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 .
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