Muckenhoupt weights

In mathematics, the class of Muckenhoupt weights A_p consists of those weights ω for which the Hardy–Littlewood maximal operator is bounded on L^p(dω). Specifically, we consider functions  f  on Rⁿ and their associated maximal functions M( f ) defined as

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

where B_r(x) is the ball in Rⁿ with radius r and centre x. Let 1 ≤ p < ∞, we wish to characterise the functions ω : Rⁿ → [0, ∞) for which we have a bound

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

where C depends only on p and ω. This was first done by Benjamin Muckenhoupt.[1]

Definition

For a fixed 1 < p < ∞, we say that a weight ω : Rⁿ → [0, ∞) belongs to A_p if ω is locally integrable and there is a constant C such that, for all balls B in Rⁿ, we have

${\displaystyle \left({\frac {1}{|B|}}\int _{B}\omega (x)\,dx\right)\left({\frac {1}{|B|}}\int _{B}\omega (x)^{-{\frac {q}{p}}}\,dx\right)^{\frac {p}{q}}\leq C<\infty ,}$

where |B| is the Lebesgue measure of B, and q is a real number such that: 1/p + 1/q = 1.

We say ω : Rⁿ → [0, ∞) belongs to A₁ if there exists some C such that

${\displaystyle {\frac {1}{|B|}}\int _{B}\omega (y)\,dy\leq C\omega (x),}$

for all x ∈ B and all balls B.[2]

Equivalent characterizations

This following result is a fundamental result in the study of Muckenhoupt weights.

Theorem. A weight ω is in A_p if and only if any one of the following hold.[2]

(a) The Hardy–Littlewood maximal function is bounded on L^p(ω(x)dx), that is

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

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 Rⁿ, and all balls B:

${\displaystyle (f_{B})^{p}\leq {\frac {c}{\omega (B)}}\int _{B}f(x)^{p}\,\omega (x)\,dx,}$

where:

${\displaystyle f_{B}={\frac {1}{|B|}}\int _{B}f,\qquad \omega (B)=\int _{B}\omega (x)\,dx.}$

Equivalently:

Theorem. Let 1 < p < ∞, then w = e^φ ∈ A_p if and only if both of the following hold:

${\displaystyle \sup _{B}{\frac {1}{|B|}}\int _{B}e^{\varphi -\varphi _{B}}dx<\infty }$

${\displaystyle \sup _{B}{\frac {1}{|B|}}\int _{B}e^{-{\frac {\varphi -\varphi _{B}}{p-1}}}dx<\infty .}$

This equivalence can be verified by using Jensen's Inequality.

Reverse Hölder inequalities and A_∞

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

1. ω ∈ A_p for some 1 ≤ p < ∞.
2. There exist 0 < δ, γ < 1 such that for all balls B and subsets E ⊂ B, |E| ≤ γ |B| implies ω(E) ≤ δ ω(B).
3. There exist 1 < q and c (both depending on ω) such that for all balls B we have:

${\displaystyle {\frac {1}{|B|}}\int _{B}\omega ^{q}\leq \left({\frac {c}{|B|}}\int _{B}\omega \right)^{q}.}$

We call the inequality in the third formulation 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_∞.

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 ∈ A_p, (p ≥ 1), then log(w) ∈ BMO (i.e. log(w) has bounded mean oscillation).

(b) If  f  ∈ BMO, then for sufficiently small δ > 0, we have e^δf ∈ A_p for some p ≥ 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 δ > 0 in part (b) is necessary for the result to be true, as −log|x| ∈ BMO, but:

${\displaystyle e^{-\log |x|}={\frac {1}{e^{\log |x|}}}={\frac {1}{|x|}}}$

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:

${\displaystyle A_{1}\subseteq A_{p}\subseteq A_{\infty },\qquad 1\leq p\leq \infty .}$

${\displaystyle A_{\infty }=\bigcup _{p<\infty }A_{p}.}$

If w ∈ 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) ≤ Cw(B) where C > 1 is a constant depending on w.

If w ∈ A_p, then there is δ > 1 such that w^δ ∈ A_p.

If w ∈ A_∞, then there is δ > 0 and weights ${\displaystyle w_{1},w_{2}\in A_{1}}$ such that ${\displaystyle w=w_{1}w_{2}^{-\delta }}$.[3]

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.[4] 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

${\displaystyle \forall f\in C_{c}^{\infty }:\qquad \|T(f)\|_{L^{2}}\leq C\|f\|_{L^{2}}.}$

Suppose also that we can realise T as convolution against a kernel K in the following sense: if  f , g are smooth with disjoint support, then:

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

Finally we assume a size and smoothness condition on the kernel K:

${\displaystyle \forall x\neq 0,\forall |\alpha |\leq 1:\qquad \left|\partial ^{\alpha }K\right|\leq C|x|^{-n-\alpha }.}$

Then, for each 1 < p < ∞ and ω ∈ A_p, T is a bounded operator on L^p(ω(x)dx). That is, we have the estimate

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

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 u₀

${\displaystyle |K(x)|\geq a|x|^{-n}}$

whenever ${\displaystyle x=t{\dot {u}}_{0}}$ with −∞ < t < ∞, then we have a converse. If we know

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

for some fixed 1 < p < ∞ and some ω, then ω ∈ A_p.[2]

Weights and quasiconformal mappings

For K > 1, a K-quasiconformal mapping is a homeomorphism  f  : Rⁿ →Rⁿ such that

${\displaystyle f\in W_{loc}^{1,2}(\mathbf {R} ^{n}),\quad {\text{ and }}\quad {\frac {\|Df(x)\|^{n}}{J(f,x)}}\leq K,}$

where Df (x) is the derivative of  f  at x and J( f , x) = det(Df (x)) is the Jacobian.

A theorem of Gehring[5] states that for all K-quasiconformal functions  f  : Rⁿ →Rⁿ, we have J( f , x) ∈ A_p, where p depends on K.

Harmonic measure

If you have a simply connected domain Ω ⊆ C, we say its boundary curve Γ = ∂Ω is K-chord-arc if for any two points z, w in Γ there is a curve γ ⊆ Γ connecting z and w whose length is no more than K|z − w|. For a domain with such a boundary and for any z₀ in Ω, the harmonic measure w( ⋅ ) = w(z₀, Ω, ⋅) is absolutely continuous with respect to one-dimensional Hausdorff measure and its Radon–Nikodym derivative is in A_∞.[6] (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

• Garnett, John (2007). Bounded Analytic Functions. Springer.

1. Muckenhoupt, Benjamin (1972). "Weighted norm inequalities for the Hardy maximal function". Transactions of the American Mathematical Society, vol. 165: 207–226.
2. Stein, Elias (1993). "5". Harmonic Analysis. Princeton University Press.
3. Jones, Peter W. (1980). "Factorization of A_p weights". Ann. of Math. 2. 111 (3): 511–530. doi:10.2307/1971107.
4. Grafakos, Loukas (2004). "9". Classical and Modern Fourier Analysis. New Jersey: Pearson Education, Inc.
5. Gehring, F. W. (1973). "The L^p-integrability of the partial derivatives of a quasiconformal mapping". Acta Math. 130: 265–277. doi:10.1007/BF02392268.
6. Garnett, John; Marshall, Donald (2008). Harmonic Measure. Cambridge University Press.