Muckenhoupt weights

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In mathematics, the class of Muckenhoupt weights $A_{p}$ consists of those weights $\omega$ for which the Hardy–Littlewood maximal operator is bounded on $L^{p}(d\omega )$ . Specifically, we consider functions $f$ on ${\mathbb {R}}^{n}$ and their associated maximal functions $M(f)$ defined as

$M(f)(x)=\sup _{{r>0}}{\frac {1}{r^{n}}}\int _{{B_{r}}}|f|,$

where $B_{r}$ is a ball in ${\mathbb {R}}^{n}$ with radius $r$ and centre $x$ . We wish to characterise the functions $\omega \colon {\mathbb {R}}^{n}\to [0,\infty )$ for which we have a bound

$\int |M(f)(x)|^{p}\,\omega (x)dx\leq C\int |f|^{p}\,\omega (x)\,dx,$

where $C$ depends only on $p\in [1,\infty )$ and $\omega$ . This was first done by Benjamin Muckenhoupt.^[1]

Definition

For a fixed $1<p<\infty$ , we say that a weight $\omega \colon {\mathbb {R}}^{n}\to [0,\infty )$ belongs to $A_{p}$ if $\omega$ is locally integrable and there is a constant $C$ such that, for all balls $B$ in ${\mathbb {R}}^{n}$ , we have

$\left({\frac {1}{|B|}}\int _{B}\omega (x)\,dx\right)\left({\frac {1}{|B|}}\int _{B}\omega (x)^{{\frac {-p'}{p}}}\,dx\right)^{{\frac {p}{p'}}}\leq C<\infty ,$

where $1/p+1/p'=1$ and $|B|$ is the Lebesgue measure of $B$ . We say $\omega \colon {\mathbb {R}}^{n}\to [0,\infty )$ belongs to $A_{1}$ if there exists some $C$ such that

${\frac {1}{|B|}}\int _{B}\omega (x)\,dx\leq C\omega (x),$

for all $x\in B$ and all balls $B$ .^[2]

Equivalent characterizations

This following result is a fundamental result in the study of Muckenhoupt weights. A weight $\omega$ is in $A_{p}$ if and only if any one of the following hold.^[2]

(a) The Hardy–Littlewood maximal function is bounded on $L^{p}(\omega (x)dx)$ , that is

$\int |M(f)(x)|^{p}\,\omega (x)\,dx\leq C\int |f|^{p}\,\omega (x)\,dx,$

for some $C$ which only depends on $p$ and the constant $A$ in the above definition.

(b) There is a constant $c$ such that for any locally integrable function $f$ on ${\mathbb {R}}^{n}$

$(f_{B})^{p}\leq {\frac {c}{\omega (B)}}\int _{B}f(x)^{p}\,\omega (x)\,dx$

for all balls $B$ . Here

$f_{B}={\frac {1}{|B|}}\int _{B}f$

is the average of $f$ over $B$ and

$\omega (B)=\int _{B}\omega (x)\,dx.$

Equivalently, $w=e^{\phi }\in A_{{p}}$ , where $p\in (1,\infty )$ , if and only if

$\sup _{{B}}{\frac {1}{|B|}}\int _{{B}}e^{{\phi -\phi _{{B}}}}dx<\infty$

and

$\sup _{{B}}{\frac {1}{|B|}}\int _{{B}}e^{{-{\frac {\phi -\phi _{{B}}}{p-1}}}}dx<\infty .$

This equivalence can be verified by using Jensen's Inequality.

Reverse Hölder inequalities and $A_{{\infty }}$

The main tool in the proof of the above equivalence is the following result.^[2] The following statements are equivalent

(a) $\omega$ belongs to $A_{p}$ for some $p\in [1,\infty )$

(b) There exists an $q>1$ and a $c$ (both depending on $\omega$ ) such that

${\frac {1}{|B|}}\int _{{B}}\omega ^{q}\leq \left({\frac {c}{|B|}}\int _{{B}}\omega \right)^{q}$

for all balls $B_{r}$

$|E|\leq \gamma |B|\implies \omega (E)\leq \delta \omega (B)$

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 $\omega$ belongs to $A_{\infty }$ .

Weights and BMO

The definition of an $A_{p}$ 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 $w\in A_{{p}},\;\;p\geq 1,$ , then $\log w\in BMO$ (i.e. $\log w$ has bounded mean oscillation).

(b) If $f\in BMO$ , then for sufficiently small $\delta >0$ , we have $e^{{\delta f}}\in A_{{p}}$ for some $p\geq 1$ .

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 $\delta >0$ in part (b) is necessary for the result to be true, as $\log {\frac {1}{|x|}}$ is a BMO function, but $e^{{\log {\frac {1}{|x|}}}}={\frac {1}{|x|}}$ is not in any $A_{{p}}$ .

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) $A_{1}\subseteq A_{p}\subseteq A_{\infty }{\text{ for }}1\leq p\leq \infty .$

(ii) $A_{\infty }=\bigcup _{{p<\infty }}A_{p}.$

(iii) If $w\in A_{p}$ , then $w\,dx$ defines a doubling measure: for any ball B, if 2B is the ball of twice the radius, then $w(2B)\leq Cw(B)$ where C > 1 is a constant depending on $w$ .

(iv) If $w\in A_{p}$ , then there is $\delta >1$ such that $w^{\delta }\in A_{p}$ .

(v) If $w\in A_{{\infty }}$ then there is $\delta >0$ and weights $w_{1},w_{2}\in A_{1}$ such that $w=w_{1}w_{2}^{{-\delta }}$ .^[3]

Boundedness of singular integrals

It is not only the Hardy–Littlewood maximal operator that is bounded on these weighted $L^{p}$ spaces. In fact, any Calderón-Zygmund singular integral operator is also bounded on these spaces.^[4] Let us describe a simpler version of this here.^[2] Suppose we have an operator $T$ which is bounded on $L^{2}(dx)$ , so we have

$\|T(f)\|_{{L^{2}}}\leq C\|f\|_{{L^{2}}},$

for all smooth and compactly supported $f$ . Suppose also that we can realise $T$ as convolution against a kernel $K$ in the sense that, whenever $f$ and $g$ are smooth and have disjoint support

$\int g(x)T(f)(x)\,dx=\iint g(x)K(x-y)f(y)\,dy\,dx.$

Finally we assume a size and smoothness condition on the kernel $K$ :

$|{\partial ^{\alpha }}K|\leq C|x|^{{-n-\alpha }}$

for all $x\neq 0$ and multi-indices $|\alpha |\leq 1$ . Then, for each $p\in (1,\infty )$ and $\omega \in A_{p}$ , we have that $T$ is a bounded operator on $L^{p}(\omega (x)\,dx)$ . That is, we have the estimate

$\int |T(f)(x)|^{p}\,\omega (x)\,dx\leq C\int |f(x)|^{p}\,\omega (x)\,dx,$

for all $f$ 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 $K$ : For a fixed unit vector $u_{0}$

$|K(x)|\geq a|x|^{{-n}}$

whenever $x=t{\dot u}_{0}$ with $-\infty <t<\infty$ , then we have a converse. If we know

$\int |T(f)(x)|^{p}\,\omega (x)\,dx\leq C\int |f(x)|^{p}\,\omega (x)\,dx,$

for some fixed $p\in (1,\infty )$ and some $\omega$ , then $\omega \in A_{p}$ .^[2]

Weights and quasiconformal mappings

For $K>1$ , a K-quasiconformal mapping is a homeomorphism $f:{\mathbb {R}}^{{n}}\rightarrow {\mathbb {R}}^{{n}}$ with $f\in W_{{loc}}^{{1,2}}({\mathbb {R}}^{{n}})$ and

${\frac {||Df(x)||^{{n}}}{J(f,x)}}\leq K$

where $Df(x)$ is the derivative of $f$ at $x$ and $J(f,x)={\mbox{det}}(Df(x))$ is the Jacobian.

A theorem of Gehring^[5] states that for all K-quasiconformal functions $f:{\mathbb {R}}^{{n}}\rightarrow {\mathbb {R}}^{{n}}$ , we have $J(f,x)\in A_{{p}}$ where $p$ depends on $K$ .

Harmonic measure

If you have a simply connected domain $\Omega \subseteq {\mathbb {C}}$ , we say its boundary curve $\Gamma =\partial \Omega$ is K-chord-arc if for any two points $z,w\in \Gamma$ there is a curve $\gamma \subseteq \Gamma$ connecting $z$ and $w$ whose length is no more than $K|z-w|$ . For a domain with such a boundary and for any $z_{{0}}\in \Omega$ , the harmonic measure $w(\cdot )=w(z_{{0}},\Omega ,\cdot )$ is absolutely continuous with respect to one-dimensional Hausdorff measure and its Radon–Nikodym derivative is in $A_{{\infty }}$ .^[6] (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

Garnett, John (2007). Bounded Analytic Functions. Springer.

↑ Muckenhoupt, Benjamin (1972). "Weighted norm inequalities for the Hardy maximal function". Transactions of the American Mathematical Society, vol. 165: 207–226.
↑ 2.0 2.1 2.2 2.3 2.4 Stein, Elias (1993). "5". Harmonic Analysis. Princeton University Press.
↑ Jones, Peter W. (1980). "Factorization of A_p weights". Ann. Of Math. (2) 111 (3): 511–530. doi:10.2307/1971107.
↑ Grakakos, Loukas (2004). "9". Classical and Modern Fourier Analysis. New Jersey: Pearson Education, Inc.
↑ Gehring, F. W. (1973). "The L^p-integrability of the partial derivatives of a quasiconformal mapping". Acta Math. 130: 265–277. doi:10.1007/BF02392268.
↑ Garnett, John; Marshall, Donald (2008). Harmonic Measure. Cambridge University Measure. Cite uses deprecated parameters (help)

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