Boundedness of certain convolution operators.

摘要

Let {dollar}psi{dollar} be a positive function defined near the origin such that lim{dollar}sb{lcub}tto0sp+{rcub}psi(t)=0.{dollar} We consider the operator{dollar}{dollar} Tsb{lcub}z{rcub}f(x)=limsb{lcub}epsilonto0sp+{rcub}intsbsp{lcub}epsilon{rcub}{lcub}1{rcub} esp{lcub}igamma(t){rcub}f(x-t){lcub}tsp{lcub}-z{rcub}over psi(t)sp{lcub}1-z{rcub}{rcub}dt,{dollar}{dollar}where z is a complex number with {dollar}0leq Re(z)leq1{dollar} and {dollar}gamma{dollar} is a real function. Assuming certain regularity conditions on {dollar}psi{dollar} and {dollar}gamma{dollar} we show that if there is a constant C such that {dollar}{lcub}1oververtgammasp{lcub}primeprime{rcub}(t)vert{rcub}leq Cpsi(t)sp2,{dollar} then {dollar}Tsb{lcub}theta{rcub}{dollar} is a bounded operator on {dollar}Lsp{lcub}p{rcub}(R){dollar} for {dollar}{lcub}1over p{rcub}={lcub}1+thetaover2{rcub}{dollar} and {dollar}0leqtheta<1,{dollar} and {dollar}Tsb1{dollar} is bounded from {dollar}Hsp1(R){dollar} to {dollar}Lsp1(R).{dollar}; We also show that if {dollar}{lcub}limlimitssb{lcub}tto0sp+{rcub}{rcub} {lcub}1oververtgammasp{lcub}primeprime{rcub}(t)vertpsi(t)sp2{rcub} = infty,{dollar} then {dollar}Tsb0{dollar} is not bounded on {dollar}Lsp2(R).{dollar}