Gagliardo–Nirenberg inequalities in logarithmic spaces

Agnieszka Ka/lamajska ¹   ; Katarzyna Pietruska-Pa/luba ¹

¹ Institute of Mathematics Warsaw University Banacha 2 02-097 Warszawa, Poland

Colloquium Mathematicum, Tome 106 (2006) no. 1, pp. 93-107

Résumé

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.

Détail

DOI : 10.4064/cm106-1-8

Keywords: obtain interpolation inequalities derivatives begin multline* int alpha nabla leq bigg int beta mit phi nabla int gamma mit phi nabla bigg end multline* their counterparts expressed orlicz norms nabla alpha leq mit phi nabla beta mit phi nabla gamma where cdot kappa orlicz norm relative function kappa kappa parameters alpha beta gamma carath odory functions mit phi mit phi supposed satisfy certain consistency conditions classical gagliardo nirenberg inequalities follow special gagliardo nirenberg inequalities logarithmic spaces higher order gradients considered