Equivalent norms onL p spaces of harmonic functions

This note gives necessary and sufficient conditions for a measurable setG in the unit ballB in ? ⁿ to satisfy the following property: there exists a constantC>0 such that $$intlimits_B {|f|^2 } dm leqslant Cintlimits_G {|f|^2 } dm$$ for everyf∈L ² (B, dm) which is harmonic inB. Herem is the Lebesgue measure of dimensionn. The same condition is sufficient if any exponentp>0 replaces 2 in (^*), and if certain weighted measures replacem. Applications to the problem of representing harmonic functions inL ² (B) as a sum of kernel functions are indicated.