Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A note on Riemann-Liouville fractional Sobolev spaces   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.

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The categories of this paper are: math-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: stat
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The categories of this paper are: acc-phys
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The categories of this paper are: cs
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The categories of this paper are: physics
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The categories of this paper are: eess
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The categories of this paper are: patt-sol
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-fin
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The categories of this paper are: funct-an
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The categories of this paper are: chem-ph