Banach space–valued Hp spaces with Ap weight

Abstract

In this research, we introduce Banach space–valued $H^p$ spaces with $A_p$ weight and prove the following results: Let $\mathbb{A}$ and $\mathbb{B}$ be Banach spaces, and let T be a convolution operator mapping $\mathbb{A}$-valued functions into $\mathbb{B}$-valued functions—that is,

$$Tf(x)={\int }_{{\mathbb{R}}^n}K(x-y)\cdot f(y)dy,$$

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

• for some $q\in [1,\mathrm{\infty }]$, 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 $f\in L_{\mathbb{A}}^q(w)$; and

• 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 $K\in L_{\mathrm{loc}}({\mathbb{R}}^n\setminus \{0\})$ 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 $f\in H_{\mathbb{A}}^1(w)$.