Hölder regularity for Nonlocal double phase equations Ⅰ; with essentially positive bounded modulating coefficient with no Hölder condition

In this paper, we obtain interior Hölder regularity of weak solutions to nonlocal double phase equations$ \begin{equation*} \begin{cases} {\mathcal L} u = 0 & \text{ in } \Omega , \\ \, \, \, \, u = g & \text{ in } {\mathbb R}^n \setminus \Omega \end{cases} \end{equation*} $where $ \Omega\subset {\mathbb R}^n $ ($ n\ge 2 $) is a bounded domain with Lipschitz boundary, $ {\mathcal L} $ is the nonlocal double phase operator given by$ \begin{equation*} \begin{split} {\mathcal L} u(x) = &{\rm p.v.}\displaystyle \int_{ {\mathbb R}^n}|u(x)-u(y)|^{p-2}(u(x)-u(y))K_{ps}(x, y)\, dy \\ &+{\rm p.v.}\displaystyle \int_{ {\mathbb R}^n} {\mathfrak a}(x, y)|u(x)-u(y)|^{q-2}(u(x)-u(y))K_{qt}(x, y)\, dy, \end{split} \end{equation*} $$ {\mathfrak a}(x, y) = {\mathfrak a}(y, x)\ge 0, \, \| {\mathfrak a}\|_{L^ \infty( {\mathbb R}^n\times {\mathbb R}^n)}< \infty $, $ \text{ess inf } {\mathfrak a}>0 $, and $ ps\ge qt $ for $ 0<s, t<1 $ and $ 1<p<q\le p_s^* $ with $ p_s^* = \frac{np}{n-ps} $ for $ ps