Muckenhoupt weights

::(b) There is a constant c such that for any locally integrable function  *f*  on **R***n*, and all balls B: :::(f_B)^p \leq \frac{c}{\omega(B)} \int_B f(x)^p \, \omega(x)\,dx, ::where: :::f_B = \frac{1}{|B|}\int_B f, \qquad \omega(B) = \int_B \omega(x)\,dx. Equivalently: :**Theorem.** Let 1 φ* ∈ *Ap* if and only if both of the following hold: :: \sup_{B}\frac{1}{|B|}\int_{B}e^{\varphi-\varphi_B}dx :: \sup_{B}\frac{1}{|B|}\int_{B}e^{-\frac{\varphi-\varphi_B}{p-1}}dx This equivalence can be verified by using Jensen's Inequality. ## Reverse Hölder inequalities and {{math|''A''_∞}} The main tool in the proof of the above equivalence is the following result. The following statements are equivalent 1. *ω* ∈ *Ap* for some {{math|1 ≤ *p* 1. There exist {{math|0 1. There exist {{math|1 :: \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 *Ap* 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* ∈ *Ap*, (*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 *eδf* ∈ *Ap* 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: :e^{-\log|x|}=\frac{1}{e^{\log|x|}} = \frac{1}{|x|} is not in any *Ap*. ## 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: :A_1 \subseteq A_p \subseteq A_\infty, \qquad 1\leq p\leq\infty. :A_\infty = \bigcup_{p :If *w* ∈ *Ap*, then *w* *dx* defines a doubling measure: for any ball B, if 2*B* is the ball of twice the radius, then *w*(2*B*) ≤ *Cw*(*B*) where *C* 1 is a constant depending on w. :If *w* ∈ *Ap*, then there is *δ* 1 such that *wδ* ∈ *Ap*. :If *w* ∈ *A*∞, then there is *δ* 0 and weights w_1,w_2\in A_1 such that w=w_1 w_2^{-\delta}. ## 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. 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 : \forall f \in C^{\infty}_c : \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: : \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: : \forall x \neq 0, \forall |\alpha| \leq 1 : \qquad \left |\partial^{\alpha} K \right | \leq C |x|^{-n-\alpha}. Then, for each 1 p*, T is a bounded operator on *Lp*(*ω*(*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 {{math|−∞ : \int |T(f)(x)|^p \, \omega(x)\,dx \leq C \int |f(x)|^p \, \omega(x)\, dx, for some fixed 1 p*. ## Weights and quasiconformal mappings For *K* 1, a K-quasiconformal mapping is a homeomorphism  *f*  : **R***n* →**R***n* such that :f\in W^{1,2}_{loc}(\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 is the Jacobian. A theorem of Gehring states that for all K-quasiconformal functions  *f*  : **R***n* →**R***n*, we have *J*( *f* , *x*) ∈ *Ap*, where p depends on K. ## Harmonic measure If you have a simply connected domain Ω ⊆ **C**, we say its boundary curve 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 *z*0 in Ω, the harmonic measure is absolutely continuous with respect to one-dimensional Hausdorff measure and its Radon–Nikodym derivative is in *A*∞. (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 - ## References 1. Muckenhoupt, Benjamin. (1972). "Weighted norm inequalities for the Hardy maximal function". *Transactions of the American Mathematical Society*. 2. Stein, Elias. (1993). "Harmonic Analysis". *Princeton University Press*. 3. Jones, Peter W.. (1980). "Factorization of {{math". *Ann. of Math.*. 4. Grafakos, Loukas. (2004). "Classical and Modern Fourier Analysis". *Pearson Education, Inc.*. 5. Gehring, F. W.. (1973). "The L^p-integrability of the partial derivatives of a quasiconformal mapping". *Acta Math.*. 6. Garnett, John. (2008). "Harmonic Measure". *Cambridge University Press*. ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Muckenhoupt_weights) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Muckenhoupt_weights?action=history). ::