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In number theory Z_p would be the p-adics and not the integers mod p
 
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In [[mathematics]], the class of '''Muckenhoupt weights''' <math>A_p</math> consists of those weights <math>\omega</math> for which the [[Hardy–Littlewood maximal operator]] is bounded on <math>L^p(d\omega)</math>. Specifically, we consider functions <math>f</math> on <math>\mathbb{R}^n</math> and their associated [[maximal function]]s <math>M(f)</math> defined as


:<math> M(f)(x) = \sup_{r>0} \frac{1}{r^n} \int_{B_r} |f|,</math>
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where <math>B_r</math> is a ball in <math>\mathbb{R}^n</math> with radius <math>r</math> and centre <math>x</math>. We wish to characterise the functions <math>\omega \colon \mathbb{R}^n \to [0,\infty)</math> for which we have a bound
 
:<math> \int |M(f)(x)|^p \, \omega(x) dx \leq C \int |f|^p \, \omega(x)\, dx,</math>
 
where <math>C</math> depends only on <math>p \in [1,\infty)</math> and <math>\omega</math>. This was first done by [[Benjamin Muckenhoupt]].<ref>{{cite journal | last = Muckenhoupt | first = Benjamin| coauthors = | title = Weighted norm inequalities for the Hardy maximal function | work = Transactions of the American Mathematical Society, vol. 165| pages = 207–226| language = | publisher = | year = 1972 | url = | accessdate = }}</ref>
 
==Definition==
 
For a fixed <math>1 < p < \infty</math>, we say that a weight <math>\omega \colon \mathbb{R}^n \to [0,\infty)</math> belongs to <math>A_p</math>  if <math>\omega</math> is locally integrable and there is a constant <math>C</math> such that, for all balls <math>B</math> in <math>\mathbb{R}^n</math>, we have
 
: <math>\left(\frac{1}{|B|} \int_B \omega(x) \, dx \right)\left(  \frac{1}{|B|} \int_B \omega(x)^\frac{-p'}{p} \, dx \right)^\frac{p}{p'} \leq C < \infty,</math>
 
where <math>1/p + 1/p' = 1</math> and <math>|B|</math> is the [[Lebesgue measure]] of <math>B</math>. We say <math>\omega \colon \mathbb{R}^n \to [0,\infty)</math> belongs to <math>A_1</math> if there exists some <math>C</math> such that
 
: <math>\frac{1}{|B|} \int_B \omega(x) \, dx \leq C\omega(x), </math>
 
for all <math>x \in B</math> and all balls <math>B</math>.<ref name = bible>{{cite book | last = Stein | first = Elias | coauthors = | title = Harmonic Analysis | chapter = 5 | work = | pages = | language = | publisher = Princeton University Press| year = 1993 | url = | accessdate =  }}</ref>
 
==Equivalent characterizations==
 
This following result is a fundamental result in the study of Muckenhoupt weights. A weight <math>\omega</math> is in <math>A_p</math> if and only if any one of the following hold.<ref name = bible/>
 
(a) The [[Hardy–Littlewood maximal function]] is bounded on <math>L^p(\omega(x)dx)</math>, that is
 
:<math> \int |M(f)(x)|^p \, \omega(x)\, dx \leq C \int |f|^p \, \omega(x)\, dx,</math>
 
for some <math>C</math> which only depends on <math>p</math> and the constant <math>A</math> in the above definition.
 
(b) There is a constant <math>c</math> such that for any locally integrable function <math>f</math> on <math>\mathbb{R}^n</math>
 
: <math>(f_B)^p \leq \frac{c}{\omega(B)} \int_B f(x)^p \, \omega(x)\,dx</math>
 
for all balls <math>B</math>. Here
 
: <math>f_B = \frac{1}{|B|}\int_B f</math>
 
is the average of <math>f</math> over <math>B</math> and
 
: <math>\omega(B) = \int_B \omega(x)\,dx.</math>
 
Equivalently, <math>w=e^\phi\in A_{p}</math>, where <math>p\in (1,\infty)</math>, if and only if
: <math> \sup_{B}\frac{1}{|B|}\int_{B}e^{\phi-\phi_{B}}dx<\infty </math>
and
: <math> \sup_{B}\frac{1}{|B|}\int_{B}e^{-\frac{\phi-\phi_{B}}{p-1}}dx<\infty. </math>
 
This equivalence can be verified by using [[Jensen's Inequality]].
 
==Reverse Hölder inequalities and <math>A_{\infty}</math>==
 
The main tool in the proof of the above equivalence is the following result.<ref name=bible/> The following statements are equivalent
 
(a) <math>\omega</math> belongs to <math>A_p</math> for some <math>p \in [1,\infty)</math>
 
(b) There exists an <math>q > 1</math> and a <math>c</math> (both depending on <math>\omega</math>) such that
 
: <math>\frac{1}{|B|} \int_{B} \omega^q \leq \left(\frac{c}{|B|} \int_{B} \omega \right)^q</math>
 
for all balls <math>B_r</math>
 
(c) There exists <math>\delta, \gamma \in (0,1)</math> so that for all balls <math>B</math> and subsets <math>E \subset B</math>
 
: <math>|E| \leq \gamma|B| \implies \omega(E) \leq \delta\omega(B)</math>
 
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 <math>\omega</math> belongs to <math>A_\infty</math>.
 
==Weights and BMO==
 
The definition of an <math>A_p</math> weight and the reverse Hölder inequality indicate that such a weight cannot degenerate or grow too quickly. This property can be phrased equivalently in terms of how much the logarithm of the weight oscillates:
 
(a) If <math>w\in A_{p},\;\; p\geq 1,</math>, then <math>\log w\in BMO</math> (i.e. <math>\log w</math> has [[bounded mean oscillation]]).
 
(b) If <math>f \in BMO</math>, then for sufficiently small <math>\delta>0</math>, we have <math>e^{\delta f}\in A_{p}</math> for some <math>p\geq 1</math>.
 
This equivalence can be established by using the exponential characterization of weights above, Jensen's inequality, and the [[John–Nirenberg inequality]]. Note that the smallness assumption on <math>\delta>0</math> in part (b) is necessary for the result to be true, as <math>\log\frac{1}{|x|}</math> is a BMO function, but <math>e^{\log\frac{1}{|x|}}=\frac{1}{|x|}</math> is not in any <math>A_{p}</math>.
 
==Further properties==
 
Here we list a few miscellaneous properties about weights, some of which can be verified from using the definitions, others are nontrivial results:
 
(i) <math>A_1 \subseteq A_p \subseteq A_\infty\text{ for }1\leq p\leq\infty.</math>
 
(ii) <math>A_\infty = \bigcup_{p<\infty}A_p.</math>
 
(iii) If <math>w\in A_p</math>, then <math>w \, dx</math> defines a [[doubling measure]]: for any ball ''B'', if 2''B'' is the ball of twice the radius, then <math>w(2B)\leq Cw(B)</math> where ''C''&nbsp;>&nbsp;1 is a constant depending on <math>w</math>.
 
(iv) If <math>w\in A_p</math>, then there is <math>\delta>1</math> such that <math>w^\delta \in A_p</math>.
 
(v) If <math>w\in A_{\infty}</math> then there is <math>\delta>0</math> and weights <math>w_1,w_2\in A_1</math> such that <math>w=w_1 w_2^{-\delta}</math>.<ref>{{cite journal
  | last = Jones | first = Peter W.
  | title = Factorization of A<sub>p</sub> weights
  | journal = Ann. Of Math. (2) | volume = 111 | pages = 511–530 | year = 1980
  | doi = 10.2307/1971107
  | issue = 3
}}</ref>
 
==Boundedness of singular integrals==
 
It is not only the Hardy–Littlewood maximal operator that is bounded on these weighted <math>L^p</math> spaces. In fact, any [[singular integrals|Calderón-Zygmund singular integral operator]] is also bounded on these spaces.<ref name = grafakos>{{cite book | last = Grakakos | first = Loukas | coauthors = | title = Classical and Modern Fourier Analysis | chapter = 9 | work = | pages = | language = | publisher = Pearson Education, Inc. | place = New Jersey| year = 2004 | url = | accessdate =  }}</ref> Let us describe a simpler version of this here.<ref name = bible/> Suppose we have an operator <math>T</math> which is bounded on <math>L^2(dx)</math>, so we have
 
: <math>\|T(f)\|_{L^2} \leq C\|f\|_{L^2},</math>
 
for all smooth and compactly supported <math>f</math>. Suppose also that we can realise <math>T</math> as convolution against a kernel <math>K</math> in the sense that, whenever <math>f</math> and <math>g</math> are smooth and have disjoint support
 
: <math>\int g(x) T(f)(x) \, dx = \iint g(x) K(x-y) f(y) \, dy\,dx.</math>
 
Finally we assume a size and smoothness condition on the kernel <math>K</math>:
 
: <math>|{\partial^\alpha}K| \leq C |x|^{-n-\alpha}</math>
 
for all <math>x \neq 0</math> and multi-indices <math>|\alpha| \leq 1</math>. Then, for each <math>p \in (1,\infty)</math> and <math>\omega \in A_p</math>, we have that <math>T</math> is a bounded operator on <math>L^p(\omega(x)\,dx)</math>. That is, we have the estimate
 
: <math>\int |T(f)(x)|^p \, \omega(x)\,dx \leq C \int |f(x)|^p \, \omega(x)\, dx,</math>
 
for all <math>f</math> for which the right-hand side is finite.
 
===A converse result===
 
If, in addition to the three conditions above, we assume the non-degeneracy condition on the kernel <math>K</math>: For a fixed unit vector <math>u_0</math>
 
: <math>|K(x)| \geq a |x|^{-n}</math>
 
whenever <math>x = t \dot u_0</math> with <math>-\infty<t<\infty</math>, then we have a converse. If we know
 
: <math>\int |T(f)(x)|^p \, \omega(x)\,dx \leq C \int |f(x)|^p \, \omega(x)\, dx,</math>
 
for some fixed <math>p \in (1,\infty)</math> and some <math>\omega</math>, then <math>\omega \in A_p</math>.<ref name = bible/>
 
==Weights and quasiconformal mappings==
 
For <math>K> 1</math>, a K-[[quasiconformal mapping]] is a homeomorphism <math>f:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}</math> with <math>f\in W^{1,2}_{loc}(\mathbb{R}^{n})</math> and
 
: <math>\frac{||Df(x)||^{n}}{J(f,x)}\leq K</math>
 
where <math>Df(x)</math> is the [[derivative]] of <math>f</math> at <math>x</math> and <math>J(f,x)=\mbox{det}(Df(x))</math> is the [[Jacobian]].
 
A theorem of Gehring<ref>{{cite journal
  | last = Gehring | first = F. W.  
  | title = The L<sup>p</sup>-integrability of the partial derivatives of a               quasiconformal mapping
  | journal = Acta Math. | volume = 130 | pages = 265–277 | year = 1973
  | doi = 10.1007/BF02392268
}}</ref>  states that for all K-quasiconformal functions <math>f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}</math>, we have <math> J(f,x)\in A_{p}</math> where <math>p</math> depends on <math>K</math>.
 
==Harmonic measure==
 
If you have a simply connected domain <math>\Omega\subseteq\mathbb{C}</math>, we say its boundary curve <math>\Gamma=\partial \Omega</math> is K-chord-arc if for any two points <math>z,w\in \Gamma</math> there is a curve <math>\gamma\subseteq\Gamma</math> connecting <math>z</math> and <math>w</math> whose length is no more than <math>K|z-w|</math>. For a domain with such a boundary and for any <math>z_{0}\in \Omega</math>, the [[harmonic measure]] <math>w(\cdot)=w(z_{0},\Omega,\cdot)</math> is absolutely continuous with respect to one-dimensional [[Hausdorff measure]] and its [[Radon–Nikodym derivative]] is in <math>A_{\infty}</math>.<ref>{{cite book | last=Garnett | first=John| coauthors=Marshall, Donald
  | title = Harmonic Measure
  | publisher = Cambridge University Measure | year = 2008
}}</ref> (Note that in this case, one needs to adapt the definition of weights to the case where the underlying measure is one-dimensional Hausdorff measure).
 
==References==
* {{cite book
  | last = Garnett | first = John | authorlink = John B. Garnett
  | title = Bounded Analytic Functions
  | publisher = Springer | year = 2007}}
 
<references/>
 
[[Category:Real analysis]]
[[Category:Harmonic analysis]]

Latest revision as of 15:23, 12 January 2015

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