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 $R n$ and their associated maximal functions $M (f)$ defined as

M(f)(x) = \sup_{r>0} \frac{1}{r^n} \int_{B_r(x)} |f|,

where $B r (x)$ is the ball in $R n$ with radius $r$ and centre $x$ . Let $1 \leq p < \infty$ , we wish to characterise the functions $ω : R n \to [0, \infty)$ 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 $p$ and $ω$ . This was first done by Benjamin Muckenhoupt.^[1]

Definition

For a fixed $1 < p < \infty$ , we say that a weight $ω : R n \to [0, \infty)$ belongs to $A p$ if $ω$ is locally integrable and there is a constant $C$ such that, for all balls $B$ in $R n$ , we have

\left(\frac{1}{|B|} \int_B \omega(x) \, dx \right)\left(\frac{1}{|B|} \int_B \omega(x)^{-\frac{q}{p}} \, dx \right)^\frac{p}{q} \leq C < \infty,

where $| B |$ is the Lebesgue measure of $B$ , and $q$ is a real number such that: $1 / p + 1 / q = 1$ .

We say $ω : R n \to [0, \infty)$ 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.

Theorem. 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

R n

, and all balls

B

(f_B)^p \leq \frac{c}{\omega(B)} \int_B f(x)^p \, \omega(x)\,dx,

where:

f_B = \frac{1}{|B|}\int_B f, \qquad \omega(B) = \int_B \omega(x)\,dx.

Equivalently:

Theorem. Let

1 < p < \infty

, then

w = e φ \in A p

if and only if both of the following hold:

\sup_{B}\frac{1}{|B|}\int_{B}e^{\varphi-\varphi_B}dx<\infty

\sup_{B}\frac{1}{|B|}\int_{B}e^{-\frac{\varphi-\varphi_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

$ω \in A p$ for some $1 \leq p < \infty$ .
There exist $0 < δ, γ < 1$ such that for all balls $B$ and subsets $E \subset B$ , $| E | \leq γ | B |$ implies $ω (E) \leq δ ω (B)$ .
There exist $1 < q$ and $c$ (both depending on $ω$ ) such that for all balls $B$ we have:

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

We call the inequality in the third formulation 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 δf \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 $δ > 0$ in part (b) is necessary for the result to be true, as $-log| x | \in BMO$ , but:

e^{-\log|x|}=\frac{1}{e^{\log|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:

A_1 \subseteq A_p \subseteq A_\infty, \qquad 1\leq p\leq\infty.

A_\infty = \bigcup_{p<\infty}A_p.

w \in A p

, then

w dx

defines a doubling measure: for any ball

B

, if

2 B

is the ball of twice the radius, then

w (2 B) \leq Cw (B)

where

C > 1

is a constant depending on

w

w \in A p

, then there is

δ > 1

such that

w δ \in A p

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 $L 2 (dx)$ , so we have

\forall f \in C^{\infty}_c : \qquad \|T(f)\|_{L^2} \leq C\|f\|_{L^2}.

Suppose also that we can realise $T$ as convolution against a kernel $K$ in the following sense: if $f, g$ are smooth with disjoint support, then:

\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 $K$ :

\forall x \neq 0, \forall |\alpha| \leq 1 : \qquad \left |\partial^{\alpha} K \right | \leq C |x|^{-n-\alpha}.

Then, for each $1 < p < \infty$ and $ω \in A p$ , $T$ is a bounded operator on $L p (ω (x) dx)$ . 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 $1 < p < \infty$ and some $ω$ , then $ω \in A p$ .^[2]

Weights and quasiconformal mappings

For $K > 1$ , a $K$ -quasiconformal mapping is a homeomorphism $f : R n \to R n$ such that

f\in W^{1,2}_{loc}(\mathbf{R}^n), \quad \text{ and } \quad \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 $f : R n \to R n$ , we have $J (f, x) \in A p$ , where $p$ depends on $K$ .

Harmonic measure

If you have a simply connected domain $Ω \subseteq C$ , we say its boundary curve $Γ = \partialΩ$ is $K$ -chord-arc if for any two points $z, w$ in $Γ$ there is a curve $γ \subseteq Γ$ 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 $Ω$ , the harmonic measure $w ( \cdot ) = w (z 0, Ω, \cdot)$ 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.
1 2 3 4 5 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.
↑ Grafakos, 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.

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