论文标题
关于Riemann-Liouville分数Sobolev空间的注释
A note on Riemann-Liouville fractional Sobolev spaces
论文作者
论文摘要
从Bergounioux，Leaci，Nardi和Tomarelli的最新论文中汲取灵感，我们研究了Riemann-Liouville分数Sobolev Space $ w^{s，p} _ {rl，a+}（i）$，对于$ i =（a，b）的$ a，b）$ a，b）$ a，b \ b），和$ p \ [1，\ infty] $;也就是说，函数空间$ u \ in l^{p}（i）$，使得左riemann -liouville $（1- s）$ - 分数积分$ i_ {a+}^{1 -s} [u] $属于$ w^{1，p}（i）$。我们证明，有限变化和分数Sobolev空间的功能空间，$ bv（i）$和$ w^{s，1}（i）$，连续嵌入$ w^{s，1} _ {rl，a+}（a+}（i）$。此外，我们以左riemann-liouville $ s $ s-fractional有界变化定义功能空间，$ bv^{s} _ {rl，a+}（i）$作为函数$ u \ in l^{1}（1}（1}（i））$ i^（I）这些功能的良好特性。最后，我们证明了一些分数Sobolev型嵌入结果，并分析了高阶Riemann-Liouville分数衍生物的情况。
Taking inspiration from a recent paper by Bergounioux, Leaci, Nardi and Tomarelli we study the Riemann-Liouville fractional Sobolev space $W^{s, p}_{RL, a+}(I)$, for $I = (a, b)$ for some $a, b \in \mathbb{R}, a < b$, $s \in (0, 1)$ and $p \in [1, \infty]$; that is, the space of functions $u \in L^{p}(I)$ such that the left Riemann-Liouville $(1 - s)$-fractional integral $I_{a+}^{1 - s}[u]$ belongs to $W^{1, p}(I)$. We prove that the space of functions of bounded variation and the fractional Sobolev space, $BV(I)$ and $W^{s, 1}(I)$, continuously embed into $W^{s, 1}_{RL, a+}(I)$. In addition, we define the space of functions with left Riemann-Liouville $s$-fractional bounded variation, $BV^{s}_{RL,a+}(I)$, as the set of functions $u \in L^{1}(I)$ such that $I^{1 - s}_{a+}[u] \in BV(I)$, and we analyze some fine properties of these functions. Finally, we prove some fractional Sobolev-type embedding results and we analyze the case of higher order Riemann-Liouville fractional derivatives.
