The authors study the Bessel potential space \(L^{\alpha,p(\cdot)}(\mathbb{R}^n)\) with variable exponent \(p(x)\), defined in the usual way as the range of the Bessel potential operators of order \(\alpha>0\) over the variable exponent Lebesgue space \(L^{p(\cdot)}(\mathbb{R}^n)\). They show (in Theorem 3.1) the coincidence of the space \(L^{\alpha,p(\cdot)}(\mathbb{R}^n)\) with the Sobolev space \(W^{m,p(\cdot)}(\mathbb{R}^n)\) in the case where \(\alpha = m\) is an integer, and that the Schwartz class \(\mathcal{S}\) is dense in \(L^{\alpha,p(\cdot)}(\mathbb{R}^n)\) under some natural assumptions on \(p(x)\). They also consider capacity in the space \(L^{\alpha,p(\cdot)}(\mathbb{R}^n)\) which is applied to the study of quasi-continuity and Hölder properties of functions \(u\in L^{\alpha,p(\cdot)}(\mathbb{R}^n)\) in terms of the capacity.
As the authors note in the final remark in connection with the referee’s comments, Theorem 3.1 was proved in the paper “Characterization of Riesz and Bessel potentials on variable Lebesgue spaces” by A.Almeida and the reviewer [J. Funct.Spaces Appl.4, No.2, 113–144 (2006; Zbl 1129.46022), reviewed above], where the denseness of \(C_0^\infty\) was also obtained in \(L^{\alpha,p(\cdot)}(\mathbb{R}^n)\), but under more restrictive assumptions. That paper also contains a characterization of the space \(L^{\alpha,p(\cdot)}(\mathbb{R}^n)\) in terms of convergence of the corresponding hypersingular integrals, as a generalization of the result of Stein–Lizorkin, known for the case of constant \(p\).