## An introduction to generalized fractional Sobolev Space with variable exponent. (arXiv:1901.05687v1 [math.AP])

In this paper, we extend the fractional Sobolev spaces with variable exponents $W^{s,p(x,y)}$ to include the general fractional case $W^{K,p(x,y)}$, where $p$ is a variable exponent, $s\in (0,1)$ and $K$ is a suitable kernel. We are concerned with some qualitative properties of the space $W^{K,p(x,y)}$ (completeness, reflexivity, separability and density). Moreover, we prove a continuous embedding theorem of these spaces into variable exponent Lebesgue spaces. As an application we establish the existence and uniqueness of a solution for a nonlocal problem involving the nonlocal integrodifferential operator of elliptic type.查看全文

## Solidot 文章翻译

 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 In this paper, we extend the fractional Sobolev spaces with variable exponents $W^{s,p(x,y)}$ to include the general fractional case $W^{K,p(x,y)}$, where $p$ is a variable exponent, $s\in (0,1)$ and $K$ is a suitable kernel. We are concerned with some qualitative properties of the space $W^{K,p(x,y)}$ (completeness, reflexivity, separability and density). Moreover, we prove a continuous embedding theorem of these spaces into variable exponent Lebesgue spaces. As an application we establish the existence and uniqueness of a solution for a nonlocal problem involving the nonlocal integrodifferential operator of elliptic type.
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