Gagliardo–Nirenberg inequalities in logarithmic spaces

Abstract

We obtain interpolation inequalities for derivatives: ∫ Mq,α(|∇f(x)|)dx ≤ C [∫ Mp,β(Φ1(x, |f |, |∇f |))dx+ ∫ Mr,γ(Φ2(x, |f |, |∇f |))dx ] , and their counterparts expressed in Orlicz norms: ‖∇f‖(q,α) ≤ C‖Φ1(x, |f |, |∇ f |)‖(p,β) ‖Φ2(x, |f |, |∇f |)‖(r,γ), where ‖ · ‖(s,κ) is the Orlicz norm relative to the function Ms,κ(t) = ts(ln(2 + t))κ. The parameters p, q, r, α, β, γ, as well as the Caratheodory functions Φ1,Φ2 are supposed to satisfy certain consistency conditions. Some of the classical Gagliardo-Nirenberg inequalities follow as a special case. Gagliardo-Nirenberg inequalities in logarithmic spaces with higher order gradients are also considered. MSC (2000): Primary 26D10, Secondary 46E35.