Boundedness of Hardy operators in the unit ball of double phase

We establish Hardy-Sobolev inequalities in the unit ball B in the framework of general double phase functionals given by \[ φ_p(x,t) = φ_1(t^p) + φ_2((b(x)t)^p), x∈ B, t ≥ 0, \] where $p>1$, φ₁, φ₂ are positive convex functions on (0,∞) and b is a non-negative function on B which is radially Hölder continuous of order θ ∈ (0,1].