Carleson measures and the BMO space on the p ‐adic vector space

For a prime number p , let Q p be the p ‐adic field and let Q p d denote a vector space over Q p which consists of all d ‐tuples of Q p . Then we study the p ‐adic version of the Calderón–Zygmund decomposition, Carleson measures on the vector space Q p d +1 and the space BMO ( Q p d ) of functions of bounded mean oscillation on Q p d . In particular, it turns out that the operator norms of various oncoming operators are independent of the dimension d and the prime number p , which is one of the big differences from that of the Euclidean case. Interestingly, the independence of the dimension d and p makes it possible to develop Harmonic Analysis on the infinite dimensional p ‐adic vector space as the importance had already been pointed out in the Euclidean case (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)