# Various sharp estimates for semi-discrete Riesz transforms of the second order

## TL;DR

This paper establishes sharp $L^p$ and related estimates for second order Riesz transforms on certain Lie groups, combining stochastic integral representations with martingale inequalities to improve understanding of these operators.

## Contribution

It provides new sharp $L^p$ estimates and extends the analysis of second order Riesz transforms to semi-discrete Lie groups, integrating stochastic methods with harmonic analysis.

## Key findings

- Derived sharp $L^p$ estimates using Choi type constants.
- Established weak-type, logarithmic, and exponential bounds.
- Proved optimal $L^q o L^p$ estimates for the transforms.

## Abstract

We give several sharp estimates for a class of combinations of second order Riesz transforms on Lie groups ${G}={G}_{x} \times {G}_{y}$ that are multiply connected, composed of a discrete abelian component ${G}_{x}$ and a connected component ${G}_{y}$ endowed with a biinvariant measure. These estimates include new sharp $L^p$ estimates via Choi type constants, depending upon the multipliers of the operator. They also include weak-type, logarithmic and exponential estimates. We give an optimal $L^q \to L^p$ estimate as well. It was shown recently by Arcozzi, Domelevo and Petermichl that such second order Riesz transforms applied to a function may be written as conditional expectation of a simple transformation of a stochastic integral associated with the function. The proofs of our theorems combine this stochastic integral representation with a number of deep estimates for pairs of martingales under strong differential subordination by Choi, Banuelos and Osekowski. When two continuous directions are available, sharpness is shown via the laminates technique. We show that sharpness is preserved in the discrete case using Lax-Richtmyer theorem.