Today I would like to introduce an important inequality from the theory of martingales that will be the subject of a few more posts. This inequality will further provide the opportunity to introduce a very interesting and powerful result of Tao and Wright – a sort of square-function characterisation for the Orlicz space .
1. The Chang-Wilson-Wolff inequality
Consider the collection of standard dyadic intervals that are contained in . We let for each denote the subcollection of intervals such that . Notice that these subcollections generate a filtration of , that is , where denotes the sigma-algebra generated by the collection . We can associate to this filtration the conditional expectation operators
and therefore define the martingale differences
With this notation, we have the formal telescopic identity
Demystification: the expectation is simply , where is the unique dyadic interval in such that .
Letting for brevity, the sequence of functions is called a martingale (hence the name “martingale differences” above) because it satisfies the martingale property that the conditional expectation of “future values” at the present time is the present value, that is
In the following we will only be interested in functions with zero average, that is functions such that . Given such a function then, we can define its martingale square function to be
With these definitions in place we can state the Chang-Wilson-Wolff inequality as follows.
An important point about the above inequality is the behaviour of the constant in the Lebesgue exponent , which is sharp. This can be seen by taking a “lacunary” function (essentially one where , a constant) and randomising the signs using Khintchine’s inequality (indeed, is precisely the asymptotic behaviour of the constant in Khintchine’s inequality; see Exercise 5 in the 2nd post on Littlewood-Paley theory).
It should be remarked that the inequality extends very naturally and with no additional effort to higher dimensions, in which is replaced by the unit cube and the dyadic intervals are replaced by the dyadic cubes. We will only be interested in the one-dimensional case here though.