The $H^1(\T^d)$-multiplier norm of Rajchman measures, Zafran-type phenomena, and Walsh--Stieltjes series

Abstract

We study convolution by complex Rajchman measures as Fourier multipliers on the mean-zero part of the classical isotropic Hardy space $H^1(\T^d)$. Our main technical result gives a lower bound for the multiplier norm in terms of the amount of mass carried by sets of small lower Minkowski dimension. The estimate holds for every $d\ge1$; the argument is first established for real measures and then extended to complex measures by a finite phase decomposition. Since the testing functions have mean zero, a corresponding lower bound also holds on the full isotropic Hardy space. The estimate yields multidimensional Zafran-type criteria for non-natural spectra and, in particular, shows that positive Rajchman probability measures carried by sets of lower Minkowski dimension zero induce convolution operators with non-natural spectrum. In dimension one, for real measures, we also obtain the analogous estimate on the analytic space $H^1_0(\T)$ and frequency-localized failures of the natural-spectrum property. Finally, the binary form of the underlying combinatorial lemma gives Haar block estimates, which transfer to uncertainty principles and carrier-dimension results for Walsh--Stieltjes coefficients.