Nonlocal Harnack inequalities for nonlocal double phase equations I ; with positive bounded modulating coefficient with no H\"older condition

In this paper, by applying the De Giorgi-Nash-Moser theory we prove nonlocal Harnack inequalities for (locally nonnegative in $\mathrm{\Omega }$) weak solutions to nolocal double phase equations \begin{equation*}\begin{cases}\cL u =0 & $\text{ in }\mathrm{\Omega }\text{,}$ \ u=g & $\text{ in }{\text{BR}}^n\text{s}\mathrm{\Omega }$ \end{cases}\end{equation*} where $\mathrm{\Omega }\subset {\text{BR}}^n$ ($n\ge 2$) is a bounded domain with Lipschitz boundary, $\text{cL}$ is the nonlocal double phase operator $\text{cL}$ given by \begin{equation*}\begin{split}\cL u(x)=&\pv\int_{\BR^n}|u(x)-u(y)|^{p-2}(u(x)-u(y))K_{ps}(x,y),dy \ &+\pv\int_{\BR^n}\fa(x,y)|u(x)-u(y)|^{q-2}(u(x)-u(y))K_{qt}(x,y),dy, \end{split} \end{equation*} $0<\text{fa}(x,y)=\text{fa}(y,x)\le \mathrm{\parallel }\text{fa}{\mathrm{\parallel }}_{L^{\text{iy}}({\text{BR}}^n\times {\text{BR}}^n)}<\text{iy}$ and $ps\ge qt$ for $0<s,t<1<p\le q<\text{iy}$. In addition, we get local boundedness with explicit formula and weak Harnack inequalities for their weak supersolutions.