Dimension-independent weak-type bounds for Gaussian maximal truncations

Establish a dimension-independent weak-type $(1,1)$ bound for the maximal truncation of the Gaussian Riesz transform, and likewise for the maximal truncation of the standard Euclidean Riesz transform.

Background

The paper proves a dimension-free weak-type $(1,1)$ estimate for the untruncated Gaussian vector Riesz transform, both in its spectral realization and in its principal-value realization. The proof depends critically on the exact identity $L^{1/2}u=Du$, which localizes the transformed bad part arising from the obstacle decomposition.

The authors explain that hard spatial truncation does not preserve this identity. Consequently, their obstacle argument does not yield a dimension-independent endpoint estimate for the Gaussian maximal truncation. They also note that the analogous dimension-free question for the maximal truncation of the standard Euclidean Riesz transform remains unresolved by the cited untruncated results, while only fixed-dimensional maximal-truncation estimates are available for the Gaussian case.