Logarithmic Sobolev Trace Inequality
نویسنده
چکیده
A logarithmic Sobolev trace inequality is derived. Bounds on the best constant for this inequality from above and below are investigated using the sharp Sobolev inequality and the sharp logarithmic Sobolev inequality. Logarithmic Sobolev inequalities capture the spirit of classical Sobolev inequalities with the logarithm function replacing powers, and they can be considered as limiting cases of the classical Sobolev inequalities. In the original analysis of the logarithmic Sobolev inequality, Gross [13] emphasized its infinite-dimensional character and the dimension-independent nature of its constants. Recent arguments by Beckner [7], [8], [9] used the product structure of the domain and asymptotics for the sharp Sobolev embedding to derive the logarithmic Sobolev inequality with more explicit geometric character: for a smooth function f ∈ S(R) with ‖f‖L2(Rn) = 1, (1) ∫ Rn |f(x)| ln |f(x)|dx ≤ n 4 ln [ 2 πen ∫
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