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 , 1 ( R n ) , R is an arbitrary set invariant with respect to external and internal dilations. We show that the above inequality implies its certain nonlinear variant involving the expressions ∫ R n H ( | u | ) μ ( d x ) and ∫ R n J ( | ∇ ( 2 ) u | ) μ ( d x ) . Various generalizations of this inequality to the more general class of N-functions, measures and to higher order derivatives are also discussed and the examples are presented.