Muckenhoupt weights

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In mathematics, the class of Muckenhoupt weights Ap are those weights ω for which the Hardy-Littlewood maximal operator is bounded on Lp(dω). Specifically, we consider functions f on \mathbb{R}^n and there associated maximal functions M(f) defined as

M(f)(x) = \sup_{r>0} \frac{1}{r^n} \int_{B_r} |f|,

where Br 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 ω. This was first done by Benjamin Muckenhoupt[1].

Contents

[edit] Definition

For a fixed 1 < p < \infty, we say that a weight \omega \colon \mathbb{R}^n \to [0,\infty) belongs to Ap if ω 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 A1 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.[2]

[edit] Equivalent characterisations

This following result is a fundamental result in the study of Muckenhoupt weights. A weight ω is in Ap if and only if any one of the following hold.[2]

(a) The Hardy-Littlewood maximal function is bounded on Lp(ω(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.

[edit] Reverse Hölder inequalities

The main tool in the proof of the above equivalence is the following result.[2] The following statements are equivalent

(a) ω belongs to Ap for some p \in [1,\infty)

(b) There exists an r > 1 and a c (both depending on ω such that

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

for all balls Br

(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 ω belongs to A_\infty.

[edit] Boundedness of singular integrals

It is not only the Hardy-Littlewood maximal operator that is bounded on these weighted Lp spaces. In fact, any Calderón-Zygmund singular integral operator is also bounded on these spaces.[3] Let us describe a simpler version of this here.[2] Suppose we have an operator T which is bounded on L2(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.

[edit] 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 u0

|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 ω, then \omega \in A_p.[2]

[edit] References

  1. ^ Munckenhoupt, Benjamin (1972). "Weighted norm inequalities for the Hardy maximal function": 207-26. 
  2. ^ a b c d e Stein, Elias (1993). "5", Harmonic Analysis. Princeton University Press. 
  3. ^ Grakakos, Loukas (2004). "9", Classical and Modern Fourier Analysis. Pearson Education, Inc..