Weak Harnack inequalities for nonlocal double phase problems

Abstract This paper is devoted to the study of weak Harnack inequalities for minimizers of nonlocal double phase functionals, whose prototype is given by ₑⁿ {Rⁿ} (|u (x) -u (y) |ᵖ|x-y|^{n+sp}+a (x, y) |u (x) -u (y) |q|x-y|^{n+tq}) \, dx\, dy, ∬ R n × R n | u (x) - u (y) | p | x - y | n + s p + a (x, y) | u (x) - u (y) | q | x - y | n + t q d x d y, with a 0 a ≥ 0 and 0 0 s, t 1 p ≤ q ∞. The core of our approach is based on expansion of positivity and several measure theoretic estimates stemming from a nonlocal Caccioppoli-type inequality. The main challenge lies in controlling the subtle interaction between the pointwise behaviour of the modulating coefficient a (, ) a (·, ·) and the structural exponents. In addition, we discuss a quantitative boundedness result for minimizers of such functionals.