A boundedness result for degenerate parabolic Riesz transforms with rough coefficients

Abstract

In this paper, we prove a boundedness result for parabolic Riesz transforms in the range $p \le 2$ associated with degenerate parabolic operators whose elliptic part is in divergence form with complex coefficients depending measurably on space and time. The degeneracy is determined by a spatial weight in the Muckenhoupt class $A_2(\mathbb{R}^n)$. The argument relies on off-diagonal estimates for the parabolic gradient of the resolvent family, together with weighted embeddings. We obtain boundedness in a range of exponents strictly below $2$, with explicit ranges quantified by the reverse Hölder class of the weight. For real coefficients, this quantification is sharp and extends down to $1$.