Certain L^p bounds for rough singular integrals

Abstract

Abstract. In this paper, we prove L p bounds for singular integrals with rough kernels associated to certain surfaces. Our results extend as well as improve previously obtained results.Mathematics subject classification (2010): 42B20, 42B15, 42B25.

Boundedness of certain singular integrals along surfaces on Triebel–Lizorkin spaces

“…Following the notations in [1], let ¹ i 1 ; : : : ; i l º be a maximal linearly independent subset of ¹ 1 ; : : : ; d º, where 1 Ä l Ä d , 1 Ä i r Ä d and r D 1; : : : ; l. Thus, for j … ¹i 1 ; : : : ; i l º, there exists an a .j / D .a j;1 ; : : : ; a j;l / 2 R l such that Since t=.log t/ˇis increasing in .eˇ; 1/ for anyˇ> 0, and j Q .z 0 /j Ä B with B > 1 for any z 0 2 S n 1 , using (3.31) and the trivial estimate j Q I k;s; .z 0 /j Ä C , …”

Integral operators of Marcinkiewicz type on Triebel–Lizorkin spaces

“…Remark We remark that when , the surfaces given as in Theorems 1.3 and 1.5 were originally introduced by Al‐Balushi and Al‐Salman in the study of bounds for singular integrals associated to certain surfaces. Obviously, Theorem represents a generalization of part (i) of Theorem .…”