In this work we mainly consider a class of generalized Calderón–Zygmund type singular integrals \(\begin{aligned} T_\epsilon f(x) =\int _{|x-y|>\epsilon }\frac{\Omega (x-y)}{|x-y|^{n-\beta }}f(y)dy \quad \, \text{ for } \ \epsilon >0, \end{aligned}\) where the constant \(\beta \in (0, n)\) whose existence is also the difference between the traditional Calderón–Zygmund type singular integral operators and the current problems, and at the same time makes the problems more difficult. This type of singular integrals arises from the approximation of the SQG equation (surface quasi-geostrophic equation) which can describe geophysical flows in atmosphere and oceanography. Our main purpose is to study regularity estimates in Lorentz spaces for the generalized Calderón–Zygmund singular integral operators under some proper conditions.
