Related papers: $L^p$-Poincar\'e inequalities on nested fractals

We introduce a notion of "gradient at a given scale" of functions defined on a metric measure space. We then use it to define Sobolev inequalities at large scale and we prove their invariance under large-scale equivalence (maps that…

This paper investigates functional inequalities involving Besov spaces and functions of bounded variation, when the underlying metric measure space displays different local and global structures. Particular focus is put on the $L^1$ theory…

We study $L^p$ Besov critical exponents and isoperimetric and Sobolev inequalities associated with fractional Laplacians on metric measure spaces. The main tool is the theory of heat semigroup based Besov classes in Dirichlet spaces that…

We investigate the validity, as well as the failure, of Sobolev-type inequalities on Cartan-Hadamard manifolds under suitable bounds on the sectional and the Ricci curvatures. We prove that if the sectional curvatures are bounded from above…

In this note, we establish a $L^p-$version of the Poincar\'e--Sobolev inequalities in the hyperbolic spaces $\mathbb H^n$. The interest of this result is that it relates both the Poincar\'e (or Hardy) inequality and the Sobolev inequality…

The article deals with a simplified proof of the Sobolev embedding theorem for Lizorkin--Triebel spaces (that contain the $L_p$-Sobolev spaces $H^s_p$ as special cases). The method extends to a proof of the corresponding fact for general…

We develop an intrinsic, heat-kernel based fractional Sobolev framework on closed Riemannian manifolds and study the critical fractional Sobolev embedding. We determine the optimal coefficient of the lower-order $L^{p}$ term and prove that…

In recent work with Berndtsson and Rubinstein, a notion of $L^p$-polarity was introduced, with classical polarity recovered in the limit $p\to\infty$, and $L^1$-polarity closely related to Bergman kernels of tube domains. A Santal\'o…

We study Poincar\'e type $L^p$ inequality on a compact semialgebraic subset of $\R^n$ for $p>>1$. First we derive a local inequality by using a Lipschitz deformation retraction with estimates on its derivatives. Then, we extend the local…

This paper is devoted to optimal functional inequalities for fractional Laplace operators on the sphere. Based on spectral properties, subcritical inequalities are established. Their consequences for fractional heat flows are considered.…

We explore connections between covariance representations, Bismut-type formulas and Stein's method. First, using the theory of closed symmetric forms, we derive covariance representations for several well-known probability measures on…

In this paper we will establish different weighted Poincar\'{e} inequalities with variable exponents on Carnot-Carath\'{e}odory spaces or Carnot groups. We will use different techniques to obtain these inequalities. For vector fields…

Using hyperbolic form convolution with doubly isometry-invariant kernels, the explicit expression of the inverse of the de Rham laplacian acting on m-forms in the Poincar\'{e} space is found. Also, by means of some estimates for hyperbolic…

We study the $L^p$-convergence of Fourier expansions in terms of non-symmetric Heckman-Opdam polynomials of type $A_1$. Using kernel estimates and duality arguments, we prove that the partial sums converge in $ L^p([-\pi,\pi],dm_k)$ for…

This note is devoted to the proof of convex Sobolev (or generalized Poincar\'{e}) inequalities which interpolate between spectral gap (or Poincar\'{e}) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of…

We obtain new oscillation inequalities in metric spaces in terms of the Peetre $K-$functional and the isoperimetric profile. Applications provided include a detailed study of Fractional Sobolev inequalities and the Morrey-Sobolev embedding…

Firstly, we derive in dimension one a new covariance inequality of $L_{1}-L_{\infty}$ type that characterizes the isoperimetric constant as the best constant achieving the inequality. Secondly, we generalize our result to $L_{p}-L_{q}$…

Poincar\'{e}-Sobolev-type inequalities involving rearrangement-invariant norms on the entire $\mathbb{R}^n$ are provided. Namely, inequalities of the type $\|u-P\|_{Y(\mathbb{R}^n)}\leq C\|\nabla^m u\|_{X(\mathbb{R}^n)}$, where $X$ and $Y$…

By using Bernstein-type inequality we define analogs of spaces of entire functions of exponential type in $L_{p}(X), 1\leq p\leq \infty$, where $X$ is a symmetric space of non-compact. We give estimates of $L_{p}$-norms, $1\leq p\leq…

We investigate the best constants for the regional fractional $p$-Poincar\'e inequality and the fractional $p$-Poincar\'e inequality in cylindrical domains. For the special case $p=2$, the result was already known due to…