Gagliardo–Nirenberg inequalities in regular Orlicz spaces involving nonlinear expressions

We consider a triple of N-functions ( M , H , J ) that satisfy the Δ ′ -condition, μ = | x | α d x and suppose that an additive variant of interpolation inequality holds ∫ R n M ( | ∇ u | ) μ ( d x ) ⩽ C ( ∫ R n H ( | u | ) μ ( d x ) + ∫ R n J ( | ∇ ( 2 ) u | ) μ ( d x ) ) , where u ∈ R ⊆ W loc 2 ,...

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