![]() | This user page or section is in a state of significant expansion or restructuring. You are welcome to assist in its construction by editing it as well. If this user page has not been edited in several days, please remove this template. If you are the editor who added this template and you are actively editing, please be sure to replace this template with {{in use}} during the active editing session. Click on the link for template parameters to use.
This page was last edited by Potahto (talk | contribs) 16 years ago. (Update timer) |
The class of Muckenhoupt weights are those weights for which the Hardy-Littlewood maximal operator is bounded on . Specifically, we consider functions on and there associated maximal function defined as
- ,
where is a ball in with radius and centre . We wish to characterise the functions for which we have a bound
where depends only on and . This was first done by Benjamin Muckenhoupt[1].
Definition
editFor a fixed , we say that a weight belongs to if is locally integrable and there is a constant such that, for all balls in , we have
where and is the Lebesgue measure of . We say belongs to if there exists some such that
for all and all balls .[2]
Equivalent characterisations
editThis following result is a fundamental result in the study of Muckenhoupt weights. A weight is in if and only if any one of the following hold.[2]
(a) The Hardy-Littlewood maximal function is bounded on , that is
for some which only depends on and the constant in the above definition.
(b) There is a constant such that for any locally integrable function on
for all balls . Here
is the average of over and
Reverse Hölder inequalities
editThe main tool in the proof of the above equivalence is the following result.[2] The following statements are equivalent
(a) belongs to for some
(b) There exists an and a (both depending on such that
for all balls
(c) There exists so that for all balls and subsets
We call the inequality in (b) a reverse Hölder inequality as the reverse inequality follows for any non-negative function directly from Hölder's inequality. If any of the three equivalent conditions above hold we say belongs to .
Boundedness of singular integrals
editIt is not only the Hardy-Littlewood maximal operator that is bounded on these weighted spaces. In fact, any Calderón-Zygmund singular integral operator is also bounded on these spaces.[3] Let us describe a simpler version of this here.[2] 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 :
for all and multi-indices . Then, for each , we have that is a bounded operator on . That is, we have the estimate
for all for which the right-hand side is finite.
A converse result
editIf, in addition to the three conditions above, we assume the non-degeneracy condition on the kernel : For a fixed unit vector
whenever with , then we have a converse. If we know
for some fixed and some , then .[2]
References
edit- ^ Munckenhoupt, Benjamin (1972). "Weighted norm inequalities for the Hardy maximal function". Transactions of the American Mathematical Society, vol. 165: 207–26.
{{cite journal}}
: Check date values in:|date=
(help); Cite has empty unknown parameter:|coauthors=
(help) - ^ a b c d e Stein, Elias (1993). "5". Harmonic Analysis. Princeton University Press.
{{cite book}}
: Check date values in:|date=
(help); Cite has empty unknown parameter:|coauthors=
(help) - ^ Grakakos, Loukas (2004). "9". Classical and Modern Fourier Analysis. New Jersey: Pearson Education, Inc.
{{cite book}}
: Check date values in:|date=
(help); Cite has empty unknown parameter:|coauthors=
(help)