  This paper is devoted to studying the weak Harnack inequalities for nonlocal
double phase functionals by using expansion of positivity, whose prototype is
$$ \iint_{\mathbb{R}^n\times\mathbb{R}^n}
\left(\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}+a(x,y)\frac{|u(x)-u(y)|^q}{|x-y|^{n+tq}}\right)
\,dxdy $$ with $a\ge0$ and $0<s\le t<1<p\le q$. The core of our approach is to
establish several measure theoretical estimates based on the nonlocal
Caccioppoli-type inequality, where the challenges consist in controlling subtle
interaction between the pointwise behaviour of modulating coefficient and the
growth exponents. Meanwhile, a quantitative boundedness result on the minimizer
of such functionals is also discussed.
