M~,..atsheffefiir Mh. Math. 96, 133--141 (1983) Malhemalik 9 by Springer-Verlag 1983 Equivalent Norms on L p Spaces of Harmonic Functions By Daniel H. Luecking*, Fayettevitle, Arkansas (Received 8 November 1982; in revisedform 22 April 1983) Abstract. This note gives necessary and sufficient conditions for a measurable set G in the unit ball B in R n to satisfy the following property: there exists a constant C > 0 such that I lj]2 dm <~ C I lj]2 dm (*) B G for every fe L 2(B, din) which is harmonic in B. Here m is the Lebesgue measure of dimension n. The same condition is sufficient if any exponent p > 0 replaces 2 in (*), and if certain weighted measures replace m. Applications to the problem of representing harmonic functions in L 2(B) as a sum of kernel functions are indicated. Introduction. Let U denote the unit disk in C and let dm be Lebesgue area measure on U. In [3] the following theorem was proved Theorem A. If G is a measurable subset of U and p > 0 then the following are equivalent: (1) There is a constant C > 0 such that [17 p am ~ C ~ ~ p dm u G for every analytic function in L p (m). (2) There is a constant d > 0 such that m (G n D) >~ d m (U c~ D)for every disk D whose center lies on OU. Condition (2) can be viewed as a "reversed Carleson condition" on the measure Zc din. Here we extend Theorem A by changing the word "analytic" to "harmonic" and the disks U and D to balls in ~n. We can also change * Supported by NSF Grant MCS 8201603.