Heat-flow monotonicity related to the Hausdorff-Young inequality
Heat-flow monotonicity related to the Hausdorff-Young inequality
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与 Hausdorff-Young 不等式相关的热流单调性
DOI：
10.1112/blms/bdp073
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发表时间：
2009
影响因子：
0.9
通讯作者：
Bennett J
中科院分区：
文献类型：
--
作者：
Bennett J
众所周知，如果q是一个偶数，那么一个固定的高斯平移的叠加的傅里叶变换的Lq（xrd）范数是单调增加的，因为它们的中心“同时滑动”到原点。我们提供了明确的例子表明，这种单调性的性质显着失败，如果q> 2是不是一个偶数。这些结果是等价的，重新标度后，类似的声明涉及解决热方程。这种考虑是自然的著名定理贝克纳关于高斯极值的豪斯多夫-杨不等式。
It is known that ifqis an even integer, then theLq(ℝd) norm of the Fourier transform of a superposition of translates of a fixed gaussian is monotone increasing as their centres ‘simultaneously slide’ to the origin. We provide explicit examples to show that this monotonicity property fails dramatically ifq> 2 is not an even integer. These results are equivalent, upon rescaling, to similar statements involving solutions to heat equations. Such considerations are natural given the celebrated theorem of Beckner concerning the gaussian extremisability of the Hausdorff–Young inequality.