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Almost Lipschitz-continuous wavelets in metric spaces via a new randomization of dyadic cubes

In any quasi-metric space of homogeneous type, Auscher and Hytönen recently gave a construction of orthonormal wavelets with Hölder-continuity exponent $η>0$. However, even in a metric space, their exponent is in general quite small. In this paper, we show that the Hölder-exponent can be taken arbitrarily close to 1 in a metric space. We do so by revisiting and improving the underlying construction of random dyadic cubes, which also has other applications.