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...

We obtain interpolation inequalities for derivatives: $\int M_{q,\alpha }\left(\right|\nabla f\left(x\right)\left|\right)dx\le C[\int M_{p,\beta }\left(\Phi ₁\right(x,\left|f\right|,|{\nabla }^{\left(2\right)}f\left|\right))dx+\int M_{r,\gamma }\left(\Phi ₂\right(x,\left|f\right|,|{\nabla }^{\left(2\right)}f\left|\right)\left)dx\right]$, and their counterparts expressed in Orlicz norms: ||∇f||²(q,α) ≤ C||Φ₁(x,|f|,|∇(2)f|)||(p,β) ||Φ₂(x,|f|,|∇(2)f|)||(r,γ)$,$where ${\left|\right|·\left|\right|}_{(s,\kappa )}$ is the Orlicz norm relative to the function $M_{s,\kappa }\left(t\right)=t^s{\left(ln(2+t)\right)}^{\kappa }$. The parameters p,q,r,α,β,γ and the Carathéodory functions Φ₁,Φ₂ are supposed to satisfy certain consistency conditions. Some of the classical Gagliardo-Nirenberg inequalities follow as a special case. Gagliardo-Nirenberg inequalities in logarithmic spaces with higher...

We derive inequalities of Gagliardo-Nirenberg type in weighted Orlicz spaces on ℝⁿ, for maximal functions of derivatives and for the derivatives themselves. This is done by an application of pointwise interpolation inequalities obtained previously by the first author and of Muckenhoupt-Bloom-Kerman-type theorems for maximal functions.

Let $\Omega \subset {ℝ}^n$, $n\ge 2$, be a bounded connected domain of the class $C^{1,\theta }$ for some $\theta \in (0,1]$. Applying the generalized Moser-Trudinger inequality without boundary condition, the Mountain Pass Theorem and the Ekeland Variational Principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem $$u\in W^1{L}^{\Phi }\left(\Omega \right),-div\left({\Phi }^{\text{'}}\left(\right|\nabla u\left|\right)\frac{\nabla u}{\left|\nabla u\right|}\right)+V\left(x\right){\Phi }^{\text{'}}\left(\right|u\left|\right)\frac{u}{\left|u\right|}=f(x,u)+\mu h\left(x\right)\text{in}\Omega ,\frac{\partial u}{\partial 𝐧}=0\text{on}\partial \Omega ,$$ where $\Phi$ is a Young function such that the space $W^1{L}^{\Phi }\left(\Omega \right)$ is embedded into exponential or multiple exponential Orlicz space, the nonlinearity $f(x,t)$ has the corresponding critical growth, $V\left(x\right)$ is a continuous potential,...

Let n ≥ 2 and let Ω ⊂ ℝn be an open set. We prove the boundedness of weak solutions to the problem $$u\in W_0^1{L}^{\Phi }\left(\Omega \right)and-div\left({\Phi }^{\text{'}}\left(\left|\nabla u\right|\right)\frac{\nabla u}{\left|\nabla u\right|}\right)+V\left(x\right){\Phi }^{\text{'}}\left(\left|u\right|\right)\frac{u}{\left|u\right|}=f\left(x,u\right)+\mu h\left(x\right)in\Omega ,$$ where ϕ is a Young function such that the space W 01 L Φ(Ω) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h ∈ L Φ(Ω) is a non-trivial continuous function and µ ≥ 0 is a small parameter. We consider two classical cases: the case of Ω being an open bounded set and the case of Ω =...