In this research we introduce the Banach space valued $H^p$ spaces with $A_p$ weight, and prove the following results: Let $\mathbb{ A}$ and $\mathbb{ B}$ Banach spaces, and $T$ be a convolution operator mapping $\mathbb{ A}$-valued functions into $\mathbb{ B}$-valued functions, i.e., $$Tf(x)=\int_{ \mathbb{ R}^n}K(x-y)\cdot f(y)\, dy,$$ where $K$ is a strongly measurable function defined on $\mathbb{ R}^n$ such that $\|K(x)\|_{ \mathbb{ B}}$ is locally integrable away from the origin. Suppose that $w$ is a positive weight function defined on $\mathbb{ R}^n$, and that i) For some $q\in [1, \infty ]$, there exists a positive constant $C_1$ such that $$\int_{ \mathbb{ R}^n}\|Tf(x)\|^q_{ \mathbb{ B}}w(x)\, dx\leq C_1\int_{ \mathbb{ R}^n}\|f(x)\|_{ \mathbb{ A}}^q w(x)\,dx$$ for all $f\in L^q_{ \mathbb{ A}}(\mathbb{ R}^n)$. ii) There exists a positive constant $C_2$ independent of $y\in\mathbb{ R}^n$ such that $$\int_{ |x|>2|y|}\|K(x-y)-K(x)\|_{ \mathbb{ B}}\, dx<C_2.$$ Then there exists a positive constant $C_3$ such that $$\|Tf\|_{ L^1_{ \mathbb{ B}}(w)}\leq C_3\|f\|_{ H^1_{ \mathbb{ A}}(w)}$$ for all $f\in H^1_{ \mathbb{ A}}(w)$. Let $w\in A_1$. Assume that $K\in L_{ \rm{ loc}}(\mathbb{ R}^n\backslash \{ 0\})$ satisfies $$\|K\ast f\|_{ L^2_{ \mathbb{ B}}(w)}\leq C_1\|f\|_{ L^2_{ \mathbb{ A}}(w)}$$ and $$\int_{ |x|\geq C_2|y|}\|K(x-y)-K(x)\|_{ \mathbb{ B}}w(x+h)\, dx\leq C_3w(y+h)\;\;\;(\forall y\neq 0, \forall h\in\mathbb{ R}^n) $$ for certain absolute constants $C_1$, $C_2$, and $C_3$. Then there exists a positive constant $C$ independent of $f$ such that $$\|K\ast f\|_{ L^1_{ \mathbb{ B}}(w)}\leq C\|f\|_{ H^1_{ \mathbb{ A}}(w)}$$ for all $f\in H^1_{ \mathbb{ A}}(w)$.
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