JournalsrmiVol. 36, No. 7pp. 2217–2236

On measures that improve LqL^q dimension under convolution

  • Eino Rossi

    University of Helsinki, Finland
  • Pablo Shmerkin

    Universidad Torcuato di Tella, Buenos Aires, Argentina
On measures that improve $L^q$ dimension under convolution cover
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Abstract

The LqL^q dimensions, for 1<q<1 < q < \infty, quantify the degree of smoothness of a measure. We study the following problem on the real line: when does the LqL^q dimension improve under convolution? This can be seen as a variant of the well-known LpL^p-improving property. Our main result asserts that uniformly perfect measures (which include Ahlfors-regular measures as a proper subset) have the property that convolving with them results in a strict increase of the LqL^q dimension. We also study the case q=q = \infty, which corresponds to the supremum of the Frostman exponents of the measure. We obtain consequences for repeated convolutions and for the box dimension of sumsets. Our results are derived from an inverse theorem for the LqL^q norms of convolutions due to the second author.

Cite this article

Eino Rossi, Pablo Shmerkin, On measures that improve LqL^q dimension under convolution. Rev. Mat. Iberoam. 36 (2020), no. 7, pp. 2217–2236

DOI 10.4171/RMI/1198