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 $$\omega \in A_p$$.