Regularity Estimates For Schr?dinger Type Operators On Carnot Groups

In 1970,Stein proposed an idea to study H?rmander type partial differential operators by using the group analysis in the “Sixteen International Congress of Mathematicians”.Since then,the research of various partial differential operators on Carnot groups has gradually become a new hotspot.This thesis is mainly concerned with Orlicz estimates for Schr?dinger type operators and parabolic Schr?dinger type operators on Carnot groups,and L^p estimates for Schr?dinger type operators with discontinuous coefficients.This work consists of three parts.The first part?Chapter 2 and Chapter 3?is concerned with the following Schr?dinger type operator on Carnot groups,where {X₁,X₂,...,X_m} is the basis in the space of horizontal vector fields of the Carnot group G and the potential V?x?belongs to a reverse H?lder class.For the potential V?x?belonging to the reverse H?lder class RH_q?G?,we generalize the iteration-covering method in the Euclidean space to the Carnot group.The iterationcovering lemma related to the operator L is proved,and the global Orlicz estimate for L is obtained.For the potential V?x?in the reverse H?lder class RH_??G?,a weighted version of iteration-covering lemma related to the operator L is proved.And a weighted version of iteration-covering lemma related to the sub-Laplace operator on Carnot groups is proved.Combining the properties of Muckenhoupt weights,we obtain the weighted global Orlicz estimates for the Schr?dinger type operator L and the subLaplace operator A,respectively.In the second part?Chapter 4?,we study Orlicz estimates for the parabolic Schr?dinger type operator on Carnot groups,where the potential V?x?is independent of time variable and belongs to the reverse H?lder class.First of all,an iteration-covering lemma related to the operator P is given.Then we prove a local L^? estimate for the solutions of the homogeneous equation P u = 0.And the global Orlicz estimate for the operator P is established.In the third part?Chapter 5 and Chapter 6?,L^p estimates for Schr?dinger type operator with discontinuous coefficients on Carnot groups,and higher order elliptic equation with a potential and discontinuous coefficients in the Euclidean space are investigated.We first introduce a generalized Stummel-Kato type class S_p?G?on the Carnot group G and note that the set S_p?G?includes some singular potentials.Then a second order Fefferman-Poincaré type inequality on the metric ball in G is established.As applications,we derive L^p?G? estimates for the following Schr?dinger type operator with discontinuous coefficients,where the potential V belongs to the generalized StummelKato type class.Finally,the generalized higher order Stummel-Kato type class in the Euclidean space Rⁿ is introduced,and a higher order Fefferman-Poincaré type inequality on the weak Boman chain domain in Rⁿ is obtained.As an application,we prove L^p?Rⁿ?estimates for the solutions of higher order elliptic equation with discontinuous coefficients and a potential in the Euclidean space.