Hardy spaces associated with a pair of commuting operators

Let L-1, L-2 be a pair of one-to-one commuting sectorial operators such that each L-i for i is an element of {1, 2} satisfies the m(i) order L-2 off-diagonal estimates and m(1) >= m(2) > 0. Let H-Li(p) (R-n), i is an element of { 1; 2}, and H-L1+(L) over tilde2(p) (R-n) be the Hardy spaces associated, respectively, to the operators L-i and L-1 +(L) over tilde (2), where (L) over tilde (2) := L-2(m1/m2) 2 is a fractional power of L-2. In this paper, the authors give out some real-variable properties of these Hardy spaces. More precisely, the authors first establish the bounded joint H-infinity functional calculus in these Hardy spaces and prove that the abstract Riesz transform D-mi (L-1 + L-2) (-1/2) is bounded from H-Li(p) (R-n) to the classical Hardy space H-p (R-n) for all p is an element of (n/n+m(i) ,1], where i is an element of {1,2}, Moreover, for all p is an element of(0, 1], the authors show that H-L1+(L) over tilde2(p) (R-n) = H-L1(p) (R-n) + H-L2(p) (R-n) and give a sufficient condition to guarantee H-L1(p) (R-n) subset of H-L2(p) (R-n).