Preliminaries Here we review some notation and basic results, but we assume that they are mostly well known to the reader. For more information, see, for example, Rudin [14]. In general we will work in Rn. The Euclidean norm will be denoted by |-|. If xeRn and r 0, B(x, r) = {y Rn : \x - y\ r} is the ball with center x and radius r. Lebesgue measure in Rn is denoted by dx and on the unit sphere 5 n _ 1 in Rn by da. If E is a subset of Rn, IE"! denotes its Lebesgue measure and XE its characteristic function: XE(%) 1 if x £ E and 0 if x £ E. The expressions almost everywhere or for almost every x refer to properties which hold except on a set of measure 0 they are abbreviated by "a.e." and "a.e. x." If a (ai,... , an) Nn is a multi-index and / : Rn C, then Daf = —— J dx\l - dxann ' where \a\ = a\ + \- an and xa = x^1 x^n. Let (X,/i) be a measure space. LP(X,/i), 1 p oo, denotes the Banach space of functions from X to C whose p-th powers are integrable the norm of / G LP(X, ji) is n/iip=(/ x i/r^ x/p XV11
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