Let $\Omega \in L^1(S^{n-1})$ have mean value zero and satisfy the condition $$\sup_{\zeta' \in S^{n-1}} \int_{S^{n-1}} |\Omega(y')| (\ln |\zeta' \cdot y'|^{1})^{(\ln(e + \ln |\zeta' \cdot y'|^{1}))^{\beta}} \, d\sigma(y') \lt \infty \text{ for some } \beta > 0.$$ Under certain conditions on the measurable function $h$, we show that the singular integral $$Tf(x) = \text{p. v.} \int_{\mathbb{R}^n} \dfrac{h(|y|) \Omega (y')}{|y|^n} f(x y)\,dy$$ is bounded on the Triebel-Lizorkin weighted spaces $\dot{F}^{\alpha,w}_{p,q}\mathbb{R}^n$. We also study the Marcinkiewicz integral (with the same kernel $\Omega$ as above) in the $L^p$ weighted spaces.
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