Weighted Endpoint Variational and Jump Inequalites for Rough Singular Integrals

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 $$T_{\mathrm{\Omega }}^{\ast }f(x)=\underset{\epsilon >0}{\sup }\mathrm{\mid }{T}_{\epsilon ,\mathrm{\Omega }}f(x)\mathrm{\mid }=\underset{\epsilon >0}{\sup }\mid {\int }_{\mathrm{\mid }x-y\mathrm{\mid }>\epsilon }\frac{\mathrm{\Omega }(x-y)}{\mathrm{\mid }x-y{\mathrm{\mid }}^d}f(y)dy\mid ,$$ where $\mathrm{\Omega }\in L^{\mathrm{\infty }}({\mathbb{S}}^{d-1})$, homogeneous of degree zero, and satisfies the cancellation condition. We show that $$\mathrm{\parallel }{T}_{\mathrm{\Omega }}^{\ast }{\mathrm{\parallel }}_{L^1(w)\to L^{1,\mathrm{\infty }}(w)}\lesssim [w{]}_{A_1}[w{]}_{A_{\mathrm{\infty }}}\log ([w{]}_{A_{\mathrm{\infty }}}+1),$$ where $w$ belongs to the Muckenhoupt class $A_1({\mathbb{R}}^d)$. This represents an essential improvement of a result (Honzík, 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 $\{T_{\epsilon ,\mathrm{\Omega }}{\}}_{\epsilon \in 2^{\mathbb{Z}}}$ and $\{T_{\epsilon ,\mathrm{\Omega }}^{\phi }{\}}_{\epsilon \in {\mathbb{R}}^{+}}$ under the condition $\mathrm{\Omega }\in L^{\mathrm{\infty }}({\mathbb{S}}^{d-1})$, where $T_{\epsilon ,\mathrm{\Omega }}^{\phi }$ 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.