Let $T_{\gamma }f\left(x\right)={ʃ}_0^{x}k{(x,y)}^{\gamma }f\left(y\right)dy$, where k is a nonnegative kernel increasing in x, decreasing in y, and satisfying a triangle inequality. An nth-order Opial inequality has the form ${ʃ}_0^{\infty }({\prod }_{i=1}^n|T_{{\gamma }_i}{f\left(x\right)|}^{q_i}{\left|\right)\left|f\left(x\right)\right|}^{q_0}w\left(x\right)dx\le C({ʃ}_0^{\infty }{\left|f\left(x\right)\right|}^p{v\left(x\right)dx)}^{(q_0+\dots +q_n)/p}$. Such inequalities can always be simplified to nth-order reduced inequalities, where the exponent $q_0=0$. When n = 1, the reduced inequality is a standard weighted norm inequality, and characterizing the weights is easy. We also find necessary and sufficient conditions on the weights for second-order reduced Opial inequalities to hold.

We prove a Hardy inequality for the fractional Laplacian on the interval with the optimal constant and additional lower order term. As a consequence, we also obtain a fractional Hardy inequality with the best constant and an extra lower order term for general domains, following the method of M. Loss and C. Sloane [J. Funct. Anal. 259 (2010)].

We prove a fractional version of the Hardy-Sobolev-Maz’ya inequality for arbitrary domains and $L^p$ norms with p ≥ 2. This inequality combines the fractional Sobolev and the fractional Hardy inequality into a single inequality, while keeping the sharp constant in the Hardy inequality.

We develop in this paper an improvement of the method given by S. Bobkov and M. Ledoux in [BL00]. Using the Prékopa-Leindler inequality, we prove a modified logarithmic Sobolev inequality adapted for all measures on ${ℝ}^n$, with a strictly convex and super-linear potential. This inequality implies modified logarithmic Sobolev inequality, developed in [GGM05, GGM07], for all uniformly strictly convex potential as well as the Euclidean logarithmic Sobolev inequality.

In a statistical mechanics model with unbounded spins, we prove uniqueness of the Gibbs measure under various assumptions on finite volume functional inequalities. We follow Royer's approach (Royer, 1999) and obtain uniqueness by showing convergence properties of a Glauber-Langevin dynamics. The result was known when the measures on the box [-n,n]d (with free boundary conditions) satisfied the same logarithmic Sobolev inequality. We generalize this in two directions: either the constants may be...