scholarly journals Second order Sobolev type inequalities in the hyperbolic spaces

2019 ◽  
Vol 477 (2) ◽  
pp. 1157-1181
Author(s):  
Van Hoang Nguyen
Keyword(s):  
Second Order ◽  
Hyperbolic Spaces ◽  
Sobolev Type

scholarly journals Hardy–Littlewood–Sobolev inequalities for a class of non-symmetric and non-doubling hypoelliptic semigroups

2020 ◽  
Author(s):  
Nicola Garofalo ◽  
Giulio Tralli
Keyword(s):  
Brownian Motion ◽  
Evolution Equation ◽  
General Class ◽  
Semigroup Theory ◽  
Second Order ◽  
Sobolev Type ◽  
Sobolev Inequalities ◽  
Hypoelliptic Operators ◽  
Special Case ◽  
New Inequalities

Abstract In his seminal 1934 paper on Brownian motion and the theory of gases Kolmogorov introduced a second order evolution equation which displays some challenging features. In the opening of his 1967 hypoellipticity paper Hörmander discussed a general class of degenerate Ornstein–Uhlenbeck operators that includes Kolmogorov’s as a special case. In this note we combine semigroup theory with a nonlocal calculus for these hypoelliptic operators to establish new inequalities of Hardy–Littlewood–Sobolev type in the situation when the drift matrix has nonnegative trace. Our work has been influenced by ideas of E. Stein and Varopoulos in the framework of symmetric semigroups. One of our objectives is to show that such ideas can be pushed to successfully handle the present degenerate non-symmetric setting.

scholarly journals Sobolev-Type Spaces on the Dual of the Chébli-Trimèche Hypergroup and Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Mourad Jelassi ◽  
Hatem Mejjaoli
Keyword(s):  
Differential Operator ◽  
Type Theorem ◽  
Reproducing Kernels ◽  
Second Order ◽  
Order Differential Operator ◽  
Sobolev Type ◽  
Weierstrass Transform ◽  
Sobolev Type Spaces ◽  
Type Spaces

We define and study Sobolev-type spacesWAs,pℝ+associated with singular second-order differential operator on0,∞. Some properties are given; in particular we establish a compactness-type imbedding result which allows a Reillich-type theorem. Next, we introduce a generalized Weierstrass transform and, using the theory of reproducing kernels, some applications are given.

Sign in / Sign up

Export Citation Format

Share Document