CORRECTION AND ADDITION TO "L 2 -BOUNDEDNESS OF MARCINKIEWICZ INTEGRALS ALONG SURFACES WITH VARIABLE KERNELS"

Abstract

In our paper 2 -boundedness of Marcinkiewicz integrals along surfaces with variable kernels, have found that Theorems 3 and 5 are not correct. Here, reformulate Theorems 3 and 5, so that L 2 -boundedness holds as in Theorems 1 and 4. s J n +k−1(ρ) ρ n 2 Φ −1 ( ρ |ξ| )ϕ(Φ −1 ( ρ |ξ| ))dρ, where s =Φ (t)|ξ| and Φ(t) t = C2Φ � (t)ϕ(t). In our paper (8, p. 380), have taken the integration interval as (0 ,s ) in place of the correct one (s, +∞). Because of this error have led false claims in Theorems 3 and 5. So, in our paper (8), delete the sentence also show some sharp difference betweeen properties of singular integrals and the Marcinkiewicz integral with rough variable kernels. in the abstract. We delete the lines 26 through 34 in the page 371, and Theorem 5 in the page 372. We also delete the section 4. Other corrections: In the assumption of Lemma 2.3 add ν − λ> −1. In the line 9 of the page 371, exists should be exist. In the line 3 from the bottom in the page 372, we only give should be we have only to give. In the line 2 of the page 375, 1