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 $\Vert K(x){\Vert }_{\mathbb{B}}$ 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 $${\int }_{{\mathbb{R}}^n}\Vert Tf(x){\Vert }_{\mathbb{B}}^{q}w(x)dx\le C_1{\int }_{{\mathbb{R}}^n}\Vert f(x){\Vert }_{\mathbb{A}}^{q}w(x)dx$$ for all $f\in L_{\mathbb{A}}^q({\mathbb{R}}^n)$. ii) There exists a positive constant $C_2$ independent of $y\in {\mathbb{R}}^n$ such that $${\int }_{|x|>2|y|}\Vert K(x-y)-K(x){\Vert }_{\mathbb{B}}dx<C_2.$$ Then there exists a positive constant $C_3$ such that $$\Vert Tf{\Vert }_{L_{\mathbb{B}}^1(w)}\le C_3\Vert f{\Vert }_{H_{\mathbb{A}}^1(w)}$$ for all $f\in H_{\mathbb{A}}^1(w)$. Let $w\in A_1$. Assume that $K\in L_{\mathrm{loc}}({\mathbb{R}}^n\backslash \{0\})$ satisfies $$\Vert K\ast f{\Vert }_{L_{\mathbb{B}}^2(w)}\le C_1\Vert f{\Vert }_{L_{\mathbb{A}}^2(w)}$$ and $${\int }_{|x|\ge C_2|y|}\Vert K(x-y)-K(x){\Vert }_{\mathbb{B}}w(x+h)dx\le C_{3}w(y+h)(\forall y\ne 0,\forall h\in {\mathbb{R}}^n)$$ for certain absolute constants $C_1$, $C_2$, and $C_3$. Then there exists a positive constant $C$ independent of $f$ such that $$\Vert K\ast f{\Vert }_{L_{\mathbb{B}}^1(w)}\le C\Vert f{\Vert }_{H_{\mathbb{A}}^1(w)}$$ for all $f\in H_{\mathbb{A}}^1(w)$.