Harmonic analysis for some Schroedinger type operators

This work consists of two parts. In the first part, we consider the Sobolev estimates for the Schrodinger operator {dollar}-Delta + V(x),{dollar} where {dollar}Delta{dollar} is the Laplacian and {dollar}V(x){dollar} is a nonnegative polynomial in R{dollar}sp{lcub}n{rcub}.{dollar} We have proved for {dollar}1 le i,j le n{dollar} and {dollar}gammain {lcub}bf R{rcub},{dollar} the operators {dollar}partialsbsp{lcub}ij{rcub}{lcub}2{rcub}(-Delta + V)sp{lcub}-1{rcub}, partialsb{lcub}i{rcub}(-Delta + V)sp{lcub}-1{rcub}partialsb{lcub}j{rcub}, partialsb{lcub}i{rcub}(-Delta + V)sp{lcub}-1/2{rcub}, (-Delta + V)sp{lcub}-1/2{rcub}partialsb{lcub}i{rcub}{dollar} and {dollar}(-Delta + V)sp{lcub}igamma{rcub}{dollar} are Calderon-Zygmund operators. All the bounds depend only on the degree of V and the dimension n. Also we have proved some Sobolev estimates related to {dollar}-Delta + V(x), (-Delta + V(x))sp{lcub}m{rcub}{dollar} and {dollar}(-Delta)sp{lcub}m{rcub} + V(x)sp{lcub}m{rcub}, m in {lcub}bf N{rcub}.{dollar}; In the second part, we consider {dollar}Lsp{lcub}p{rcub} - Lsp{lcub}q{rcub}{dollar} estimates for the Cauchy problem (UNFORMATTED TABLE OR EQUATION FOLLOWS){dollar}{dollar}left{lcub}eqalign{lcub}&(partialsbsp{lcub}tt{rcub}{lcub}2{rcub} - Deltasb{lcub}x{rcub} + V(x))u(x,t) = F(x,t)cr&u(x,0) = g(x), usb{lcub}t{rcub}(x,0) = f(x)cr{rcub}right. , x in {lcub}bf R{rcub}sp{lcub}n{rcub}, t ge 0.eqno(0.1){dollar}{dollar}(TABLE/EQUATION ENDS)Here {dollar}V(x) ge 0.{dollar}; We have proved for {dollar}nge3,{dollar} (UNFORMATTED TABLE OR EQUATION FOLLOWS){dollar}{dollar}eqalign{lcub}&Vert u(cdot,t)Vertsb{lcub}Lsp q({lcub}bf R{rcub}sp n){rcub} le C(t)(Vert fVertsb{lcub}Lsp p({lcub}bf R{rcub}sp n){rcub} + Vert gVertsb{lcub}Lsbsp{lcub}1{rcub}{lcub}p{rcub}({lcub}bf R{rcub}sp n){rcub} +cr&sk{lcub}20{rcub}Vert Vsp{lcub}1/2{rcub}gVertsb{lcub}Lsp p({lcub}bf R{rcub}sp n){rcub} + intsbsp{lcub}0{rcub}{lcub}t{rcub}Vert F(cdot,tau)Vertsb{lcub}Lsp p({lcub}bf R{rcub}sp n){rcub}dtau)cr{rcub}{dollar}{dollar}(TABLE/EQUATION ENDS)if {dollar}(1/p,1/q){dollar} belongs to the closed quadrilateral with the vertices {dollar}Asb1 = (1/2 + 1/n, 1/2 + 1/n), Asb2 = (1/2 - 1/n, 1/2 - 1/n), Asb3 = (1/2,1/2 - 1/n){dollar} and {dollar}Asb4 = (1/2 + 1/n, 1/2).{dollar} Here {dollar}C(t){dollar} is a polynomial of t and is independent of V. Similarly we also got some results for n = 1 or 2. We gave an application to a nonlinear problem.