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 the sharp, sharp and/or function:
“The A B C of being,
The ruddy temper, the hammer
Of red and blue, the hard sound
Steel against intimation the sharp flash,
The vital, arrogant, fatal, dominant X.”
—Wallace Stevens (1879–1955)
“West Germans are tall, pert and orthodontically corrected, with hands, teeth and hair as clean as their clothes and clothes as sharp as their looks. Except for the fact that they all speak English pretty well, they’re indistinguishable from Americans.”
—P.J. (Patrick Jake)
“Think of the tools in a tool-box: there is a hammer, pliers, a saw, a screwdriver, a rule, a glue-pot, nails and screws.—The function of words are as diverse as the functions of these objects.”
—Ludwig Wittgenstein (1889–1951)