The classical Gagliardo-Nirenberg interpolation inequality is a well-known estimate which gives, in particular, an estimate for the Lebesgue norm of intermediate derivatives of functions in Sobolev spaces. We present an extension of this estimate into the scale of the general rearrangement-invariant Banach function spaces with the proof based on the Maz'ya's pointwise estimates. As corollaries, we present the Gagliardo-Nirenberg inequality for intermediate derivatives in the case of triples of Orlicz spaces and triples of Lorentz spaces. Finally, we promote the scaling argument to validate the optimality of the Gagliardo-Nirenberg inequality and show that the presented estimate in Orlicz scale is optimal.
We characterize the associate spaces of weighted Lorentz spaces GΓ(p, m, w) and present some applications of this result including necessary and sufficient conditions for a Sobolev-type embedding into L ∞ .
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