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)
“When her guests were awash with champagne and with gin,
She was recklessly sober, as sharp as a pin.
An abstemious man would reel at her look,
As she rolled a bright eye and praised his last book.”
—William Plomer (1903–1973)
“To look backward for a while is to refresh the eye, to restore it, and to render it the more fit for its prime function of looking forward.”
—Margaret Fairless Barber (1869–1901)