Subgaussian estimates in probability and harmonic analysis

We will discuss subgaussian estimates in harmonic analysis involving the non-tangential maximal function N_αf and the area function A_β f of a function f on ℝ^n. We will first introduce subgaussian estimates in the setting of martingales; these then lead to analogous estimates for harmonic functions. Among the consequences of these are sharp L^p inequalities ‖ N_αf‖ _p ≤ C_p ‖ A_βf‖ _p ; here C_p= O(√(p)) as p →∞ and this order is sharp. The subgaussian estimates produce Laws of the Iterated Logarithm (LILs) involving the non-tangential maximal function and area function. These ideas are also applied to lacunary series of more general functions to yield LILs.