Brownian Martingales and Some New Results on Interpolation of Hardy Spaces

摘要

Let H_1 (R~n) be the usual Hardy space on R~n. We show that the couple (H_1(R~n), L_(infinity)(R~n)) is a Calderon couple. This result immediately follows from the following stronger one: Given any f implied by H_1(R~n) + L_(infinity)(R~n) there exist two linear operators U and V satisfying the properties: (i) Uf = Nf (Nf being the nontangential maximal function of f) and U is contractive from H_1(R~n) to L_1(R~n) and also from L_(infinity)(R~n) to L_(infinity)(R~n) to L_(infinity)(R~n); (ii) V (Nf) = f, V is similtaneously bounded from L_1(R~n) to H_1(R~n) and from L_(infinity)(R~n) to L_(infinity)(R~n) and the norms of V on these spaces are controlled by a universal constant. We also have similar results on the couple (L_p(R~n), BMO(R~n)) for every 1 < p < infinity. Our approach to these results is via Brownian motion.