On the closure of the Lizorkin space in spaces of Beppo Levi type (Q2773431)

Let \(\Phi\) be the Lizorkin space consisting of Schwartzian functions orthogonal to polynomials. As is known, this space is invariant with respect to the Riesz potential operator and other operators whose symbol has a singularity at the origin. The author describes the closure of \(\Phi\) with respect to the Sobolev type norm defined by higher derivatives: NEWLINE\[NEWLINE\|u\|=|u|_{\mathcal{L}^p_m}+ \int_{|x|<1}|u(x)|dx,NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\mathcal{L}^p_m =\Biggl\{|u|_{\mathcal{L}^p_m}=\sum_{|j|=m}\|D^j u\|_{L^p(R^n)}< \infty\Biggr\}. NEWLINE\]NEWLINE An auxiliary proposition is proved wich states that convolutions, whose symbol is infinitely differentiable beyond the origin and has a power-type behaviour at the origin and infinity, preserve the Lizorkin space invariant. On the base of this fact, the main statement is proved that the closure of \(\Phi\) coincides with \(\mathcal{L}^p_m\bigcup L^{p,-m}\) in the case where \(m-\frac{n}{p}\) is not an integer, where NEWLINE\[NEWLINEL^{p,-m}=\Biggl\{u: \int_{R^n}|u(x)|p(1+|x|)^{-mp} dx < \infty\Biggr\}. \tag{1}NEWLINE\]NEWLINE The case where \(m -\frac{n}{p}\) is an integer, is also considered, with a logarithmic factor appearing in this case in the weight function in (1).