Some Estimates For Schr(?)dinger Operators With Potentials Satisfying Reverse H(?)lder Inequality

Several problems associated with Schro¨dinger operators with potentials satisfy-ing reverse Ho¨lder inequality are investigated in this dissertation,which focus on fourcontents: weighted Lp boundedness of commutators composed of Riesz transformsassociated with Schro¨dinger operators with potentials satisfying reverse Ho¨lder in-equality and weighted BMO functions and weighted Lipschitz functions respectively;BLOL spaces associated with Schro¨dinger operators with potentials satisfying re-verse Ho¨lder inequality; Lp estimate for some operator associated with parabolicSchro¨dinger operator with potentials satisfying reverse Ho¨lder inequality; estimatesfor uniformly parabolic differential operators with potentials satisfying reverse Ho¨lderinequality on weighted Lp(Rn+1) spaces and Morrey spaces.This dissertation comprise five chapters. Chapter One presents the research back-ground, questions and significance, preliminary knowledge and outline of this disser-tation. The purpose of Chapter Two is devoted to studying weighted Lp bounded-ness of commutators composed of Riesz transforms associated with Schro¨dinger op-erators with potentials satisfying reverse Ho¨lder inequality (T1 = (-â–³+ V )-1V,T2 =(-â–³+ V )^1/2V^1/2,T₃ = (-â–³+ V )^1/2▽和T₄ = (-â–³+ V )^-1â–½²) and weighted BMO func-tions and weighted Lipschitz functions respectively, by applying Sto¨mberge'idea andH(m) condition introduced by Zihua Guo, Pengtao Li and Lizhong Peng. The resultproved above includes non-weighted cases.In Chapter Three, the author gives the definition of BLO type spaces associated withSchro¨dinger operators with potentials satisfying reverse Ho¨lder inequality and gets itsone characterization. This characterization is parallel with that of the classical BLOspaces. The author also investigates boundedness of Hardy-Littlewood maximal func-tion , natural maximal function, two kinds of maximal singular operators and maximalRiesz transforms associated with Schro¨dinger operators with potentials satisfying re-verse Ho¨lder inequality from BMOL to BLOL.In Chapter Four, the author studies Lp(Rn+1) boundedness of a parabolic operator(?t -â–³+V )^1/2V^1/2 associated with parabolic Schro¨dinger operators with potentials sat- isfying reverse H(o|¨)lder inequality, where V∈Bq,q≥^n/2. In Chapter Five, the authorstudies estimates for some parabolic operators ( V L^-1,V^1/2â–½L^-1,V^1/2L^-1/2) associatedwith uniformly parabolic operators (?) with potentials satisfy-ing reverse H(o|¨)lder inequality and boundedness fo them on weighted Lp(Rn+1) spacesand Morrey spaces when ai,j satisfy certain conditions and V∈Bq for some q.