In this article we focus on a general real-valued continuous stationary Gaussian field X characterized by its spectral density $|g{|}^2$, where g is any even real-valued deterministic square integrable function. Our starting point consists in drawing a close connection between critical Besov regularity of the inverse Fourier transform of g and ${\alpha }_X$ the random pointwise Hölder exponent function of X, which measures local roughness/smoothness of its sample paths at each point. Then, thanks to Littlewood-Paley methods and Hausdorff-Young inequalities, under weak conditions on g, we show that the random function ${\alpha }_X$ is actually a deterministic constant which does not change from point to point. This result means that the field X is of monofractal nature. Also, it is worth mentioning that such a result can easily be extended to the case where X is no longer stationary but has stationary increments.

在这篇文章中，我们关注一个一般实值连续平稳高斯场X，其特征在于它的谱密度$|G{|}^2$，其中g是任何偶数实值确定性平方可积函数。我们的出发点在于在g的逆傅立叶变换的临界 Besov 正则性和${\alpha }_X$X的随机逐点 Hölder 指数函数，它测量每个点的样本路径的局部粗糙度/平滑度。然后，感谢 Littlewood-Paley 方法和 Hausdorff-Young 不等式，在g 的弱条件下，我们证明随机函数${\alpha }_X$实际上是一个确定性常数，不会随点变化。这个结果意味着场X是单分形的。此外，值得一提的是，这样的结果可以很容易地扩展到X不再平稳但具有平稳增量的情况。