Discrete Fourier transform (general)

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

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

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

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

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

Definition

For a fixed $1<p<\mathrm{\infty }$, we say that a weight $\omega :{\mathbb{ℝ}}^n\to [0,\mathrm{\infty })$ 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

$(\frac{1}{\left|B\right|}\underset{B}{\int }\omega (x)dx){(\frac{1}{\left|B\right|}\underset{B}{\int }\omega (x{)}^{\frac{-p^{\prime }}{p}}dx)}^{\frac{p}{p^{\prime }}}\le C<\mathrm{\infty },$

where $1/p+1/p^{\prime }=1$ and $\left|B\right|$ is the Lebesgue measure of $B$. We say $\omega :{\mathbb{ℝ}}^n\to [0,\mathrm{\infty })$ belongs to $A_1$ if there exists some $C$ such that

$\frac{1}{\left|B\right|}\underset{B}{\int }\omega (x)dx\le C\omega (x),$

for all $x\in B$ and all balls $B$.

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.

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

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

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 ${\mathbb{ℝ}}^n$

$(f_B{)}^p\le \frac{c}{\omega (B)}\underset{B}{\int }f(x{)}^p\omega (x)dx$

for all balls $B$. Here

$f_B=\frac{1}{\left|B\right|}\underset{B}{\int }f$

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

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

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

${\sup }_B\frac{1}{\left|B\right|}\underset{B}{\int }{e}^{\varphi -{\varphi }_B}dx<\mathrm{\infty }$

and

${\sup }_B\frac{1}{\left|B\right|}\underset{B}{\int }{e}^{-\frac{\varphi -{\varphi }_B}{p-1}}dx<\mathrm{\infty }.$

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

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

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

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

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

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

for all balls $B_r$

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

$\left|E\right|\le \gamma \left|B\right|⟹\omega (E)\le \delta \omega (B)$

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 $A_{\mathrm{\infty }}$.

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 $w\in A_p,p\ge 1,$, 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 $p\ge 1$.

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 $\log \frac{1}{\left|x\right|}$ is a BMO function, but $e^{\log \frac{1}{\left|x\right|}}=\frac{1}{\left|x\right|}$ 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) $A_1\subseteq A_p\subseteq A_{\mathrm{\infty }}\text{ for }1\le p\le \mathrm{\infty }.$

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

(iii) If $w\in A_p$, then $wdx$ defines a doubling measure: for any ball B, if 2B is the ball of twice the radius, then $w(2B)\le Cw(B)$ 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 $w\in A_{\mathrm{\infty }}$ then there is $\delta >0$ and weights $w_1,w_2\in A_1$ such that $w=w_1{w}_2^{-\delta }$.

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. Let us describe a simpler version of this here. Suppose we have an operator $T$ which is bounded on $L^2(dx)$, so we have

$\Vert T(f){\Vert }_{L^2}\le C\Vert f{\Vert }_{L^2},$

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

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

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

$\left|{\partial }^{\alpha }K\right|\le C|x{|}^{-n-\alpha }$

for all $x\ne 0$ and multi-indices $\left|\alpha \right|\le 1$. Then, for each $p\in (1,\mathrm{\infty })$ 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

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

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 $u_0$

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

whenever $x=t{\dot{u}}_0$ with $-\mathrm{\infty }<t<\mathrm{\infty }$, then we have a converse. If we know

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

for some fixed $p\in (1,\mathrm{\infty })$ and some $\omega$, then $\omega \in A_p$.

Weights and quasiconformal mappings

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

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

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

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

Harmonic measure

If you have a simply connected domain $\mathrm{\Omega }\subseteq \mathbb{ℂ}$, we say its boundary curve $\mathrm{\Gamma }=\partial \mathrm{\Omega }$ is K-chord-arc if for any two points $z,w\in \mathrm{\Gamma }$ there is a curve $\gamma \subseteq \mathrm{\Gamma }$ connecting $z$ and $w$ whose length is no more than $K|z-w|$. For a domain with such a boundary and for any $z_0\in \mathrm{\Omega }$, the harmonic measure $w(\cdot )=w(z_0,\mathrm{\Omega },\cdot )$ is absolutely continuous with respect to one-dimensional Hausdorff measure and its Radon–Nikodym derivative is in $A_{\mathrm{\infty }}$. (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

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