A sharp Blaschke–Santaló inequality for \alpha -concave functions

We define a new transform on \(\alpha \) -concave functions, which we call the \(\sharp \) -transform. Using this new transform, we prove a sharp Blaschke–Santaló inequality for \(\alpha \) -concave functions, and characterize the equality case. This extends the known functional Blaschke–Santaló inequality of Artstein-Avidan, Klartag and Milman, and strengthens a result of Bobkov. Finally, we prove that the \(\sharp \) -transform is a duality transform when restricted to its image. However, this transform is neither surjective nor injective on the entire class of \(\alpha \) -concave functions.     

Stability of the functional forms of the Blaschke-Santaló inequality

Stability versions of the functional forms of the Blaschke-Santaló inequality due to Ball, Artstein–Klartag–Milman, Fradelizi–Meyer and Lehec are proved.     

Increasing functions and inverse Santaló inequality for unconditional functions

Let be a convex function and be its Legendre tranform. It is proved that if is invariant by changes of signs, then . This is a functional version of the inverse Santaló inequality for unconditional convex bodies due to J. Saint Raymond. The proof involves a general result on increasing functions on together with a functional form of Lozanovskii’s lemma. In the last section, we prove that for some c > 0, one has always . This generalizes a result of B. Klartag and V. Milman.      

Stability of some versions of the Prékopa–Leindler inequality

Two consequences of the stability version of the one dimensional Prékopa–Leindler inequality are presented. One is the stability version of the Blaschke–Santaló inequality, and the other is a stability version of the Prékopa– Leindler inequality for even functions in higher dimensions, where a recent stability version of the Brunn–Minkowski inequality is also used in an essential way.     

Stability of some versions of the Prékopa–Leindler inequality

We study the behavior of positive solutions of the following Dirichlet problem

$$\left \{ \begin{array}{ll} -\Delta_{p}u=\lambda u^{s-1}+u^{q-1} &\quad {\rm in}\enspace \Omega \\ u_{\mid\partial \Omega}=0 \end{array}\right. $$

when s → p ^?. Here \({p >1 , s\,{\in}\,]1,p]}\) and q > p with \({q\leq\frac{Np}{N-p}}\) if N > p.

     

Proof of Stability in the Brower–Paul Problem

Doklady Mathematics - We study the stability of equilibrium in the problem known as “a ball on a rotating saddle,” which was first considered by the famous Dutch mathematician Brauer in...     

On an extended Hardy-Hilbert‘s inequality in the whole plane

By means of weight coefficients, a complex integral formula and Hermite-Hadamard’s inequality, a new extended Hardy-Hilbert‘s inequality in the whole plane with multi-parameters and a best possible constant factor is given. The equivalent forms, the operator expressions and a few particular cases are considered.     

Robustness of the Gaussian concentration inequality and the Brunn–Minkowski inequality

We provide a sharp quantitative version of the Gaussian concentration inequality: for every \(r>0\), the difference between the measure of the r-enlargement of a given set and the r-enlargement of a half-space controls the square of the measure of the symmetric difference between the set and a suitable half-space. We also prove a similar estimate in the Euclidean setting for the enlargement with a general convex set. This is equivalent to the stability of the Brunn–Minkowski inequality for the Minkowski sum between a convex set and a generic one.     

Stability of Planar Waves in the Allen–Cahn Equation

We study the asymptotic stability of planar waves for the Allen–Cahn equation on ? ⁿ , where n ≥ 2. Our first result states that planar waves are asymptotically stable under any—possibly large—initial perturbations that decay at space infinity. Our second result states that the planar waves are asymptotically stable under almost periodic perturbations. More precisely, the perturbed solution converges to a planar wave as t → ∞. The convergence is uniform in ? ⁿ . Lastly, the existence of a solution that oscillates permanently between two planar waves is shown, which implies that planar waves are not asymptotically stable under more general perturbations.     

On Bernstein–Walsh-type lemmas in regions of the complex plane

Let G ì \mathbb C G \subset {\mathbb C} be a finite region bounded by a Jordan curve L: = ?G L: = \partial G , let W: = \textext[`(G)] \Omega : = {\text{ext}}\bar{G} (with respect to [`(\mathbb C)] {\overline {\mathbb C}} ), $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} , and let w = F(z) w = \Phi (z) be a univalent conformal mapping of Ω onto Δ normalized by $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 . By A _p (G); p > 0; we denote a class of functions f analytic in G and satisfying the condition

Inversion of the Cauchy–Bunyakovskii–Schwarz inequality

In the present paper, the inequality inverse to the Cauchy–Bunyakovskii–Schwarz inequality and generalizing other well-known inversions of this inequality is proved.     

A refinement of the Brascamp–Lieb–Poincaré inequality in one dimension

In this short note, we give a refinement of the Brascamp–Lieb inequality in the style of the Houdré–Kagan extension for the Poincaré inequality in one dimension. This is inspired by works by Helffer and by Ledoux.     

Space-time asymptotics of the two dimensional Navier–Stokes flow in the whole plane

We consider the space-time behavior of the two dimensional Navier–Stokes flow. Introducing some qualitative structure of initial data, we succeed to derive the first order asymptotic expansion of the Navier–Stokes flow without moment condition on initial data in $L^1({\mathbb{R}}^2)\cap L_{\sigma }^2({\mathbb{R}}^2)$. Moreover, we characterize the necessary and sufficient condition for the rapid energy decay ${6u(t)6}_2=o(t^{?1})$ as $t\to \infty$ motivated by Miyakawa–Schonbek [21]. By weighted estimated in Hardy spaces, we discuss the possibility of the second order asymptotic expansion of the Navier–Stokes flow assuming the first order moment condition on initial data. Moreover, observing that the Navier–Stokes flow $u(t)$ lies in the Hardy space $H^1({\mathbb{R}}^2)$ for $t>0$, we consider the asymptotic expansions in terms of Hardy-norm. Finally we consider the rapid time decay ${6u(t)6}_2=o(t^{?\frac{3}{2}})$ as $t\to \infty$ with cyclic symmetry introduced by Brandolese [2].     

Stability of Pólya–Szegő inequality for log-concave functions

A quantitative version of Pólya–Szeg? inequality is proven for log-concave functions in the case of Steiner and Schwarz rearrangements.     

Spectrum of the Kerzman–Stein operator for a family of smooth regions in the plane

The Kerzman–Stein operator is the skew-hermitian part of the Cauchy operator defined with respect to an unweighted hermitian inner product on the boundary. For bounded regions with smooth boundary, the Kerzman–Stein operator is compact on the Hilbert space of square integrable functions. Here we give an explicit computation of its Hilbert–Schmidt norm for a family of simply connected regions. We also give an explicit computation of the Cauchy operator acting on an orthonormal basis, and we give estimates for the norms of the Kerzman–Stein and Cauchy operators on these regions. The regions are the first regions that display no apparent Möbius symmetry for which there now is explicit spectral information.     

Landau levels on the hyperbolic plane in the presence of Aharonov–Bohm fields

We consider the magnetic Schrödinger operators on the Poincaré upper half plane with constant Gaussian curvature ?1. We assume the magnetic field is given by the sum of a constant field and the Dirac δ measures placed on some lattice. We give a sufficient condition for each Landau level to be an infinitely degenerated eigenvalue. We also prove the lowest Landau level is not an eigenvalue if the above condition fails. In particular, the infinite degeneracy of the lowest Landau level is equivalent to the infiniteness of the zero-modes of the two-dimensional Pauli operator.