| In this thesis, we obtain a priori estimates in harmonic analysis, i.e. estimates with bounds independent of the domain under consideration, using probabilistic techniques. Our approach is to relate our estimates to harmonic functions and then to martingales. We then prove the estimates using martingale techniques, in particular, the properties of spare functions (quadratic variation). Estimates are obtained for the boundary values of conjugate harmonic functions, Riesz transforms, and a generalization of the g*-function.;We also consider the general martingale transform. We show that it is possible to obtain probabilistic representations for multipliers of Laplace transform type and Riesz transforms of such operators. We also show that some martingale transforms correspond to pseudodifferential operators. |
