Weighted weak type (1,1) estimates for singular integrals and Littlewood-Paley functions
We prove some weighted weak type inequalities for certain singular integrals and Littlewood Paley functions Introduction Let K be a locally integrable function on R n f g which satis es Z a jxj b K x dx for all a b such that a b We assume that n We consider a singular integral operator which can be de ned by T f x p v Z R n K x y f y dy lim Z jx yj K x y f y dy where f C R n the space of in nitely di erentiable functions with compact support De ne V sup R VR where VR Z R R jK r jr dr Also put for t q q t sup jsj tR R n Z Sn Z R R jR K r s K r jr dr d q where d denotes the Lebesgue surface measure on the unit sphere S of R and the supremum is taken over all s and R such that jsj tR When q we can de ne t by usual modi cation Mathematics Subject Classi cation Primary B B