Sobolev inequalities for weight spaces and supercontractivity

Abstract:

For $\phi \in {C^2}({{\mathbf {R}}^n})$ with $\phi (x) = a|x{|^{1 + s}}$ for $|x| \geqslant {x_0},a,s > 0$, define the measure $d\mu = \exp ( - 2\phi ){d^n}x$ on ${{\mathbf {R}}^n}$. We show that for any $k \in {{\mathbf {Z}}^ + }$ \[ \begin {array}{*{20}{c}} {\int {|f{|^2}|\lg (|f|){|^{2sk/(s + 1)}}d\mu } } \hfill \\ { \leqslant c\left \{ {\sum \limits _{|\alpha | = 0}^k {\left \|{D^\alpha }f\right \|_{{L_2}(d\mu )}^2 + \left \|f\right \|_{{L_2}(d\mu )}^2 \bullet |\lg (\left \|f\right \|_{{L_2}(d\mu )}){|^{2sk/(s + 1)}}} } \right \}} \hfill \\ \end {array} \] As a consequence we prove ${e^{ - t{\nabla ^\ast } \cdot \nabla }}:{L_q}({{\mathbf {R}}^n},d\mu ) \to {L_p}({{\mathbf {R}}^n},d\mu ),p,q \ne 1,\infty$, is bounded for all $t > 0$.