Dyadic analysis meets number theory

Abstract

We unite two themes in dyadic analysis and number theory by studying an analogue of the failure of the Hasse principle in harmonic analysis. Explicitly, we construct an explicit family of measures on the real line that are p-adic doubling for any finite set of primes, yet not doubling, and we apply these results to show analogous statements about the reverse Hölder and Muckenhoupt Ap classes of weights. The proofs involve a delicate interplay among several geometric and number theoretic properties. µ(Jp) µ(Jp−1) 27 6.1. Computation of µ(Jp) µ(Jp−1) if K does not exist. 28 6.2. Computation of µ(Jp) µ(Jp−1) if K exists. 28 7. The analysis part IV: Other cases and the proof of Theorem 1.2 32 7.1. Z in on the left hand side of J. 32 7.2. Z ∈ J. 33 7.3. Proof of Theorem 1.2. 33 8. Applications 34 References 45