Weighted Norm Inequalities for Homogeneous Singular Integrals

Abstract

Abstract. We prove weighted norm inequalities for homogeneous singular integrals when only a size condition is assumed on the restriction of the kernel to the unit sphere. The same results hold for the operator obtained by modifying the centered Hardy-Littlewood maximal operator over balls with a degree zero homogeneous function and also for the maximal singular integral.

A unifying approach for certain class of maximal functions

“…The results in this paper can be easily extended to the radial weights setting introduced by Duoandikoetxea [9]. In order to state our weighted L p estimates, we recall the definition of the radial weights [9,13].…”

Lp(Rn) boundedness for the commutator of a homogeneous singular integral operator

“…By a well-known result of Duoandikoetxea [5] and Watson [10] ), then for any fixed 1 < p, q < ∞, we do not know whether the operator T is bounded on L p (R n , w(x) dx) for all w ∈ A q , and the Alvarez-Bagby-Kurtz-Pérez theorem does not apply. In this case, the L p (R n ) boundedness for T b,k is not known.…”

On the commutators of singular integrals related to block spaces

“…( [6,2,10]) Let Ω, a, k be as above and h ≡ 1, 1 < p < ∞. If Ω ∈ ∪ q>1 L q (S n−1 ), then T a,k is bounded on L p (R n ).…”

“…Afterwards, by a well-known result of Duoandikoetxea [6] and the boundedness criterion of Alvarez-Bagby-KurtzPérez for the commutators of linear operator (see [2]), we have obtained the following theorem (see also [10]):…”