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

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

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.

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 $B$.

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

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

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. 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 $B_{r}$

(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 $\omega$ 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 $w=w_{1}w_{2}^{-\delta }$.

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

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

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 $\omega \in A_{p}$.

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 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 $A_{\infty }$. (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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