Maximal Function - The Sharp Maximal Function

The Sharp Maximal Function

For a locally integrable function on, the sharp maximal function is defined as

for each, where the supremum is taken over all balls .

The sharp function can be used to obtain a point-wise inequality regarding singular integrals. Suppose we have an operator which is bounded on, so we have

for all smooth and compactly supported . Suppose also that we can realise as convolution against a kernel in the sense that, whenever and are smooth and have disjoint support

Finally we assume a size and smoothness condition on the kernel :

when . Then for a fixed, we have

for all .