论文标题
球campanato型功能空间的分数积分的界限
Boundedness of Fractional Integrals on Ball Campanato-Type Function Spaces
论文作者
论文摘要
令$ x $为$ {\ mathbb r}^n $满足某些温和假设的$ {\ mathbb r}^n $上的球$α\ in（0，n）$和$β\ in（1，\ infty）$。在本文中，当$α\ in（0,1）$中时，作者首先找到了一个合理的版本$ \ widetilde {i}_α$的分数积分$i_α$在球campanato-type type函数$ \ MATHCAL {LATCAL型函数$ \ MATHCAL {l} _ $ q \ in [1，\ infty）$，$ s \ in \ mathbb {z} _+^n $和$ d \ in（0，\ infty）$。然后，作者证明了$ \ widetilde {i}_α$从$ \ nathcal {l} _ {x^β，q，q，s，d}（\ m athbb {r}^n）$到$ \ \ \ \ \ m natercal {L}常数$ c $使得，对于任何球$ b \ subset \ mathbb {r}^n $，$ | b |^{\fracα{n}} \ leq c \ | \ | \ mathbf {1} _b \ \ | _x^^{\ frac {\ frac {β-β-1bivile $ x $。此外，作者将$α\ in（0,1）$ in $ \ widetilde {i}_α$扩展到（0,1）$ in（0,1）$ in（0，n）$，并在这种情况下也获得相应的界限。此外，事实证明，$ \ widetilde {i}_α$是$i_α$的伴随运营商。所有这些结果都有广泛的应用。特别是，即使分别将它们应用于莫里空间，混合 - 勒布斯格空间，局部概括的HERZ空间和混合 - 纳尔兹HERZ空间，所有获得的结果都是新的。这些结果的证据在很大程度上取决于$ \ MATHCAL {l} _ {x，q，q，s，d}（\ Mathbb {r}^n）$，以及$ h_x的分子的特殊原子分解（\ nathbb {r}^n）$（r}^n）$（$ h_x $），该$（r}^n）$（$） $ \ MATHCAL {L} _ {X，Q，S，D}（\ Mathbb {r}^n）$。
Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ satisfying some mild assumptions and let $α\in(0,n)$ and $β\in(1,\infty)$. In this article, when $α\in(0,1)$, the authors first find a reasonable version $\widetilde{I}_α$ of the fractional integral $I_α$ on the ball Campanato-type function space $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ with $q\in[1,\infty)$, $s\in\mathbb{Z}_+^n$, and $d\in(0,\infty)$. Then the authors prove that $\widetilde{I}_α$ is bounded from $\mathcal{L}_{X^β,q,s,d}(\mathbb{R}^n)$ to $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ if and only if there exists a positive constant $C$ such that, for any ball $B\subset \mathbb{R}^n$, $|B|^{\fracα{n}}\leq C \|\mathbf{1}_B\|_X^{\frac{β-1}β}$, where $X^β$ denotes the $β$-convexification of $X$. Furthermore, the authors extend the range $α\in(0,1)$ in $\widetilde{I}_α$ to the range $α\in(0,n)$ and also obtain the corresponding boundedness in this case. Moreover, $\widetilde{I}_α$ is proved to be the adjoint operator of $I_α$. All these results have a wide range of applications. Particularly, even when they are applied, respectively, to Morrey spaces, mixed-norm Lebesgue spaces, local generalized Herz spaces, and mixed-norm Herz spaces, all the obtained results are new. The proofs of these results strongly depend on the dual theorem on $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ and also on the special atomic decomposition of molecules of $H_X(\mathbb{R}^n)$ (the Hardy-type space associated with $X$) which proves the predual space of $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$.
