The $L^q$ dimensions, for $1 < q < \infty$, quantify the degree of smoothness of a measure. We study the following problem on the real line: when does the $L^q$ dimension improve under convolution? This can be seen as a variant of the well-known $L^p$-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 $L^q$ dimension. We also study the case $q = \infty$, 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 $L^q$ norms of convolutions due to the second author.

中文翻译：

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关于卷积下提高$ L ^ q $维数的措施

对于$ 1 <q <\ infty $，$ L ^ q $维度量化了度量的平滑度。我们实际研究以下问题：卷积下的$ L ^ q $维何时会改善？这可以看作是众所周知的$ L ^ p $ -improving属性的变体。我们的主要结果断言，统一的完美测度（包括Ahlfors-常规测度作为适当的子集）具有以下性质：与之卷积会导致$ L ^ q $维度的严格增加。我们还研究了$ q = \ infty $的情况，它对应于该度量的Frostman指数的最大值。我们得到重复卷积和和集的盒维数的结果。我们的结果来自于第二作者的卷积$ L ^ q $范数的逆定理。