On the weak Harnack inequalities for nonlocal double phase problems

This paper is devoted to studying the weak Harnack inequalities for nonlocal double phase functionals by using expansion of positivity, whose prototype is $${\iint }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}(\frac{\mathrm{\mid }u(x)-u(y){\mathrm{\mid }}^p}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+sp}}+a(x,y)\frac{\mathrm{\mid }u(x)-u(y){\mathrm{\mid }}^q}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+tq}})dxdy$$ with $a\ge 0$ 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.