Banach space valued $H^p$ spaces with $A_p$ weight

Abstract

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 $f\in H^1_{\mathbb{A}}(w)$.