Banach space–valued Hp spaces with Ap weight

In this research, we introduce Banach space–valued Hp spaces with Ap weight and prove the following results: Let A and B be Banach spaces, and let T be a convolution operator mapping A-valued functions into B-valued functions—that is, Tf(x)= ∫RnK(x−y)⋅f(y)dy, where K is a strongly measurable function defined on Rn such that ‖K(x)‖B is locally integrable away from the origin. Suppose that w is a positive weight function defined on Rn and that for some q∈[1,∞], there exists a positive constant C1 such that ∫Rn‖Tf(x)‖Bqw(x)dx≤C 1 ∫Rn‖f(x)‖Aqw(x)dx for all f∈LAq(w); and there exists a positive constant C2 independent of y∈Rn such that ∫|x|>2|y|‖K(x−y)−K(x)‖Bdx<C2. Then there exists a positive constant C3 such that ‖Tf‖LB1(w)≤C3‖f‖HA1(w) for all f∈HA1(w). Let w∈A1. Assume that K∈Lloc(Rn∖{0}) satisfies ‖K∗f‖LB2(w)≤C1‖f‖LA2(w) and ∫|x|≥C2|y|‖K(x−y)−K(x)‖Bw(x+h)dx≤C3w(y+h)(∀y≠0,∀h∈Rn) for certain absolute constants C1, C2, and C3. Then there exists a positive constant C independent of f such that ‖K∗f‖LB1(w)≤C‖f‖HA1(w) for all f∈HA1(w).