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

Abstract

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 o the Calderon-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. (C) 2009 WILEYNCH Verlag GrnbH & Co. KGaA, Weinheim