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We study stability estimates for the almost extremal functions associatedwith the $L^p$-bound for the real and imaginary parts of the Beurling-Ahlforsoperator. The proof exploits probabilistic methods and rests on analogousresults for differentially subordinate martingales which are of independentinterest. This allows us to obtain stability inequalities for a larger class ofFourier multipliers.

Stability in Burkholder's differentially subordinate martingales inequalities and applications to Fourier multipliers