Banach space-valued HP spaces with AP weight

In this research, we introduce Banach space-valued H-p spaces with A(p) 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) = integral K-n(R)(x-y)center dot f(y)dy, where K is a strongly measurable function defined on R-n such that parallel to K(x)parallel to(B) is locally integrable away from the origin. Suppose that w is a positive weight function defined on R-n and that (i)for some q is an element of [1,infinity], there exists a positive constant C-1 such that integral(n)(R)parallel to Tf(x)parallel to(q)(B)w(x)dx <= C-1 integral(n)(R)parallel to f(x)parallel to(q)(A)w(x)dx for all f is an element of L-A(q)(w); and (ii) there exists a positive constant C-2 independent of y is an element of R-n such that integral(|x|>2|y|)parallel to K(x-y)-K(x)parallel to(B)dx < C-2. Then there exists a positive constant C-3 such that parallel to Tf parallel to(1)(LB)(w) <= C-3 parallel to f parallel to(1)(HA)(w) for all f is an element of H-A(1)(w). Let w is an element of A(1). Assume that K is an element of L-loc(R-n\{0}) satisfies parallel to K*f parallel to(2)(LB)(w) <= C-1 parallel to f parallel to(2)(LA)(w) and integral(|x| >= C2|y|)parallel to K(x-y)-K(x)parallel to(B) w(x+h)dx <= C(3)w(y+h) (for all y not equal 0,for all h is an element of R-n) for certain absolute constants C-1, C-2, and C-3. Then there exists a positive constant C independent of f such that parallel to K*f parallel to(1)(LB)(w)<bold> </bold><= C parallel to f parallel to(1)(HA)(w) for all f is an element of H-A(1)(w).