Control level sets

Quick description

From Fubini's theorem one has the identity (1)

for non-negative . Thus, to estimate an integral, one way to do this is to control the measure of the level sets for different values of .

A slight variant: the norm of a function can be expressed by the formula (2)

As another variant, one can view (1) as a means to decompose any non-negative function as a superposition of indicator functions: Measure theory

Example 1

Let be a finite set, and let be a function. Show that for any .

Solution: we may as well normalize , thus If we insert this bound directly into (2) (using counting measure instead of ) we obtain a logarithmic divergence. But we can improve the bound in two ways. Firstly, when , then , since there is no other non-negative integer less than . Secondly, we also have the trivial upper bound of , which is superior when . If we then use "divide and conquer" and partition the integral on the right-hand side of (2) into the regions , , and , one obtains the claim.

Example 2

Show that if a function lies in the weak spaces and for some measure space and some exponents , then it lies in the strong spaces for all .

We need to use "divide and conquer" efficiently in order to 'interpolate' the information we have at the endpoint weak spaces. The idea is that when the function is large, say , then the norms increase when increases which is the same as saying that whenever and . This indicates that we should split the integral into two parts; one integral over the set where is large and one over its complement. Then the integral over the region where is large is controlled by the weak norm of and the integral where is small is controlled by the weak norm of .

This can be done in a very elegant fashion by using the description (2) of the norm of a function. Indeed we can write For the first term (which corresponds to the set where is small) we use the weak estimate Similarly we get for the second term Thus we have and in particular More efficiently, one could split the integral at some point instead of the point and then optimize in the parameter .

General discussion

This method is closely related to double counting and "interchange integrals or sums".

The method also combines well with dyadic decomposition. Indeed, one easily verifies that is comparable to , and more generally is comparable (up to constants depending on ) to .

The Marcinkiewicz interpolation theorem relies heavily on these sorts of level set decompositions.