Published online by Cambridge University Press: 07 December 2020
Let $\Omega $ be homogeneous of degree zero and have mean value zero on the unit sphere
${S}^{d-1}$,
$T_{\Omega }$ be the convolution singular integral operator with kernel
$\frac {\Omega (x)}{|x|^d}$. In this paper, we prove that if
$\Omega \in L\log L(S^{d-1})$, and U is an operator which is bounded on
$L^2(\mathbb {R}^d)$ and satisfies the weak type endpoint estimate of
$L(\log L)^{\beta }$ type, then the composition operator
$UT_{\Omega }$ satisfies a weak type endpoint estimate of
$L(\log L)^{\beta +1}$ type.
The research of X.T was supported by the NNSF of China under grant #11771399, and the research of G.H. (corresponding author) was supported by the NNSF of China under grant #11871108.