Weighted Endpoint Variational and Jump Inequalites for Rough Singular Integrals

Abstract

In this present paper, we study the weighted weak type $(1,\,1)$ bounds for rough maximal singular integral operators as well as the the corresponding variation and jump operators. Our first result is the weighted weak type $(1,\,1)$ bound for the maximal singular integral $$T_{\Omega}^{*}f(x)=\sup\limits_{\varepsilon>0}|T_{\varepsilon,\Omega}f(x)|=\sup\limits_{\varepsilon>0}\bigg|\int_{|x-y|>\varepsilon} \frac{\Omega(x-y)}{|x-y|^{d}}f(y)dy\bigg|,$$ where $\Omega\in L^\infty(\mathbb{S}^{d-1})$, homogeneous of degree zero, and satisfies the cancellation condition. We show that $$\|T_{\Omega}^{*}\|_{L^1(w)\rightarrow L^{1,\infty}(w)}\lesssim[w]_{A_1}[w]_{A_\infty}\log([w]_{A_\infty}+1),$$ where $w$ belongs to the Muckenhoupt class $A_1(\mathbb{R}^d)$. This represents an essential improvement of a result (Honz\'{\i}k, Inter.Math. Res. Not. 2020) and a result (Bhojak and Mohanty, J. Funct. Anal.2023). Our second one is the weighted weak type $(1,\,1)$ bounds for variation and jump operators corresponding to $\{T_{\varepsilon,\Omega}\}_{\varepsilon\in2^{\mathbb{Z}}}$ and $\{T_{\varepsilon, \Omega}^{\phi}\}_{\varepsilon\in\mathbb{R}^{+}}$ under the condition $\Omega\in L^\infty(\mathbb{S}^{d-1})$, where $T_{\epsilon,\Omega}^{\phi}$ represents a smooth truncation of rough singular integral operator.These results of this part are the first weighted weak type $(1,\,1)$ variation inequalities and jump inequalities for rough singular integrals.