## 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,

$$Tf(x)={\int}_{{\mathbb{R}}^{n}}K(x-y)\cdot f(y)\phantom{\rule{0.1667em}{0ex}}dy,$$

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

$${\int}_{{\mathbb{R}}^{n}}{\Vert Tf(x)\Vert}_{\mathbb{B}}^{q}w(x)\phantom{\rule{0.1667em}{0ex}}dx\le {C}_{1}{\int}_{{\mathbb{R}}^{n}}{\Vert f(x)\Vert}_{\mathbb{A}}^{q}w(x)\phantom{\rule{0.1667em}{0ex}}dx$$

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

$${\int}_{|x|>2|y|}{\Vert K(x-y)-K(x)\Vert}_{\mathbb{B}}\phantom{\rule{0.1667em}{0ex}}dx<{C}_{2}.$$

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

$${\int}_{|x|\ge {C}_{2}|y|}{\Vert K(x-y)-K(x)\Vert}_{\mathbb{B}}w(x+h)\phantom{\rule{0.1667em}{0ex}}dx\le {C}_{3}w(y+h)\phantom{\rule{1em}{0ex}}(\forall y\ne 0,\forall h\in {\mathbb{R}}^{n})$$

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

## Citation

Sakin Demir. "Banach space–valued ${H}^{p}$ spaces with ${A}_{p}$ weight." Illinois J. Math. 68 (2) 331 - 339, June 2024. https://doi.org/10.1215/00192082-11321393

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