We consider the generalized Wirtinger inequality \[ (\int_{0}^{T} a |u|^q )^{1/q} \le C \biggm(\int_{0}^{T} a^{1-p} |u'|^{p}\biggm)^{1/p}, \] with $p,q>1$, $T>0$, $a\in L^1[0,T]$, $a\ge0$, $a\not\equiv0$ and where $u$ is a $T$-periodic function satisfying the constraint \[ \int_{0}^{T} a |u|^{q-2}u =0. \] We provide the best constant $C>0$ as well as all extremals. Furthermore, we characterize the natural functional space where the inequality is defined.
Let $Ω$ be a smooth, convex, unbounded domain of $\R^N$. Denote by $μ_1(Ω)$ the first nontrivial Neumann eigenvalue of the Hermite operator in $Ω$; we prove that $μ_1(Ω) \ge 1$. The result is sharp since equality sign is achieved when $Ω$ is a $N$-dimensional strip. Our estimate can be equivalently viewed as an optimal Poincaré-Wirtinger inequality for functions belonging to the weighted Sobolev space $H^1(Ω,dγ_N)$, where $γ_N$ is the $N$-dimensional Gaussian measure.
We obtain a sharp estimate for the best constant $C>0$ in the Wirtinger type inequality \[ \int_0^{2π}γ^pw^2\le C\int_0^{2π}γ^qw'^2 \] where $γ$ is bounded above and below away from zero, $w$ is $2π$-periodic and such that $\int_0^{2π}γ^pw=0$, and $p+q\ge0$. Our result generalizes an inequality of Piccinini and Spagnolo.
In this paper, generalizing to the non smooth case already existing results, we prove that, for any convex planar set $Ω$, the first non-trivial Neumann eigenvalue $μ_1(Ω)$ of the Hermite operator is greater than or equal to 1. Furthermore, and this is our main result, under some additional assumptions on $Ω$, we show that $μ_1(Ω)=1$ if and only if $Ω$ is any strip. The study of the equality case requires, among other things, an asymptotic analysis of the eigenvalues of the Hermite operator in thin domains.
In this note, we characterize the equality case of the sharp $L^2$-Euclidean logarithmic Sobolev inequality with monomial weights, exploiting the idea by Bobkov and Ledoux \cite{Bob}. Our approach is new even in the unweighted case. Also, we show that the same strategy yields the equality case of the sharp $L^p$-Euclidean logarithmic Sobolev inequality for $1 < p < \infty$ with arbitrary norm on ${\mathbb R}^n$, if the inequality is unweighted.
The main aim of this note is to prove a sharp Poincaré-type inequality for vector-valued functions on $\mathbb{S}^2$, that naturally emerges in the context of micromagnetics of spherical thin films.
In this paper we deal with nonnegative distributional supersolutions for a class of linear elliptic equations involving inverse-square potentials and logarithmic weights. We prove sharp nonexistence results.
We prove a sharp quantitative form of isocapacitary inequality in the case of a general $p$. This work is a generalization of the author's paper with Guido De Philippis and Michele Marini, where we treated the case of $2$-capacity.
In this paper, we prove Hardy-Leray inequality for three-dimensional solenoidal (i.e., divergence-free) fields with the best constant. To derive the best constant, we impose the axisymmetric condition only on the swirl components. This partially complements the former work by O. Costin and V. Maz'ya \cite{Costin-Mazya} on the sharp Hardy-Leray inequality for axisymmetric divergence-free fields.
There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent $λ=n-α$ (that is for the case of $α>n$). In this paper we confirm the possibility for the extension along the first direction by establishing the sharp Hardy-Littlewood-Sobolev inequality on the upper half space (which is conformally equivalent to a ball). The existences of extremal functions are obtained; And for certain range of the exponent, we classify all extremal functions via…
Sharp $L^p$ extensions of Pitt's inequality expressed as a weighted Sobolev inequality are obtained using convolution estimates and Stein-Weiss potentials. More generally, optimal constants are obtained for the full Stein-Weiss potential as a map from $L^p$ to itself, and new proofs are given for the classical Pitt and Stein-Weiss estimates.
We prove a sharp quantitative form of the classical isocapacitary inequality. Namely, we show that the difference between the capacity of a set and that of a ball with the same volume bounds the square of the Fraenkel asymmetry of the set. This provides a positive answer to a conjecture of Hall, Hayman, and Weitsman (J. d' Analyse Math. '91).
In this paper we establish the best constant of an anisotropic Gagliardo-Nirenberg-type inequality related to the Benjamin-Ono-Zakharov-Kuznetsov equation. As an application of our results, we prove the uniform bound of solutions for such a equation in the energy space.
The relative isoperimetric inequality inside an open, convex cone $\mathcal C$ states that, at fixed volume, $B_r \cap \mathcal C$ minimizes the perimeter inside $\mathcal C$. Starting from the observation that this result can be recovered as a corollary of the anisotropic isoperimetric inequality, we exploit a variant of Gromov's proof of the classical isoperimetric inequality to prove a sharp stability result for the relative isoperimetric inequality inside $\mathcal C$. Our proof follows the line of reasoning in \cite{Fi}, though several new ideas are needed in order to deal with the lack o…
Sharp error estimates in terms of the fractional Laplacian and a weighted Besov norm are obtained for Pitt's inequality by using the spectral representation with weights for the fractional Laplacian due to Frank, Lieb and Seiringer and the sharp Stein-Weiss inequality. Dilation invariance, group symmetry on a non-unimodular group and a nonlinear Stein-Weiss lemma are used to provide short proofs of the Frank-Seiringer "Hardy inequalities" where fractional smoothness is measured by a Besov norm.
In this paper, we are concerned with the stability problem for endpoint conformally invariant cases of the Sobolev inequality on the sphere $\mathbb{S}^n$. Namely, we will establish the stability for Beckner's log-Sobolev inequality and Beckner's Moser-Onofri inequality on the sphere. We also prove that the sharp constant of global stability for the log-Sobolev inequality on the sphere $\mathbb{S}^n$ must be strictly smaller than the sharp constant of local stability for the same inequality. Furthermore, we also derive the non-existence of the global stability for Moser-Onofri inequality on th…
In this paper, we examine the boundary $L^2$ term of the sharp Sobolev trace inequality $\|u\|_{L^{q}(\pa M)}^2\leq S \|\nabla_g u\|_{L^2(M)}^2 +A(M,g)\|u\|^2_{L^2(\pa M)}$ on Riemannian manifolds $(M,g)$ with boundaries $\pa M$, where $q=\frac{2(n-1)}{n-2}$, $S$ is the best constant and $A(M,g)$ is some positive constant depending only on $M$ and $g$. We obtain a sharp trace inequality involving the mean curvature in a remainder term, which would fail in general once the mean curvature is replaced by any smaller function.
We find best constants in several dilation invariant integral inequalities involving derivatives of functions. Some of these inequalities are new and some were known without best constants. The contents: 1. Estimate for a quadratic form of the gradient, 2. Weighted Gårding inequality for the biharmonic operator, 3.Dilation invariant Hardy's inequality with remainder term, 4. Generalized Hardy-Sobolev inequality with sharp constant, 5. Hardy's inequality with sharp Sobolev remainder term.
We prove a sharp three sphere inequality for solutions to third order perturbations of a product of two second order elliptic operators with real coefficients. Then we derive various kinds of quantitative estimates of unique continuation for the anisotropic plate equation. Among these, we prove a stability estimate for the Cauchy problem for such an equation and we illustrate some applications to the size estimates of an unknown inclusion made of different material that might be present in the plate. The paper is self-contained and the Carleman estimate, from which the sharp three sphere inequ…
In this paper, we study the extremal problem for the Strichartz inequality for the Schrödinger equation on the $\mathbb{R} \times \mathbb{R}^2$; we provide a new proof to the characterization of the extremal functions. The only extremal functions are Gaussian functions up to the natural symmetry of the Strichartz inequality, which was investigated previously by Foschi \cite{Foschi:2007:maxi-strichartz-2d} and Hundertmark-Zharnitsky \cite{Hundertmark-Zharnitsky:2006:maximizers-Strichartz-low-dimensions}.
Let $N\geq 5$, $a>0$, $Ω$ be a smooth bounded domain in $\mathbb{R}^{N}$, $2^*=\frac{2N}{N-2}$, $2^\#=\frac{2(N-1)}{N-2}$ and $||u||^2=|\nabla u|_{2}^2+a|u|_{2}^2$. We prove there exists an $α_{0}>0$ such that, for all $u\in H^1(Ω)\setminus\{0\}$, $$\frac{S}{2^{\frac 2N}}\leq\frac{||u||^2}{|u|_{2^*}^2}\left(1+α_{0}\frac{|u|_{2^\#}^{2^\#}}{||u||\cdot|u|_{2^*}^{2^*/2}}\right).$$ This inequality implies Cherrier's inequality.
We establish an analog Hardy inequality with sharp constant involving exponential weight function. The special case of this inequality (for n=2) leads to a direct proof of Onofri inequality on S^2.
We give a new proof of the almost sharp Moser-Trudinger inequality on compact Riemannian manifolds based on the sharp Moser inequality on Euclidean spaces. In particular we can lower the smoothness requirement of the metric and apply the same approach to higher order Sobolev spaces and manifolds with boundary under several boundary conditions.
We prove a sharp Beckner's inequality for axially symmetric functions on $S^4$. The key ingredients of our proof is some pointwise quantitative properties of Gegenbauer polynomials.
We show that the sharp constant in the classical $n$-dimensional Hardy-Leray inequality can be improved for axisymmetric divergence-free fields, and find its optimal value. The same result is obtained for $n=2$ without the axisymmetry assumption.
The well known duality between the Sobolev inequality and the Hardy-Littlewood-Sobolev inequality suggests that the Nash inequality could also have an interesting dual form, even though the Nash inequality relates three norms instead of two. We provide such a dual form here with sharp constants. This dual inequality relates the $L^2$ norm to the infimal convolution of the $L^\infty $ and $H^{-1}$ norms. The computation of this infimal convolution is a minimization problem, which we solve explicitly, thus providing a new proof of the sharp Nash inequality itself. This proof, via duality, also y…
We provide a full quantitative version of the Gaussian isoperimetric inequality. Our estimate is independent of the dimension, sharp on the decay rate with respect to the asymmetry and with optimal dependence on the mass.
We give a simpler proof of the sharp Frank-Lieb inequality on the Heisenberg group. The proof bypasses the sophisticated argument for existence of a minimizer and is based on the study of the 2nd variation of subcritical functionals using their fundamental techniques.
Using a dimension reduction argument and a stability version of the weighted Sobolev inequality on half space recently proved by Seuffert, we establish, in this paper, some stability estimates (or quantitative estimates) for a family of the sharp Gagliardo--Nirenberg--Sobolev inequalities due to Del Pino and Dolbeault \cite{DelPino}.
We introduce a new variational method for the study of stability in the isoperimetric inequality. The method is quite general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter. Two applications are presented. First we give a new proof of the sharp quantitative isoperimetric inequality in $R^n$. Second we positively answer to a conjecture by Hall concerning the best constant for the quantitative isoperimetric inequality in $R^2$ in the small asymmetry regime.
We give a new proof of certain cases of the sharp HLS inequality. Instead of symmetric decreasing rearrangement it uses the reflection positivity of inversions in spheres. In doing this we extend a characterization of the minimizing functions due to Li and Zhu.
We classify local minimizers of $\intσ_2+\oint H_2$ among all conformally flat metrics in the Euclidean $(n+1)$-ball, $4\leq n\leq 5$, for which the boundary has unit volume, subject to an ellipticity assumption. We also classify local minimizers of the analogous functional in the critical dimension $n+1=4$. If minimizers exist, this implies a fully nonlinear sharp Sobolev trace inequality. Our proof is an adaptation of the Frank--Lieb proof of the sharp Sobolev inequality, and in particular does not rely on symmetrization or Obata-type arguments.
We establish several Poincaré--Sobolev type inequalities for the Lapalce--Beltrami operator $Δ_g$ in the hyperbolic space $\mathbb H^n$ with $n\geq 5$. These inequalities could be seen as the improved second order Poincaré inequality with remainder terms involving with the sharp Rellich inequality or sharp Sobolev inequality in $\mathbb H^n$. The novelty of these inequalities is that it combines both the sharp Poincaré inequality and the sharp Rellich inequality or the sharp Sobolev inequality for $Δ_g$ in $\mathbb H^n$. As a consequence, we obtain the Poincaré--Sobolev inequality for the seco…
In this paper we prove a class of second order Caffarelli-Kohn-Nirenberg inequalities which contains the sharp second order uncertainty principle recently established by Cazacu, Flynn and Lam \cite{CFL2020} as a special case. We also show the sharpness of our inequalities for several classes of parameters. Finally, we prove two stability versions of the sharp second order uncertainty principle of Cazacu, Flynn and Lam by showing that the difference of both sides of the inequality controls the distance to the set of extremal functions in $L^2$ norm of gradient of functions.
We show that the sharp constant in the Hardy-Littlewood-Sobolev inequality can be derived using the method that we employed earlier for a similar inequality on the Heisenberg group. The merit of this proof is that it does not rely on rearrangement inequalities; it is the first one to do so for the whole parameter range.
We prove a Log Log inequality with a sharp constant in four dimensions for radially symmetric functions. We also show that the constant in the Log estimate is almost sharp.
We find a new sharp trace Gagliardo-Nirenberg-Sobolev inequality on convex cones, aswell as a weighted sharp trace Sobolev inequality on epigraphs of convex functions. This is done by using a generalized Borell-Brascamp-Lieb inequality, coming from the Brunn-Minkowski theory.
In this paper we show a new inequality which generalizes to the unit sphere the Lebedev-Milin inequality of the exponentiation of functions on the unit circle. It may also be regarded as the counterpart on the sphere of the second inequality in the Szegö limit theorem on the Toeplitz determinants on the circle. On the other hand, this inequality is also a variant of several classical inequalities of Moser-Trudinger type on the sphere. The inequality incorporates the deviation of the center of mass from the origin into the optimal inequality of Aubin for functions with mass centered at the orig…
We identify the compactness threshold for optimizing sequences of the Airy-- Strichartz inequality as an explicit multiple of the sharp constant in the Strichartz inequality. In particular, if the sharp constant in the Airy--Strichartz inequality is strictly smaller than this multiple of the sharp constant in the Strichartz inequality, then there is an optimizer for the former inequality. Our result is valid for the full range of Airy--Strichartz inequalities (except the endpoints) both in the diagonal and off-diagonal cases.
