[1] For a fixed 1 < p < ∞, we say that a weight ω : Rn → [0, ∞) belongs to Ap if ω is locally integrable and there is a constant C such that, for all balls B in Rn, we have where |B| is the Lebesgue measure of B, and q is a real number such that: 1/p + 1/q = 1.
This property can be phrased equivalently in terms of how much the logarithm of the weight oscillates: 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: is not in any Ap.
Here we list a few miscellaneous properties about weights, some of which can be verified from using the definitions, others are nontrivial results: 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.
[2] Suppose we have an operator T which is bounded on L2(dx), so we have 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: Finally we assume a size and smoothness condition on the kernel K: Then, for each 1 < p < ∞ and ω ∈ Ap, T is a bounded operator on Lp(ω(x)dx).