Subgaussian estimates in probability and harmonic analysis

Abstract

We will discuss subgaussian estimates in harmonic analysis involving the non-tangential maximal function $$N_{\alpha }f$$ and the area function $$A_{\beta } f$$ of a function f on $${\mathbb {R}}^n.$$ 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 $$\Vert N_{\alpha }f\Vert _p \le C_p \Vert A_{\beta }f\Vert _p$$ ; here $$C_p= O(\sqrt{p})$$ as $$p \rightarrow \infty $$ 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.