We study weighted local Sobolev interpolation inequalities of the form \[ 1 w 2 ( B ) ∫ B | u ( x ) | p h w 2 ( x ) d x ≤ c ( 1 v ( B ) ∫ B | u ( x ) | p v ( x ) d x ) h − 1 × ( | B | p / n w 1 ( B ) ∫ B | ∇ u ( x ) | p w 1 ( x ) d x + 1 v ( B ) ∫ B | u ( x ) | p v ( x ) d x ) , \begin {gathered} \frac {1} {{{w_2}(B)}}{\int \limits _B {|u(x){|^{ph}}{w_2}(x)dx \leq c\left ( {\frac {1} {{v(B)}}\int \limits _B {|u(x){|^p}v(x)dx} } \right )} ^{h - 1}} \hfill \\ \times \left ( {\frac {{|B{|^{p/n}}}} {{{w_1}(B)}}\int \limits _B {|\nabla u(x){|^p}{w_1}(x)dx + \frac {1} {{v(B)}}} \int \limits _B {|u(x){|^p}v(x)dx} } \right ), \hfill \\ \end {gathered} \] , where 1 > p > ∞ , h > 1 , B 1 > p > \infty ,h > 1, B is a ball in R n {{\mathbf {R}}^n} , and v v , w 1 {w_1} , and w 2 {w_2} are weight functions. The case p = 2 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 u has compact support in B B , and we derive global Landau inequalities ‖ ∇ u ‖ L w q ≤ c ‖ ∇ u ‖ L v p 1 − a ‖ ∇ 2 u ‖ L v p a , 0 > a > 1 , 1 > p ≤ q > ∞ {\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 u has compact support.