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 .
Read more about this topic: Maximal Function
Famous quotes containing the words sharp and/or function:
“Everything I ever learned as a small boy came from my father. And I never found anything he ever told me to be wrong or worthless. The simple lessons he taught me are as sharp and clear in my mind, as if I had heard them only yesterday.”
—Philip Dunne (1908–1992)
“For me being a poet is a job rather than an activity. I feel I have a function in society, neither more nor less meaningful than any other simple job. I feel it is part of my work to make poetry more accessible to people who have had their rights withdrawn from them.”
—Jeni Couzyn (b. 1942)