On relations between Bessel potential spaces and Riesz potential spaces (Q1977890)

The purpose of this paper is to improve a characterization of Riesz potential spaces \({R_\alpha}^p\) and to give relations between Bessel potential spaces \({B_\alpha}^p\) and Riesz potential spaces. The main result gives the following relation: Let \(\mathcal P\) be the set of all polynomials, and let \(\mathcal{P}_k=\{P\in\mathcal{P}\): degree of \(P\leq k\}\). If \(k=[\alpha-(n/p)]\), then \[ {B_\alpha}^p=({R_\alpha}^p\oplus\mathcal{P}_k) \cap L^p \] and \[ \|u\|_{{B_\alpha}^p}\approx\|u\|_{{R_\alpha}^p\oplus\mathcal{P}_k}+ \|u\|_p , \] where the notation \(\approx\) stands for equivalent norms.