User:Potahto/Muckenhoupt weights

The class of Muckenhoupt weights $A_{p}$ are 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 there associated maximal function $M(f)$ 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

${\frac {1}{|B|}}\int _{B}\omega (x)\,dx[{\frac {1}{|B|}}\int _{B}\omega (x)^{\frac {-p}{p'}}\,dx]^{\frac {p}{p'}}\leq A<\infty ,$

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

${\frac {1}{|B|}}\int _{B}\omega (x)\,dx\leq A\omega (x),$

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

Equivalent characterisations

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

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

Reverse Hölder inequalities

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 $r>1$ and a $c$ (both depending on $\omega$ such that

${\frac {1}{|B|}}\int _{B_{r}}\omega ^{r}\leq ({\frac {c}{|B|}}\int _{B_{r}}\omega )^{r}$

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 $A_{\infty }$.

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

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

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 $\omega \in A_{p}$.