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We establish Hardy-Sobolev inequalities in the unit ball B in the framework of general double phase functionals given by \ ₚ (x, t) = ₁ (tᵖ) + ₂ ( (b (x) t) ᵖ), x B, t 0, \ where p>1, ₁, ₂ are positive convex functions on (0, ) and b is a non-negative function on B which is radially Hölder continuous of order (0, 1].
Mizuta et al. (Sat,) studied this question.