In this paper we obtain the non - asymptotic estimations of Poincare type between function and its gradient in the so - called Bilateral Grand Lebesgue Spaces. We also give some examples to show the sharpness of these inequalities.
We extend the classical Hardy-Sobolev-Poincare-Wirtinger inequalities from the ordinary Lebesgue-Riesz spaces into the Grand Lebesgue ones, with exact constants evaluation.
We establish the bilateral exact reciprocal interrelations between a tail behavior of a measurable functions and its norm in the suitable Grand Lebesgue Space (GLS) as well as Orlicz one, builded over the set with infinite measure. We bring also some examples in order to illustrate the exactness of offered estimates.
We derive bilateral estimates for the constants appearing in the Fourier transform restricted theorems on the Euclidean sphere for the ordinary and especially radial functions belonging to the Lebesgue-Riesz spaces as well as belonging to the Grand Lebesgue Spaces. We obtain an exact estimate for the norm of the restriction Fourier transform operator acting on the radial functions.
We offer in this paper the non-asymptotical pairwise bilateral exact up to multiplicative constants interrelations between exponential decreasing tail behavior, moments (Grand Lebesgue Spaces) norm and moment generating functions norm for random variables and vectors (r.v.).
Poincaré inequalities on Carnot Groups and spectral gap of Schrödinger operators
Ghent University(根特大学);University of Bologna(博洛尼亚大学);Institut de Mathématiques de Toulouse(图卢兹数学研究所;图卢兹大学;法国国家科学研究中心);UMR5219 Université de Toulouse;CNRS
We introduce function spaces for the treatment of non-linear parabolic equations with variable $\log$-Hölder continuous exponents, which only incorporate information of the symmetric part of a gradient. As an analogue of Korn's inequality for these functions spaces is not available, the construction of an appropriate smoothing method proves itself to be difficult. To this end, we prove a point-wise Poincaré inequality near the boundary of a bounded Lipschitz domain involving only the symmetric gradient. Using this inequality, we construct a smoothing operator with convenient properties. In par…
Boxing inequalities for relative fractional perimeter and fractional Poincaré-type inequalities on John domains with the BBM factor
Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University(湖南师范大学数学与统计学院计算与随机数学教育部重点实验室);Academy of Mathematics and Systems Science, Chinese Academy of Sciences(中国科学院数学与系统科学研究院)
Let $(\mathcal{X}, ρ, μ)$ be a metric measure space of homogeneous type which supports a certain Poincaré inequality. Denote by the symbol $\mathcal{C}_{\mathrm{c}}^\ast(\mathcal{X})$ the space of all continuous functions $f$ with compact support satisfying that $\operatorname{Lip} f:=\limsup _{r \rightarrow 0} \sup_{y\in B(\cdot, r)} |f(\cdot)-f(y)|/r$ is also a continuous function with compact support and $\operatorname{Lip} f=\lim _{r \rightarrow 0} \sup_{y\in B(\cdot, r)} |f(\cdot)-f(y)|/r$ converges uniformly. Let $p \in[1,\infty)$. In this article, the authors prove that, for any $f\in\m…
Following a Maz'ya-type approach, we adapt the theory of rough traces of functions of bounded variation ($BV$) in the context of doubling metric measure spaces supporting a Poincaré inequality. This eventually allows for an integration by parts formula involving the rough trace of such a function. We then compare our analysis with the discussion done in a recent work by P. Lahti and N. Shanmugalingam, where traces of $BV$ functions are studied by means of the more classical Lebesgue-point characterization, and we determine the conditions under which the two notions coincide.
Let $X$ be a ball Banach function space on ${\mathbb R}^n$. In this article, under some mild assumptions about both $X$ and the boundedness of the Hardy--Littlewood maximal operator on the associate space of the convexification of $X$, the authors prove that, for any locally integrable function $f$ with $\|\,|\nabla f|\,\|_{X}<\infty$, $$\sup_{λ\in(0,\infty)}λ\left \|\left|\left\{y\in{\mathbb R}^n:\ |f(\cdot)-f(y)| >λ|\cdot-y|^{\frac{n}{q}+1}\right\}\right|^{\frac{1}{q}} \right\|_X\sim \|\,|\nabla f|\,\|_X$$ with the positive equivalence constants independent of $f$, where the index $q\in(0,\i…