Muckenhoupt weights

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

:$\backslash left(\backslash frac\{1\}$

where {{math|{{!}}B{{!}}}} is the Lebesgue measure of {{mvar|B}}, and {{mvar|q}} is a real number such that: {{math|{{sfrac|1|p}} + {{sfrac|1|q}} {{=}} 1}}.

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

: $\backslash frac\{1\}$

for almost every {{math|x ∈ B}} and all balls {{mvar|B}}.{{cite book|last = Stein | first = Elias | title = Harmonic Analysis | chapter = 5 | publisher = Princeton University Press| year = 1993 }}

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

:Theorem. Let {{math|1 < p < ∞}}. A weight {{mvar|ω}} is in {{math|A_p}} if and only if any one of the following hold.

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

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

::for some {{mvar|C}} which only depends on {{mvar|p}} and the constant {{mvar|A}} in the above definition.

::(b) There is a constant {{mvar|c}} such that for any locally integrable function {{math| f }} on {{math|Rⁿ}}, and all balls {{mvar|B}}:

:::$(f\_B)^p\; \backslash leq\; \backslash frac\{c\}\{\backslash omega(B)\}\; \backslash int\_B\; f(x)^p\; \backslash ,\; \backslash omega(x)\backslash ,dx,$

::where:

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

Equivalently:

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

::$\backslash sup\_\{B\}\backslash frac\{1\}$

::$\backslash sup\_\{B\}\backslash frac\{1\}$

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

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

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

:: $\backslash frac\{1\}$

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 {{mvar|ω}} belongs to {{math|A_∞}}.

The definition of an {{math|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 {{math|w ∈ A_p, (p ≥ 1),}} then {{math|log(w) ∈ BMO}} (i.e. {{math|log(w)}} has bounded mean oscillation).

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

:$e^\{-\backslash log|x|\}=\backslash frac\{1\}\{e^\{\backslash log|x|\}\}\; =\; \backslash frac\{1\}$

is not in any {{math|A_p}}.

Here we list a few miscellaneous properties about weights, some of which can be verified from using the definitions, others are nontrivial results:

:$A\_1\; \backslash subseteq\; A\_p\; \backslash subseteq\; A\_\backslash infty,\; \backslash qquad\; 1\backslash leq\; p\backslash leq\backslash infty.$

:

:If {{math|w ∈ A_p}}, then {{math|w dx}} defines a doubling measure: for any ball {{mvar|B}}, if {{math|2B}} is the ball of twice the radius, then {{math|w(2B) ≤ Cw(B)}} where {{math|C > 1}} is a constant depending on {{mvar|w}}.

:If {{math|w ∈ A_p}}, then there is {{math|δ > 1}} such that {{math|w^δ ∈ A_p}}.

:If {{math|w ∈ A_∞}}, then there is {{math|δ > 0}} and weights $w\_1,w\_2\backslash in\; A\_1$ such that $w=w\_1\; w\_2^\{-\backslash delta\}$.{{cite journal| last = Jones | first = Peter W.|title = Factorization of {{math|A_p}} weights| journal = Ann. of Math. |series = 2 | volume = 111 | pages = 511–530 | year = 1980| doi = 10.2307/1971107 | issue = 3| jstor = 1971107}}

It is not only the Hardy–Littlewood maximal operator that is bounded on these weighted {{math|L^p}} spaces. In fact, any Calderón-Zygmund singular integral operator is also bounded on these spaces.{{cite book | last = Grafakos | first = Loukas | title = Classical and Modern Fourier Analysis | chapter = 9 | publisher = Pearson Education, Inc. | place = New Jersey| year = 2004 }} Let us describe a simpler version of this here. Suppose we have an operator {{mvar|T}} which is bounded on {{math|L²(dx)}}, so we have

: $\backslash forall\; f\; \backslash in\; C^\{\backslash infty\}\_c\; :\; \backslash qquad\; \backslash |T(f)\backslash |\_\{L^2\}\; \backslash leq\; C\backslash |f\backslash |\_\{L^2\}.$

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

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

Finally we assume a size and smoothness condition on the kernel {{mvar|K}}:

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

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

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

for all {{math| f }} for which the right-hand side is finite.

If, in addition to the three conditions above, we assume the non-degeneracy condition on the kernel {{mvar|K}}: For a fixed unit vector {{math|u₀}}

: $|K(x)|\; \backslash geq\; a\; |x|^\{-n\}$

whenever $x\; =\; t\; \backslash dot\; u\_0$ with {{math|−∞ < t < ∞}}, then we have a converse. If we know

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

for some fixed {{math|1 < p < ∞}} and some {{mvar|ω}}, then {{math|ω ∈ A_p}}.

For {{math|K > 1}}, a {{mvar|K}}-quasiconformal mapping is a homeomorphism {{math| f  : Rⁿ →Rⁿ}} such that

:$f\backslash in\; W^\{1,2\}\_\{loc\}(\backslash mathbf\{R\}^n),\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; \backslash frac\{\backslash |Df(x)\backslash |^n\}\{J(f,x)\}\backslash leq\; K,$

where {{math|Df (x)}} is the derivative of {{math| f }} at {{mvar|x}} and {{math|J( f , x) {{=}} det(Df (x))}} is the Jacobian.

A theorem of Gehring{{cite journal|last = Gehring | first = F. W.| title = The L^p-integrability of the partial derivatives of a quasiconformal mapping| journal = Acta Math. | volume = 130 | pages = 265–277 | year = 1973| doi = 10.1007/BF02392268| doi-access = free}} states that for all {{mvar|K}}-quasiconformal functions {{math| f  : Rⁿ →Rⁿ}}, we have {{math|J( f , x) ∈ A_p}}, where {{mvar|p}} depends on {{mvar|K}}.

If you have a simply connected domain {{math|Ω ⊆ C}}, we say its boundary curve {{math|Γ {{=}} ∂Ω}} is {{mvar|K}}-chord-arc if for any two points {{math|z, w}} in {{math|Γ}} there is a curve {{math|γ ⊆ Γ}} connecting {{mvar|z}} and {{mvar|w}} whose length is no more than {{math|K{{!}}z − w{{!}}}}. For a domain with such a boundary and for any {{math|z₀}} in {{math|Ω}}, the harmonic measure {{math|w( ⋅ ) {{=}} w(z₀, Ω, ⋅)}} is absolutely continuous with respect to one-dimensional Hausdorff measure and its Radon–Nikodym derivative is in {{math|A_∞}}.{{cite book| last=Garnett | first=John|author2=Marshall, Donald | title = Harmonic Measure| publisher = Cambridge University Press | year = 2008}} (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).

• {{cite book|last = Garnett|first = John | author-link = John B. Garnett| title = Bounded Analytic Functions| publisher = Springer| year = 2007}}

Category:Real analysis

Category:Harmonic analysis

Category:Lp spaces