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 center at $x$ . 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: $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠1/p⁠ + ⁠1/q⁠ = 1$ .

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 (y)\,dy\leq C\omega (x),

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

Equivalent characterizations

This following result is a fundamental result in the study of Muckenhoupt weights.

Theorem. Let

1 < p < \infty

. 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

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}.

If

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

.

If

w \in A p

, then there is

δ > 1

such that

w δ \in A p

.

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 $L 2 (dx)$ , so we have

\forall f\in C_{c}^{\infty }:\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_{loc}^{1,2}(\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. 165: 207–226. doi:10.1090/S0002-9947-1972-0293384-6.
^ ^a ^b ^c ^d ^e 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. JSTOR 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 Press.

[1] Muckenhoupt, Benjamin (1972). "Weighted norm inequalities for the Hardy maximal function". Transactions of the American Mathematical Society. 165: 207–226. doi:10.1090/S0002-9947-1972-0293384-6.

[bible-2] 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. JSTOR 1971107.

[grafakos-4] Grafakos, 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 Press.

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