Sobolev interpolation inequalities with weights

Abstract:

We study weighted local Sobolev interpolation inequalities of the form \[ \begin {gathered} \frac {1} {{{w_2}(B)}}{\int \limits _B {|u(x){|^{ph}}{w_2}(x)dx \leq c\left ( {\frac {1} {{v(B)}}\int \limits _B {|u(x){|^p}v(x)dx} } \right )} ^{h - 1}} \hfill \\ \times \left ( {\frac {{|B{|^{p/n}}}} {{{w_1}(B)}}\int \limits _B {|\nabla u(x){|^p}{w_1}(x)dx + \frac {1} {{v(B)}}} \int \limits _B {|u(x){|^p}v(x)dx} } \right ), \hfill \\ \end {gathered} \], where $1 < p < \infty ,h > 1, B$ is a ball in ${{\mathbf {R}}^n}$, and $v$ ,${w_1}$, and ${w_2}$ are weight functions. The case $p = 2$ is of special importance in deriving regularity results for solutions of degenerate parabolic equations. We also study the analogous inequality without the second summand on the right in the case $u$ has compact support in $B$, and we derive global Landau inequalities ${\left \| {\nabla u} \right \|_{L_w^q}} \leq c\left \| {\nabla u} \right \|_{L_v^p}^{1 - a}\left \| {{\nabla ^2}u} \right \|_{L_v^p}^a,0 < a < 1,1 < p \leq q < \infty$, when $u$ has compact support.