Some properties of conjugate harmonic functions in a half-space

We prove a multi-dimensional analog of the theorem of Hardy and Littlewood about the logarithmic bound of the L p -average of the conjugate harmonic functions, 0 < p ⩽ 1 . We also give sufficient conditions for a harmonic vector to belong to H p ( R + n + 1 ) , 0 < p ⩽ 1 .