Stability in Burkholder's differentially subordinate martingales   inequalities and applications to Fourier multipliers
  We study stability estimates for the almost extremal functions associated with the $L^p$-bound for the real and imaginary parts of the Beurling-Ahlfors operator. The proof exploits probabilistic methods and rests on analogous results for differentially subordinate martingales which are of independent interest. This allows us to obtain stability inequalities for a larger class of Fourier multipliers. 
Rodrigo Banuelos and Adam Osekowski
math.PR math.CA math.FA