On the local boundedness of singular integral operators

Abstract

The class of singular integral operators whose kernels satisfy the usual smoothness conditions is studied. Let such an operator be denoted by K K . We establish necessary conditions that imply K K has local (weighted) L p {L^p} norm inequalities. The underlying principle is as follows. If χ Q {\chi _Q} is the characteristic function of a fixed cube Q Q of R n {R^n} , or all of R n {R^n} , then K χ Q K{\chi _Q} and (the adjoint of K K ) K ∗ χ Q {K^{\ast }}{\chi _Q} determine the boundedness properties of K K for functions supported in a proper fraction of Q Q .