Muckenhoupt weights

In mathematics, the class of Muckenhoupt weights A_p consists of those weights ω for which the Hardy–Littlewood maximal operator is bounded on L^p(dω). 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 ω. 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 ω 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 A₁ 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 ω 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(ω(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) ω belongs to A_p for some $p \in [1,\infty)$

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

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

for all balls B_r

(c) There exists $\delta, \gamma \in (0,1)$ so that for all balls B and subsets $E \subset B$

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

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 ω 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 δ > 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 δ > 0 condition 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 δ > 0 such that $w^\delta \in A_p$.

(v) If $w\in A_{\infty}$ then there is δ > 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 L²(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 u₀

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

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 ω, 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 $f\in W^{1,2}_{loc}(\mathbb{R}^{n})$ and

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

where Df(x) is the derivative of f at x and J(f,x) = 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. 

1. ^ Muckenhoupt, Benjamin (1972). "Weighted norm inequalities for the Hardy maximal function". Transactions of the American Mathematical Society, vol. 165: 207–226. 
2. ^ ^{a b c d e} Stein, Elias (1993). "5". Harmonic Analysis. Princeton University Press. 
3. ^ Jones, Peter W. (1980). "Factorization of A_p weights". Ann. Of Math. (2) 111 (3): 511–530. doi:10.2307/1971107. 
4. ^ Grakakos, Loukas (2004). "9". Classical and Modern Fourier Analysis. New Jersey: Pearson Education, Inc.. 
5. ^ 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. 
6. ^ Garnett, John; Marshall, Donald (2008). Harmonic Measure. Cambridge University Measure.