Abstract
Let a a ∈ [0, 1] and r ∈ [1, 2] satisfy relation r = 2/(2 − a). Let μ(dx)=c n r exp(-(|x 1|r+|x 2|r+...+|x n |r))dx 1 dx 2...dx n be a probability measure on the Euclidean space (R n, ‖ · ‖). We prove that there exists a universal constant C such that for any smooth real function f on R n and any p ∈ [1,2)
. We prove also that if for some probabilistic measure μ on R n the above inequality is satisfied for any p ∈ [1, 2) and any smooth f then for any h : R n → R such that |h(x)-h(y)|≤∥x-y∥ there is E μ |h| < ∞ and
for t > 1, where K > 0 is some universal constant.
Research partially supported by KBN Grant 2 P03A 043 15.
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© 2000 Springer-Verlag
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Latała, R., Oleszkiewicz, K. (2000). Between sobolev and poincaré. In: Milman, V.D., Schechtman, G. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 1745. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0107213
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DOI: https://doi.org/10.1007/BFb0107213
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