Sobolev Inequalities for Weight Spaces and Supercontractivity

For <!-- MATH $\phi \in {C^2}({{\mathbf{R}}^n})$ --> with <!-- MATH $\phi (x) = a|x{|^{1 + s}}$ --> for <!-- MATH $|x| \geqslant {x_0},a,s > 0$ --> 0$">, define the measure <!-- MATH $d\mu = \exp ( - 2\phi ){d^n}x$ --> on <!-- MATH ${{\mathbf{R}}^n}$ --> . We show that for any <!-- MATH $k \in {{\mathbf{Z}}^ + }$ --> <!-- MATH \begin{displaymath} \begin{array}{*{20}{c}} {\int {|f{|^2}|\lg(|f|){|^{2sk/(s + 1)}}d\mu } } \hfill \\{ \leqslant c\left\{ {\sum\limits_{|\alpha | = 0}^k {\left\|{D^\alpha }f\right\|_{{L_2}(d\mu )}^2 + \left\|f\right\|_{{L_2}(d\mu )}^2 \bullet |\lg(\left\|f\right\|_{{L_2}(d\mu )}){|^{2sk/(s + 1)}}} } \right\}} \hfill \\\end{array} \end{displaymath} -->