs

We show a connection between the \(CDE'\) inequality introduced in Horn et al. (Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for nonnegative curvature graphs. arXiv:1411.5087v2, 2014) and the \(CD\psi \) inequality established in Münch (Li–Yau inequality on finite graphs via non-linear curvature dimension conditions. arXiv:1412.3340v1, 2014). In particular, we introduce a \(CD_\psi ^\varphi \) inequality as a slight generalization of \(CD\psi \) which turns out to be equivalent to \(CDE'\) with appropriate choices of \(\varphi \) and \(\psi \). We use this to prove that the \(CDE'\) inequality implies the classical CD inequality on graphs, and that the \(CDE'\) inequality with curvature bound zero holds on Ricci-flat graphs.     

A Link Between the Log-Sobolev Inequality and Lyapunov Condition

We give an alternative look at the log-Sobolev inequality (LSI in short) for log-concave measures by semigroup tools. The similar idea yields a heat flow proof of LSI under some quadratic Lyapunov condition for symmetric diffusions on Riemannian manifolds provided the Bakry-Emery’s curvature is bounded from below. Let’s mention that, the general ?-Lyapunov conditions were introduced by Cattiaux et al. (J. Funct. Anal. 256(6), 1821–1841 2009) to study functional inequalities, and the above result on LSI was first proved subject to ?(?) = d²(?,x₀) by Cattiaux et al. (Proba. Theory Relat. Fields 148(1–2), 285–304 2010) through a combination of detective L² transportation-information inequality W₂I and the HWI inequality of Otto-Villani. Next, we assert a converse implication that the Lyapunov condition can be derived from LSI, which means their equivalence in the above setting.     

Semilinear elliptic variational inequalities with dependence on the gradient via Mountain Pass techniques

In this paper we consider a semilinear variational inequality with a gradient-dependent nonlinear term. Obviously the nature of this problem is non-variational. Nevertheless we study that problem associating a suitable semilinear variational inequality, variational in nature, with it, and performing an iterative technique used in De Figueiredo et al. (2004) [6] in order to treat semilinear elliptic equations when there is a gradient dependence on the nonlinearity. We prove the existence of a non-trivial non-negative weak solution u for our problem using essentially variational methods, a penalization technique and an iterative scheme. Via Lewy-Stampacchia’s estimates and regularity theory for elliptic equation we also show that u is differentiable and its gradient is α-H?lder continuous on for any α∈(0,1).     

Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality

Let $\mathbb{M}$ be a smooth connected manifold endowed with a smooth measure μ and a smooth locally subelliptic diffusion operator L which is symmetric with respect to μ. We assume that L satisfies a generalized curvature dimension inequality as introduced by Baudoin and Garofalo (2009) [9]. Our goal is to discuss functional inequalities for μ like the Poincaré inequality, the log-Sobolev inequality or the Gaussian logarithmic isoperimetric inequality.     

An Extension of a Simply Inequality Between von Neumann–Jordan and James Constants in Banach Spaces

For a non-trivial Banach space X, let J(X), CNJ(X), C_(NJ)~(p)(X) respectively stand for the James constant, the von Neumann–Jordan constant and the generalized von Neumann–Jordan constant recently inroduced by Cui et al. In this paper, we discuss the relation between the James and the generalized von Neumann–Jordan constants, and establish an inequality between them: C_(NJ)~(p)(X) ≤J(X) with p ≥ 2, which covers the well-known inequality CNJ(X) ≤ J(X). We also introduce a new constant, from which we establish another inequality that extends a result of Alonso et al.     

On a Harnack inequality for the elliptic (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-Laplacian

A special Harnack inequality is proved for solutions of nonlinear elliptic equations of the p(x)-Laplacian type with a variable exponent p(x) that takes different values on two sides of a hyperplane dividing the domain. Examples are given showing that the classical Harnack inequality does not hold in this case.     

ℚ-Fano threefolds of index 7

We show that if the inequality dim|?K_X| ≥ 15 holds for a ?-Fano threefold X of Fano index 7, then X is isomorphic to one of the following varieties: ?(1², 2, 3), X₆ ? ?(1, 2², 3, 5), or X₆ ? ?(1, 2, 3², 4).     

Wintgen Ideal Surfaces in Four-dimensional Neutral Indefinite Space Form {R^4_2(c)}

For an oriented space-like surface M in a four-dimensional indefinite space form ${R^4_2(c)}$ , there is a Wintgen type inequality; namely, the Gauss curvature K, the normal curvature K ^D and mean curvature vector H of M in ${R^4_2(c)}$ satisfy the general inequality: ${K+K^D \geq \langle H,H \rangle+c}$ . An oriented space-like surface in ${R^4_2(c)}$ is called Wintgen ideal if it satisfies the equality case of the inequality identically. In this paper, we study Wintgen ideal surfaces in ${R^4_2(c)}$ . In particular, we classify Wintgen ideal surfaces in ${R^4_2(c)}$ with constant Gauss and normal curvatures. We also completely classify Wintgen ideal surfaces in ${\mathbb E^4_2}$ satisfying |K| = |K ^D | identically.     

A complement of the Ando-Hiai inequality

In this paper, we present a complement of a generalized Ando-Hiai inequality due to Fujii and Kamei [M. Fujii, E. Kamei, Ando-Hiai inequality and Furuta inequality, Linear Algebra Appl. 416 (2006) 541-545]. Let A and B be positive operators on a Hilbert space H such that 0<m₁?A?M₁ and 0<m₂?B?M₂ for some scalars m_i?M_i (i=1,2), and let α∈[0,1]. Put for i=1,2. Then for each 0<r?1 and s?1

     

Remarks on the Gagliardo-Nirenberg Type Inequality in the Besov and the Triebel-Lizorkin Spaces in the Limiting Case

We consider the generalized Gagliardo-Nirenberg inequality in $\Bbb{R}^{n}$ including homogeneous Besov space $\dot{B}^{s}_{r,\rho}(\Bbb{R}^{n})$ with the critical order s=n/r, which describes the continuous embedding such as $L^{p}(\Bbb{R}^{n})\cap\dot{B}^{n/r}_{r,\rho}(\Bbb{R}^{n})\subset L^{q}(\Bbb{R}^{n})$ for all q with p ≦ q<∞, where 1≦ p ≦ r<∞ and 1<ρ ≦∞. Indeed, the following inequality holds: $$\|u\|_{L^{q}(\Bbb{R}^{n})}\leqq C\,q^{1-1/\rho}\|u\|_{L^{p}(\Bbb{R}^{n})}^{p/q}\|u\|_{\dot{B}^{n/r}_{r,\rho}(\Bbb{R}^{n})}^{1-p/q},$$ where C is a constant depending only on r. In this inequality, we have the exact order 1?1/ρ of divergence to the power q tending to the infinity. Furthermore, as a corollary of this inequality, we obtain the Gagliardo-Nirenberg inequality with the homogeneous Triebel-Lizorkin space $\dot{F}^{n/r}_{r,\rho}(\Bbb{R}^{n})$ , which implies the usual Sobolev imbedding with the critical Sobolev space $\dot{H}^{n/r}_{r}(\Bbb{R}^{n})$ . Moreover, as another corollary, we shall prove the Trudinger-Moser type inequality in $\dot{B}^{n/r}_{r,\rho}(\Bbb{R}^{n})$ .     

Around Choi inequalities for positive linear maps

We show an inequality for unital positive linear maps Φ interpolating Choi’s inequality (p=0) with a recent result of Bourin and Ricard (r=1):

     

L’inégalité de Melin-Hörmander en Caractéristiques Multiples

We prove here the Melin-Hörmander inequality with the optimal loss of k/2 - 1/4 derivatives. We then prove that if k ≥ 2, the inequality with k/2-1/4 loss is false in general. We correct and improve the work of [13], which has well posed the question.     

Some inequalities for <Emphasis Type="Italic">k</Emphasis>-colored partition functions

Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for k-colored partition functions \(p_{-k}(n)\) for all \(k\ge 2\). This enables us to extend the k-colored partition function multiplicatively to a function on k-colored partitions and characterize when it has a unique maximum. We conclude with one conjectural inequality that strengthens our results.