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>0η>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 11 in a metric space. We do so by revisiting and improving the underlying construction of random dyadic cubes, which also has other applications.