247A, Notes 4: Weighted estimates and pseudodifferential operators

— 1. Weighted inequalities — In the previous notes we have taken a linear or sublinear operator on a measure space and established mapping properties, such as These estimates reveal information about how the height and width of is controlled by the height and width of . However, these estimates do not indicate where the support of is located, due to the rearrangement-invariant nature of the norms. One way to establish this type of control is to start establishing weighted estimates, such as for various pairs of weight functions and . Roughly speaking, these weighted estimates are a variant of the usual estimates which assert that if avoids the regions where is large, then avoids the regions where is large. If and one can establish the above type of estimate for many pairs , with “looking similar to” , then these estimates begin to confirm an assertion that does not significantly change the support of . One can view the estimate (1) as an assertion that is at least as large as some…

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