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Register a new userMean Value Inequalities and Characterizations of Sobolev spaces on graded groups
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We develop some characterizations for Sobolev spaces on the setting of graded Lie groups. A key role is played by several mean value inequalities that may be of independent interest.
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By using, among other things, the Fourier analysis techniques on hyperbolic and symmetric spaces, we establish the Hardy-Sobolev-Mazya inequalities for higher order derivatives on half spaces. The proof relies on a Hardy-Littlewood-Sobolev inequality on hyperbolic spaces which is of its independent interest. We also give an alternative proof of Benguria, Frank and Loss work concerning the sharp constant in the Hardy-Sobolev-Mazya inequality in the three dimensional upper half space. Finally, we show the sharp constant in the Hardy-Sobolev-Mazya inequality for bi-Laplacian in the upper half space of dimension five coincides with the Sobolev constant.
Interpolation inequalities in Triebel-Lizorkin-Lorentz spaces and Besov-Lorentz spaces are studied for both inhomogeneous and homogeneous cases. First we establish interpolation inequalities under quite general assumptions on the parameters of the function spaces. Several results on necessary conditions are also provided. Next, utilizing the interpolation inequalities together with some embedding results, we prove Gagliardo-Nirenberg inequalities for fractional derivatives in Lorentz spaces, which do hold even for the limiting case when one of the parameters is equal to 1 or $infty$.
We provide new characterizations of Sobolev ad BV spaces in doubling and Poincare metric spaces in the spirit of the Bourgain-Brezis-Mironescu and Nguyen limit formulas holding in domains of R^N.
We prove the exponential law $mathcal A(E times F, G) cong mathcal A(E,mathcal A(F,G))$ (bornological isomorphism) for the following classes $mathcal A$ of test functions: $mathcal B$ (globally bounded derivatives), $W^{infty,p}$ (globally $p$-integrable derivatives), $mathcal S$ (Schwartz space), $mathcal D$ (compact sport, $mathcal B^{[M]}$ (globally Denjoy_Carleman), $W^{[M],p}$ (Sobolev_Denjoy_Carleman), $mathcal S_{[L]}^{[M]}$ (Gelfand_Shilov), and $mathcal D^{[M]}$. Here $E, F, G$ are convenient vector spaces (finite dimensional in the cases of $W^{infty,p}$, $mathcal D$, $W^{[M],p}$, and $mathcal D^{[M]})$, and $M=(M_k)$ is a weakly log-convex weight sequence of moderate growth. As application we give a new simple proof of the fact that the groups of diffeomorphisms $operatorname{Diff} mathcal B$, $operatorname{Diff} W^{infty,p}$, $operatorname{Diff} mathcal S$, and $operatorname{Diff}mathcal D$ are $C^infty$ Lie groups, and $operatorname{Diff} mathcal B^{{M}}$, $operatorname{Diff}W^{{M},p}$, $operatorname{Diff} mathcal S_{{L}}^{{M}}$, and $operatorname{Diff}mathcal D^{[M]}$, for non-quasianalytic $M$, are $C^{{M}}$ Lie groups, where $operatorname{Diff}mathcal A = {operatorname{Id} +f : f in mathcal A(mathbb R^n,mathbb R^n), inf_{x in mathbb R^n} det(mathbb I_n+ df(x))>0}$. We also discuss stability under composition.
We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity theory and in the theory of electrorheological fluids. We also get a singular limit formula extending Nguyen results to the anisotropic case.
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