Muckenhoupt weights

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

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 , 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

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

where $1/p\; +\; 1/p\text{'}\; =\; 1$ 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

$frac\{1\}$ int_B omega(x) , dx leq Aomega(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

$f\_B\; =\; frac\{1\}$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

$frac\{1\}$ int_{B_r} omega^r leq left(frac{c}{|B> int_{B_r} omega right)^r

for all balls $B\_r$

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

$|E|\; leq\; gamma|B|\; implies\; omega(E)\; leq\; deltaomega(B)$

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

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

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

$int\; |T(f)(x)|^p\; ,\; omega(x),dx\; leq\; C\; int\; |f(x)|^p\; ,\; omega(x),\; dx,$

for some fixed $p\; in\; (1,infty)$ and some $omega$, then $omega\; in\; A\_p$.

References

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Last updated on Saturday July 19, 2008 at 15:25:41 PDT (GMT -0700)
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