A convergence result for the master operator

In this paper, we establish a convergence result for the fully fractional heat operator $\text{ma}s$, also known as the master operator, stated as follows: $$If{u}_i\to uin{C}_{x,t,loc}^{2,1}({\mathbb{R}}^n\times \mathbb{R}),then\text{ma}s{u}_i\to \text{ma}su-ba\mathrm{.}e\mathrm{.}in{\mathbb{R}}^n\times \mathbb{R},$$ for some nonnegative constant $b$. This result addresses a fundamental question in the blow-up and rescaling analysis, which are essential for establishing a priori estimates for solutions of master equations. Additionally, we present examples demonstrating that in certain cases, the constant $b$ can indeed be positive. This highlights a key distinction between nonlocal and local operators: for a local heat operator, such as ${\mathrm{\partial }}_t-\text{lap}$, it is well-known that $b\equiv 0$.