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In this note we collect some results in ℝn about global Poincaré inequalities for differential forms obtained in a joint research with Pierre Pansu and presented by the authors in two seminars held in Bologna respectively in 2023 and 2024. At the end of the note we comment some very new results obtained in the Heisenberg groups ℍn.

We consider the following stochastic heat equation ∂ t u (t , x) = 1 2 ∂ x 2 u (t , x) + b (u (t , x)) + σ (u (t , x)) W ˙ (t , x) , defined for (t , x) ∈ (0 , ∞) × R , where W ˙ denotes space-time white noise. The function σ is assumed to be positive, bounded, globally Lipschitz, and bounded uniformly away from the origin, and the function b is assumed to be positive, locally Lipschitz and nondecreasing. We prove that the Osgood condition ∫ 1 ∞ d y b (y) < ∞ implies that the solution almost surely blows up everywhere and instantaneously, In other words, the Osgood condition ensures that P { u (t , x) = ∞ for all t > 0 and x ∈ R } = 1. The main ingredients of the proof involve a hitting-time bound for a class of differential inequalities (Remark 3.3), and the study of the spatial growth of stochastic convolutions using techniques from the Malliavin calculus and the Poincaré inequalities that were developed in Chen et al. (Electron J Probab 26:1–37, 2021, J Funct Anal 282(2):109290, 2022). [ABSTRACT FROM AUTHOR]

Distance covariance is a popular dependence measure for two random vectors |$X$| and |$Y$| of possibly different dimensions and types. Recent years have witnessed concentrated efforts in the literature to understand the distributional properties of the sample distance covariance in a high-dimensional setting, with an exclusive emphasis on the null case that |$X$| and |$Y$| are independent. This paper derives the first non-null central limit theorem for the sample distance covariance, and the more general sample (Hilbert–Schmidt) kernel distance covariance in high dimensions, in the distributional class of |$(X,Y)$| with a separable covariance structure. The new non-null central limit theorem yields an asymptotically exact first-order power formula for the widely used generalized kernel distance correlation test of independence between |$X$| and |$Y$|⁠. The power formula in particular unveils an interesting universality phenomenon: the power of the generalized kernel distance correlation test is completely determined by |$n\cdot \operatorname{dCor}^{2}(X,Y)/\sqrt{2}$| in the high-dimensional limit, regardless of a wide range of choices of the kernels and bandwidth parameters. Furthermore, this separation rate is also shown to be optimal in a minimax sense. The key step in the proof of the non-null central limit theorem is a precise expansion of the mean and variance of the sample distance covariance in high dimensions, which shows, among other things, that the non-null Gaussian approximation of the sample distance covariance involves a rather subtle interplay between the dimension-to-sample ratio and the dependence between |$X$| and |$Y$|⁠. [ABSTRACT FROM AUTHOR]

In this paper, we study one-dimensional hyperbolic Anderson models (HAM) driven by space-time pure-jump Lévy white noise in a finite-variance setting. Motivated by recent active research on limit theorems for stochastic partial differential equations driven by Gaussian noises, we present the first study in this Lévy setting. In particular, we first establish the spatial ergodicity of the solution and then a quantitative central limit theorem (CLT) for the spatial averages of the solution to HAM in both Wasserstein distance and Kolmogorov distance, with the same rate of convergence. To achieve the first goal (i.e. spatial ergodicity), we exploit some basic properties of the solution and apply a Poincaré inequality in the Poisson setting, which requires delicate moment estimates on the Malliavin derivatives of the solution. Such moment estimates are obtained in a soft manner by observing a natural connection between the Malliavin derivatives of HAM and a HAM with Dirac delta velocity. To achieve the second goal (i.e. CLT), we need two key ingredients: (i) a univariate second-order Poincaré inequality in the Poisson setting that goes back to Last, Peccati, and Schulte (Probab. Theory Related Fields, 2016) and has been recently improved by Trauthwein (arXiv:2212.03782); (ii) aforementioned moment estimates of Malliavin derivatives up to second order. We also establish a corresponding functional CLT by (a) showing the convergence in finite-dimensional distributions and (b) verifying Kolmogorov's tightness criterion. Part (a) is made possible by a linearization trick and the univariate second-order Poincaré inequality, while part (b) follows from a standard moment estimate with an application of Rosenthal's inequality. [ABSTRACT FROM AUTHOR]

In this paper we prove Poincaré inequalities for the Discrete de Rham (DDR) sequence on a general connected polyhedral domain Ω of R3. We unify the ideas behind the inequalities for all three operators in the sequence, deriving new proofs for the Poincaré inequalities for the gradient and the divergence, and extending the available Poincaré inequality for the curl to domains with arbitrary second Betti numbers. A key preliminary step consists in deriving “mimetic” Poincaré inequalities giving the existence and continuity of the solutions to topological balance problems useful in general discrete geometric settings. As an example of application, we study the stability of a novel DDR scheme for the magnetostatics problem on domains with general topology.

We consider kinetic Fokker-Planck (or Vlasov-Fokker-Planck) equations on the torus with Maxwellian or fat tail local equilibria. Results based on weak norms have recently been achieved by S. Armstrong and J.-C. Mourrat in case of Maxwellian local equilibria. Using adapted Poincaré and Lions-type inequalities, we develop an explicit and constructive method for estimating the decay rate of time averages of norms of the solutions, which covers various regimes corresponding to subexponential, exponential and superexponential (including Maxwellian) local equilibria. As a consequence, we also derive hypocoercivity estimates, which are compared to similar results obtained by other techniques. [ABSTRACT FROM AUTHOR]

We study boundary value problems for bounded uniform domains in R n , n ⩾ 2 , with non-Lipschitz, and possibly fractal, boundaries. We prove Poincaré inequalities with uniform constants and trace terms for (ε , ∞) -domains contained in a fixed bounded Lipschitz domain. We introduce generalized Dirichlet, Neumann, and Robin problems for Poisson-type equations and prove the Mosco convergence of the associated energy functionals along sequences of suitably converging domains. This implies a stability result for weak solutions, the norm convergence of the associated resolvents, and the convergence of the corresponding eigenvalues and eigenfunctions. We provide compactness results for parametrized classes of admissible domains, energy functionals, and weak solutions. Using these results, we can then prove the existence of optimal shapes in these classes in the sense that they minimize the initially given energy functionals. For the Robin boundary problems, this result is new. [ABSTRACT FROM AUTHOR]

For each natural number n and any bounded, convex domain \Omega \subset \mathbb {R}^n we characterize the sharp constant C(n,\Omega) in the Poincaré inequality \| f - \bar {f}_{\Omega }\|_{L^{\infty }(\Omega ;\mathbb {R})} \leq C(n,\Omega) \|\nabla f\|_{L^{\infty }(\Omega ;\mathbb {R})}. Here, \bar {f}_{\Omega } denotes the mean value of f over \Omega. In the case that \Omega is a ball B_r of radius r in \mathbb {R}^n, we calculate C(n,B_r)=C(n)r explicitly in terms of n and a ratio of the volumes of the unit balls in \mathbb {R}^{2n-1} and \mathbb {R}^n. More generally, we prove that C(n,B_{r(\Omega)}) \leq C(n,\Omega) \leq \frac {n}{n+1}\mathrm {diam}(\Omega), where B_{r(\Omega)} is a ball in \mathbb {R}^n with the same n-dimensional Lebesgue measure as \Omega. Both bounds are sharp, and the lower bound can be interpreted as saying that, among convex domains of equal measure, balls have the best, i.e. smallest, Poincaré constant. [ABSTRACT FROM AUTHOR]

In this paper we prove that several natural approaches to Sobolev spaces coincide on the Vicsek fractal. More precisely, we show that the metric approach of Korevaar-Schoen, the approach by limit of discrete p-energies and the approach by limit of Sobolev spaces on cable systems all yield the same functional space with equivalent norms for p > 1. As a consequence we prove that the Sobolev spaces form a real interpolation scale. We also obtain Lp-Poincaré inequalities for all values of p ≥ 1. [ABSTRACT FROM AUTHOR]

We study stability of the eigenvalues of the generator of a one dimensional reversible diffusion process satisfying some natural conditions. The proof is based on Stein's method. In particular, these results are applied to the Normal distribution (via the Ornstein–Uhlenbeck process), to Gamma distributions (via the Laguerre process) and to Beta distributions (via the Jacobi process). [ABSTRACT FROM AUTHOR]

Given a subset D of the Euclidean space, we study nonlocal quadratic forms that take into account tuples (x , y) ∈ D × D if and only if the line segment between x and y is contained in D. We discuss regularity of the corresponding Dirichlet form leading to the existence of a jump process with visibility constraint. Our main aim is to investigate corresponding Poincaré inequalities and their scaling properties. For dumbbell shaped domains we show that the forms satisfy a Poincaré inequality with diffusive scaling. This relates to the rate of convergence of eigenvalues in singularly perturbed domains. [ABSTRACT FROM AUTHOR]

We show that any d-Ahlfors regular subset of Rn supporting a weak (1, d)-Poincare inequality with respect to surface measure is uniformly rectifiable. [ABSTRACT FROM AUTHOR]

We study the long time behavior of the kinetic Fokker-Planck equation with mean field interaction, whose limit is often called Vlasov-Fokker-Planck equation. We prove a uniform (in the number of particles) exponential convergence to equilibrium for the solutions in the weighted Sobolev space H 1 (μ) with a rate of convergence which is explicitly computable and independent of the number of particles. The originality of the proof relies on functional inequalities and hypocoercivity with Lyapunov type conditions, usually not suitable to provide adimensional results. [ABSTRACT FROM AUTHOR]

Very general univariate and multivariate Caputo ψ-fractional integral inequalities of Poincaré, Sobolev and Hilbert–Pachpatte types are presented. Estimates are with respect to ∥ ⋅ ∥ p {\lVert\,\cdot\,\rVert_{p}} , 1 ≤ p < ∞ {1\leq p<\infty}. Applications are given. [ABSTRACT FROM AUTHOR]

This paper considers two notions of effective dimension for quadrature in weighted pre-Sobolev spaces with dominating mixed partial derivatives. We begin by finding a ball in those spaces just barely large enough to contain a function with unit variance. If no function in that ball has more than ϵ of its variance from analysis of variance (ANOVA) components involving interactions of order s or more, then the space has effective dimension at most s in the superposition sense. A similar truncation sense notion replaces the cardinality of an ANOVA component by the largest index it contains. These effective dimension definitions for the integration problem coincide with some of the definitions in information-based complexity for the function approximation problem. Some Poincar\'e-type inequalities are used to bound variance components by multiples of these space's squared norm and those in turn provide bounds on effective dimension. Very low effective dimension in the superposition sense holds for some spaces defined by product weights in which quadrature is strongly tractable. The superposition dimension is O(log(1/ϵ )/ log(log(1/ϵ ))) just like the superposition dimension used in the multivariate decomposition method. Surprisingly, even spaces where all subset weights are equal, regardless of their cardinality or included indices, have low superposition dimension in this sense. This paper does not require periodicity of the integrands. [ABSTRACT FROM AUTHOR]

The Fast Diffusion Equation (FDE) u t = Δ u m , with m ∈ (0 , 1) , is an important model for singular nonlinear (density dependent) diffusive phenomena. Here, we focus on the Cauchy–Dirichlet problem posed on smooth bounded Euclidean domains. In addition to its physical relevance, there are many aspects that make this equation particularly interesting from the pure mathematical perspective. For instance: mass is lost and solutions may extinguish in finite time, merely integrable data can produce unbounded solutions, classical forms of Harnack inequalities (and other regularity estimates) fail to be true, etc. In this paper, we first provide a survey (enriched with an extensive bibliography) focussing on the more recent results about existence, uniqueness, boundedness and positivity (i.e., Harnack inequalities, both local and global), and higher regularity estimates (also up to the boundary and possibly up to the extinction time). We then prove new global (in space and time) Harnack estimates in the subcritical regime. In the last section, we devote a special attention to the asymptotic behaviour, from the first pioneering results to the latest sharp results, and we present some new asymptotic results in the subcritical case. [ABSTRACT FROM AUTHOR]

For bounded domains Ω , we prove that the Lp -norm of a regular function with compact support is controlled by weighted Lp -norms of its gradient, where the weight belongs to a class of symmetric non-negative definite matrix-valued functions. The class of weights is defined by regularity assumptions and structural conditions on the degeneracy set , where the determinant vanishes. In particular, the weight A is assumed to have rank at least 1 when restricted to the normal bundle of the degeneracy set S. This generalization of the classical Poincaré inequality is then applied to develop a robust theory of first-order Lp -based Sobolev spaces with matrix-valued weight A. The Poincaré inequality and these Sobolev spaces are then applied to produce various results on existence, uniqueness and qualitative properties of weak solutions to boundary-value problems for degenerate elliptic , degenerate parabolic and degenerate hyperbolic partial differential equations (PDEs) of second order written in divergence form, where A is calibrated to the matrix of coefficients of the second-order spatial derivatives. The notion of weak solution is variational : the spatial states belong to the matrix-weighted Sobolev spaces with p = 2. For the degenerate elliptic PDEs, the Dirichlet problem is treated by the use of the Poincaré inequality and Lax–Milgram theorem, while the treatment of the Cauchy–Dirichlet problem for the degenerate evolution equations relies only on the Poincaré inequality and the parabolic and hyperbolic counterparts of the Lax–Milgram theorem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Moebius measures μxn indexed by dimension n and |x| < 1 on the unit sphere Sn−1 in ℝn (n ≥ 3), and provide a precise two-sided estimate on the order of the Beckner inequality constant with exponent p ∈ [1, 2) in the three parameters. As special cases for p = 1 and p tending to 2, our results cover those in Barthe et al. [Forum Math. (submitted for publication)] for n ≥ 3 and explore an interesting phenomenon. [ABSTRACT FROM AUTHOR]

Extremal functions are exhibited in Poincaré trace inequalities for functions of bounded variation in the unit ball Bn of the n-dimensional Euclidean space Rn. Trial functions are subject to either a vanishing mean value condition, or a vanishing median condition in the whole of Bn, instead of just on ∂Bn, as customary. The extremals in question take a different form, depending on the constraint imposed. In particular, under the median constraint, unusually shaped extremal functions appear. A key step in our approach is a characterization of the sharp constant in the relevant trace inequalities in any admissible domain Ω⊂Rn, in terms of an isoperimetric inequality for subsets of Ω. [ABSTRACT FROM AUTHOR]

We prove a new vectorial functional inequality of Poincaré–Beckner type. The inequality may be interpreted as an entropy–entropy production one for a gradient flow in the metric space of Radon measures. The proof uses subtle analysis of combinations of related super- and sub-level sets employing the coarea formula and the relative isoperimetric inequality. [ABSTRACT FROM AUTHOR]

On a doubling metric measure space endowed with a “carré du champ”, we consider L p estimates ( G p ) of the gradient of the heat semigroup and scale-invariant L p Poincaré inequalities ( P p ) . We show that the combination of ( G p ) and ( P p ) for p ≥ 2 always implies two-sided Gaussian heat kernel bounds. The case p = 2 is a famous theorem of Saloff-Coste, of which we give a shorter proof, without parabolic Moser iteration. We also give a more direct proof of the main result in [37] . This relies in particular on a new notion of L p Hölder regularity for a semigroup and on a characterisation of ( P 2 ) in terms of harmonic functions. [ABSTRACT FROM AUTHOR]

We study the boundedness of Riesz transforms in $$L^p$$ for $$p>2$$ on a doubling metric measure space endowed with a gradient operator and an injective, $$\omega $$ -accretive operator L satisfying Davies-Gaffney estimates. If L is non-negative self-adjoint, we show that under a reverse Hölder inequality, the Riesz transform is always bounded on $$L^p$$ for p in some interval $$[2,2+\epsilon )$$ , and that $$L^p$$ gradient estimates for the semigroup imply boundedness of the Riesz transform in $$L^q$$ for $$q \in [2,p)$$ . This improves results of Auscher et al. (Ann Sci Ecole Norm Sup 37(4):911-957, 2004) and Auscher and Coulhon (Ann Scuola Norm Sup Pisa 4:531-555, 2005), where the stronger assumption of a Poincaré inequality and the assumption $$e^{-tL}(1)=1$$ were made. The Poincaré inequality assumption is also weakened in the setting of a sectorial operator L. In the last section, we study elliptic perturbations of Riesz transforms. [ABSTRACT FROM AUTHOR]

The scope of this paper is to prove a Poincaré type inequality for a family of nonlinear vector fields, whose coefficients are only Lipschitz continuous with respect to the distance induced by the vector fields themselves. [ABSTRACT FROM AUTHOR]

We prove an optimal reverse Poincaré inequality for the heat semigroup generated by the sub-Laplacian on a Carnot group of any step. As an application we give new proofs of the isoperimetric inequality and of the boundedness of the Riesz transform in Carnot groups. [ABSTRACT FROM AUTHOR]

We consider a quantitative form of the quasi-isometry problem. We discuss several arguments which lead us to a number of results and bounds of quasi-isometric distortion: comparison of volumes, connectivity, etc. Then we study the transport of Poincaré constants by quasi-isometries and we give sharp lower and upper bounds for the homotopy distortion growth for a certain class of hyperbolic metric spaces, a quotient of a Heintze group R ⋉ R n by Z n . We also prove the linear distortion growth between hyperbolic space H n , n ≥ 3 and a tree. [ABSTRACT FROM AUTHOR]

We deal with the generalized Emden-Fowler equation f″( x) + g( x) f− θ( x) = 0, where [ABSTRACT FROM AUTHOR]

In this paper, we will establish Poincaré inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type. We obtain the first order Poincaré inequalities for vector fields satisfying Hörmander's condition in variable non-isotropic Sobolev spaces. We also set up the higher order Poincaré inequalities with variable exponents on stratified Lie groups. Moreover, we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups. These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p( x)-subLaplacian. Our results are only stated and proved for vector fields satisfying Hörmander's condition, but they also hold for Grushin vector fields as well with obvious modifications. [ABSTRACT FROM AUTHOR]

The dual purpose of this article is to establish bilinear Poincaré-type estimates associated with an approximation of the identity and to explore the connections between bilinear pseudo-differential operators and bilinear potential-type operators. The common underlying theme in both topics is their applications to Leibniz-type rules in Sobolev and Campanato-Morrey spaces under Sobolev scaling. [ABSTRACT FROM AUTHOR]

This paper is concerned with the weak solutions to a class of non-homogeneous A-Dirac equations DA( x, u, Du) + B( x, u, Du) = 0 with the Dirichlet boundary data. By means of the Poincaré inequalities of the Clifford valued function and some assumptions on operators A and B, we obtain the existence and uniqueness of solution to the scalar part of $${{\int_{\Omega}}{\overline{A(x, u, Du)}}D{\varphi}\,dx + {\int_{\Omega}}{\overline{B(x, u, Du)}}{\varphi}\,dx = 0}$$ . for each $${\varphi \in W^{1,p}_0 (\Omega, C\ell_n^k)}$$ . [ABSTRACT FROM AUTHOR]

This paper is concerned with the Poincaré inequalities and the sharp maximal inequalities for differential forms with [InlineEquation not available: see fulltext.]-norm, where φ satisfies nonstandard growth conditions. These results can be used to estimate the norms of classical operators and analyze integral properties of differential forms. [ABSTRACT FROM AUTHOR]

Recently, many new features of Sobolev spaces W (ℝ) were studied in [4-6, 32]. This paper is devoted to giving a brief review of some known characterizations of Sobolev spaces in Euclidean spaces and describing our recent study of new characterizations of Sobolev spaces on both Heisenberg groups and Euclidean spaces obtained in [12] and [13] and outlining their proofs. Our results extend those characterizations of first order Sobolev spaces in [32] to the Heisenberg group setting. Moreover, our theorems also provide different characterizations for the second order Sobolev spaces in Euclidean spaces from those in [4, 5]. [ABSTRACT FROM AUTHOR]

Equivalence of the spectral gap, exponential integrability of hitting times and Lyapunov conditions is well known. We give here the correspondence (with quantitative results) for reversible diffusion processes. As a consequence, we generalize results of Bobkov in the one dimensional case on the value of the Poincaré constant for log-concave measures to superlinear potentials. Finally, we study various functional inequalities under different hitting times integrability conditions (polynomial,...). In particular, in the one dimensional case, ultracontractivity is equivalent to a bounded Lyapunov condition. [ABSTRACT FROM AUTHOR]

We point out some of the differences between the consequences of p-Poincaré inequality and that of ∞-Poincaré inequality in the setting of doubling metric measure spaces. Based on the geometric characterization of ∞-Poincaré inequality given in Durand-Cartagena et al. (Mich Math J 60, ), we obtain a geometric property implied by the support of a p-Poincaré inequality, and demonstrate by examples that an analogous geometric characterization for finite p is not possible. The examples we give are metric measure spaces which are doubling and support an ∞-Poincaré inequality, but support no finite p-Poincaré inequality. In particular, these examples show that one cannot expect a self-improving property for ∞-Poincaré inequality in the spirit of Keith-Zhong (Ann Math 167(2):575-599, ). We also show that the persistence of Poincaré inequality under measured Gromov-Hausdorff limits fails for ∞-Poincaré inequality. [ABSTRACT FROM AUTHOR]

We study the connection between the p-Talagrand inequality and the q-logarithmic Sololev inequality for conjugate exponents p ≥ 2, q ≤ 2 in proper geodesic metric spaces. By means of a general Hamilton-Jacobi semigroup we prove that these are equivalent, and moreover equivalent to the hypercontractivity of the Hamilton-Jacobi semigroup. Our results generalize those of Lott and Villani. They can be applied to deduce the p-Talagrand inequality in the sub-Riemannian setting of the Heisenberg group. [ABSTRACT FROM AUTHOR]

We present geometric conditions on a metric space (Y,dY) ensuring that, almost surely, any isometric action on Y by Gromov’s expander-based random group has a common fixed point. These geometric conditions involve uniform convexity and the validity of nonlinear Poincaré inequalities, and they are stable under natural operations such as scaling, Gromov–Hausdorff limits, and Cartesian products. We use methods from metric embedding theory to establish the validity of these conditions for a variety of classes of metric spaces, thus establishing new fixed point results for actions of Gromov’s ‘wild groups’. [ABSTRACT FROM AUTHOR]

Abstract: We give a new proof of the fact that Gaussian concentration implies the logarithmic Sobolev inequality when the curvature is bounded from below, and also that exponential concentration implies Poincaré inequality under null curvature condition. Our proof holds on non-smooth structures, such as length spaces, and provides a universal control of the constants. We also give a new proof of the equivalence between dimension free Gaussian concentration and Talagrand''s transport inequality. [Copyright &y& Elsevier]

Abstract: We prove a fractional version of Poincaré inequalities in the context of endowed with a fairly general measure. Namely we prove a control of an norm by a non-local quantity, which plays the role of the gradient in the standard Poincaré inequality. The assumption on the measure is the fact that it satisfies the classical Poincaré inequality, so that our result is an improvement of the latter inequality. Moreover we also quantify the tightness at infinity provided by the control on the fractional derivative in terms of a weight growing at infinity. The proof goes through the introduction of the generator of the Ornstein–Uhlenbeck semigroup and some careful estimates of its powers. To our knowledge this is the first proof of fractional Poincaré inequality for measures more general than Lévy measures. [Copyright &y& Elsevier]

Abstract: We show that the Laplacian on the loop space over a class of Riemannian manifolds has a spectral gap. The Laplacian is defined using the Levi-Civita connection, the Brownian bridge measure and the standard Bismut tangent spaces. [Copyright &y& Elsevier]

Let Ω be an arbitrary bounded domain of n. We study the right invertibility of the divergence on Ω in weighted Lebesgue and Sobolev spaces on Ω, and rely this invertibility to a geometric characterization of Ω and to weighted Poincare inequalities on Ω. We recover, in particular, well-known results on the right invertibility of the divergence in Sobolev spaces when Ω is Lipschitz or, more generally, when Ω is a John domain, and focus on the case of s-John domains. [ABSTRACT FROM AUTHOR]