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 equationsequation* cases L u = 0 & in Ω , \\ \, \, \, \, u = g & in Rⁿ Ω cases equation*where Ω⊂ Rⁿ (n≥ 2) is a bounded domain with Lipschitz boundary, L is the nonlocal double phase operator given by$ {equation*} {split} { L} u(x) = &{ p.v.} ∫_{ { R}^n}|u(x)-u(y)|ᵖ⁻²(u(x)-u(y))Kₚₛ(x, y)\, dy \\ &+{ p.v.} ∫_{ { R}^n} { a}(x, y)|u(x)-u(y)|q-2(u(x)-u(y))Kqt(x, y)\, dy, {split} {equation*} a(x, y) = a(y, x)≥ 0, \, \| a\|_L^ ∞( Rⁿ× Rⁿ)< ∞, ess inf a>0, and ps≥ qt for $ 0<s, t<1 $ and 1<p<q≤ pₛ^* with pₛ^* = np/n-ps for $ ps