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Abstract

We obtain interpolation inequalities for derivatives: \begin{multline*} \int M_{q,\alpha}(|\nabla f (x)|)\,dx \\\leq C\bigg[ \int M_{p,\beta }({\mit\Phi} _1(x, |f|,|\nabla^{(2)}f| ))\, dx+ \int M_{r,\gamma } ({\mit\Phi} _2(x, |f|,|\nabla^{(2)}f| ))\, dx\bigg] ,\end{multline*} and their counterparts expressed in Orlicz norms: \[ \|\nabla f\|_{(q,\alpha)}^2\leq C\| {\mit\Phi} _1(x, |f|,|\nabla^{(2)}f|) \|_{(p,\beta)}\, \| {\mit\Phi} _2(x, |f|,|\nabla^{(2)}f|) \|_{(r,\gamma)}, \] where $\|\cdot \|_{(s,\kappa)}$ is the Orlicz norm relative to the function $M_{s,\kappa}(t)=t^s(\ln(2+t))^{\kappa}.$ The parameters $p,q,r,\alpha,\beta,\gamma$ and the Carathéodory functions ${\mit\Phi} _1,{\mit\Phi} _2$ 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.