论文标题
Banach空间价值$ h^p $空间，$ a_p $重量
Banach space valued $H^p$ spaces with $A_p$ weight
论文作者
论文摘要
在这项研究中，我们介绍了带有$ a_p $重量的$ h^p $空间的Banach空间，并证明了以下结果：让$ \ Mathbb {a} $和$ \ Mathbb {b} $ banach Space，$ t $ a envolution cnolution crenolution Operator绘图$ \ nathBb {a} $ - 估价$ - $ \ \ \ \ \ \ \ \ \ - $$ tf（x）= \ int _ {\ mathbb {r}^n} k（x-y）\ cdot f（y）\，dy，$ $，其中$ k $是在$ \ mathbb {r}^n $上定义的一个很强可测量的函数假设$ w $是在$ \ mathbb {r}^n $上定义的正权重函数， i）对于[1，\ infty] $中的某些$ q \，存在一个正常数$ c_1 $，因此$ \ int _ {\ mathbb {r}^n}^n}^n} \ | tf（x）\ |^q _ { C_1 \ int _ {\ Mathbb {r}^n} \ | f（x）\ | _ {\ Mathbb {a}}^q w（x）\，dx $$，dx $$，dx $$，in l^q _ {\ q _ {\ s. ii）存在一个正常数$ c_2 $独立于$ y \ in \ mathbb {r}^n $，以至于$$ \ int_ {| x |> 2 |> 2 | y |} \ | k（x-y）-k（x）-k（x）\ | ___________ 然后存在一个正常数$ C_3 $，以至于$$ \ | tf \ | _ {l^1 _ {\ Mathbb {b}}}}（w）} \ leq c_3 \ | f \ | _ {h^1 _ {h^1 _ { H^1 _ {\ Mathbb {a}}（w）$。令$ w \在A_1 $中。假设$ k \ in L _ {\ rm {loc}}（\ Mathbb {r}^n \ backslash \ {0 \}）$满足$ \ | k \ | k \ ast f \ | ________________ {l^2 _ {l^2 _ {l^2 _ { C_1 \ | f \ | _ {l^2 _ {\ Mathbb {a}}（w）} $$和$$ \ int_ {| x | x | \ geq c_2 | y | y |} \ | k（x-y）-K（x-y） c_3w（y+h）\; \; \;（\ forall y \ neq 0，\ forall h \ in \ mathbb {r}^n）$$对于某些绝对常数$ c_1 $，$ c_2 $，$ c_2 $和$ c_3 $。然后存在一个正常数$ c $独立于$ f $，因此$ | k \ ast f \ | _ {l^1 _ {\ Mathbb {b}}}（w）} \ leq c \ | f \ | f \ | _ _ {h^1 _ {h^1 _ { H^1 _ {\ Mathbb {a}}（w）$。
In this research we introduce the Banach space valued $H^p$ spaces with $A_p$ weight, and prove the following results: Let $\mathbb{A}$ and $\mathbb{B}$ Banach spaces, and $T$ be a convolution operator mapping $\mathbb{A}$-valued functions into $\mathbb{B}$-valued functions, i.e., $$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 $f\in H^1_{\mathbb{A}}(w)$.
