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User:Potahto/Muckenhoupt weights

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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

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For 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

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This 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

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The 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

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It 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

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If, 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

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  1. ^ 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)
  2. ^ 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)
  3. ^ 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)