The sharp $L^p$ bound for the imaginary part of the Beurling-Ahlfors operator on the lattice

Abstract

We prove the sharp ${\ell }^p$ bound $p^{\star }-1$ for certain second-order discrete Riesz transforms on products of integers, in all dimensions. The family includes the imaginary part of the discrete Beurling-Ahlfors operator in dimension two as a special case. While these operators admit a martingale representation via compound Poisson jump processes and heat extensions, the martingale attached to a function and the martingale attached to its transform do not enjoy strong differential subordination, so the operator-norm estimate is not accessible through prior work.

20 pages. Added extension of the result to all dimensions. Corrected typos and added references