| We study a problem of finding the best constant in a weak type inequality for martingale transforms extending the result of Burkholder (1966). First, we study the inequality for the discrete-time martingale case. We present examples of martingales that give good lower estimates of the best constant. We then find a biconcave function to prove that the supremum of these lower estimates is in fact the best constant. We use this biconcave function to prove a sharp weak type inequality for differentially subordinate martingales with the same best constant, and by approximation a similar inequality for stochastic integrals. We generalize these results to the continuous-time case and give an application to harmonic functions. |
