Gagliardo-Nirenberg inequalities in regular Orlicz spaces involving nonlinear expressions

摘要

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 holdsunder(∫, R~n) M (| ? u |) μ (d x) ≤ C (under(∫, R~n) H (| u |) μ (d x) + under(∫, 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 ∫_(Rn) H (| u |) μ (d x) and ∫_(Rn) 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.