Variational inequalities for the commutator of some singular integral with rough kernel
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In this paper, we consider variational inequalities for the commutator of the truncated singular integral operator $T_{\Omega,\beta,\varepsilon}$ with rough kernel \begin{align*} [b, T_{\Omega,\beta,\varepsilon}] f(x)= \int_{|x-y|>\varepsilon} \frac{\Omega(x-y)}{|x-y|^{n-\beta}}(b(x)-b(y)) f(y) d y, \end{align*} where $b\in BMO(\mathbb{R}^n)$ and the kernel $\Omega\in{(L(log^+L)^3)^{\frac{n}{n-\beta}}(\mathbb{S}^{n-1})}$ satisfies the vanishing condition and the homogeneous condition of degree $0$. We establish the $(L^p,L^q)$ estimate of the variation of the families $\{[b, T_{\Omega,\beta,\varepsilon}]\}_{\varepsilon>0}$ for $\frac{1}{q}=\frac{1}{p}-\frac{\beta}{n}$ and $0<\beta<1/2$. Moreover, one can get the $L^p$ boundedness of the commutator for Calder\'{o}n-Zygmund operator by letting $\beta\rightarrow0^+$.
Abstract
In this paper, we consider variational inequalities for the commutator of the truncated singular integral operator $T_{\Omega,\beta,\varepsilon}$ with rough kernel \begin{align*} [b, T_{\Omega,\beta,\varepsilon}] f(x)= \int_{|x-y|>\varepsilon} \frac{\Omega(x-y)}{|x-y|^{n-\beta}}(b(x)-b(y)) f(y) d y, \end{align*} where $b\in BMO(\mathbb{R}^n)$ and the kernel $\Omega\in{(L(log^+L)^3)^{\frac{n}{n-\beta}}(\mathbb{S}^{n-1})}$ satisfies the vanishing condition and the homogeneous condition of degree $0$. We establish the $(L^p,L^q)$ estimate of the variation of the families $\{[b, T_{\Omega,\beta,\varepsilon}]\}_{\varepsilon>0}$ for $\frac{1}{q}=\frac{1}{p}-\frac{\beta}{n}$ and $0<\beta<1/2$. Moreover, one can get the $L^p$ boundedness of the commutator for Calder\'{o}n-Zygmund operator by letting $\beta\rightarrow0^+$.