论文标题
Lebesgue和Morrey空间，Hardy-Littlewood-s-obolev和Olsen-type不平等现象的均匀分数积分运营商
Homogeneous fractional integral operators on Lebesgue and Morrey spaces, Hardy--Littlewood--Sobolev and Olsen-type inequalities
论文作者
论文摘要
令$ t_ {ω，α} $为均质的分数积分运算符，定义为\ begin {equination*} t_ {ω，α} f（x）：= \ int _ {\ int _ {\ mathbb r^n} \ end {equation*}和相关的分数最大运算符$ m_ {ω，α} $由\ begin {equation*}给出m_ {ω，α} f（x）：= \ sup_ {r> 0} \ frac {1} {| b（x，x，r）|^{1--α/n}} \ int_ {| x-y | x-y | <r} | em <r} |Ω \ end {equation*}在本文中，我们将使用hedberg的想法来谴责操作员$ t_ {ω，α} $和$ m_ {ω，α} $从$ l^p（\ mathbb r^n）$ to $ l^q（$ l^q（ l^s（\ mathbf {s}^{n-1}）$，$ s'<p <n/α$和$ 1/q = 1/p-α/n $，这是由Muckenhoupt和Wheeden获得的。我们还遵守假设$ω\ in l^s（\ Mathbf {s}^{n-1}）$，$ s'\ leq p <n/α$和$ 1/q = 1/p-α/n $，操作员$ t_ {Ω r^n）$ to $ l^{q，\ infty}（\ mathbb r^n）$，由Chanillo，Watson和Wheeden获得。我们将利用Adams的想法表明$ t_ {ω，α} $和$ m_ {ω，α} $从$ l^{p，κ}（\ m athbb r^n）$ to $ l^n）$ l^{q {q，κ}（\ mathbb r r^n）$ n时$ 1/q = 1/p-α/{n（1-κ）} $，并从$ l^{p，κ}（\ mathbb r^n）$到$ wl^{q，κ}（\ mathbb r^n）$时还建立了一些新的估计值。获得的结果是一些已知结果的实质性改进和扩展。此外，我们将将这些结果应用于一些众所周知的不平等现象，例如硬木 - 小木 - 贝贝尔和奥尔森型不平等。
Let $T_{Ω,α}$ be the homogeneous fractional integral operator defined as \begin{equation*} T_{Ω,α}f(x):=\int_{\mathbb R^n}\frac{Ω(x-y)}{|x-y|^{n-α}}f(y)\,dy, \end{equation*} and the related fractional maximal operator $M_{Ω,α}$ is given by \begin{equation*} M_{Ω,α}f(x):=\sup_{r>0}\frac{1}{|B(x,r)|^{1-α/n}}\int_{|x-y|<r}|Ω(x-y)f(y)|\,dy. \end{equation*} In this article, we will use the idea of Hedberg to reprove that the operators $T_{Ω,α}$ and $M_{Ω,α}$ are bounded from $L^p(\mathbb R^n)$ to $L^q(\mathbb R^n)$ provided that $Ω\in L^s(\mathbf{S}^{n-1})$, $s'<p<n/α$ and $1/q=1/p-α/n$, which was obtained by Muckenhoupt and Wheeden. We also reprove that under the assumptions that $Ω\in L^s(\mathbf{S}^{n-1})$, $s'\leq p<n/α$ and $1/q=1/p-α/n$, the operators $T_{Ω,α}$ and $M_{Ω,α}$ are bounded from $L^p(\mathbb R^n)$ to $L^{q,\infty}(\mathbb R^n)$, which was obtained by Chanillo, Watson and Wheeden. We will use the idea of Adams to show that $T_{Ω,α}$ and $M_{Ω,α}$ are bounded from $L^{p,κ}(\mathbb R^n)$ to $L^{q,κ}(\mathbb R^n)$ whenever $s'<p<n/α$ and $1/q=1/p-α/{n(1-κ)}$, and bounded from $L^{p,κ}(\mathbb R^n)$ to $WL^{q,κ}(\mathbb R^n)$ whenever $s'\leq p<n/α$ and $1/q=1/p-α/{n(1-κ)}$. Some new estimates in the limiting cases are also established. The results obtained are substantial improvements and extensions of some known results. Moreover, we will apply these results to several well-known inequalities such as Hardy--Littlewood--Sobolev and Olsen-type inequalities.
