Fractional smoothness of images of logarithmically concave measures under polynomials. (English) Zbl 1390.28002
Summary: We show that a measure on the real line, that is the image of a log-concave measure under a polynomial of degree \(d\), possesses a density from the Nikolskii-Besov class of fractional order \(1 / d\). This result is used to prove an estimate for the total variation distance between such measures in terms of the Fortet-Mourier distance.
MSC:
| 28A10 | Real- or complex-valued set functions |
| 28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |
Keywords:
logarithmically concave measure; total variation distance; Fortet-Mourier distance; distribution of a polynomial; Nikolskii-Besov classReferences:
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