Integral representations, differentiability properties and limits at infinity for Beppo Levi functions
For an open subset \(G\) in \(\mathbb{R}^n\) and \(1\leq p <\infty\), the set of Beppo Levi functions on \(G\), denoted \(BL_m(L^p_{\text{loc}}(G))\), is the set of functions \(u\in L^p_{\text{loc}}(G)\) for which \(|\nabla_m u(x)|= (\sum_{|\mu|=m}|D^{\mu}u(x)|^2)^{1/2}\) is in \(L^p_{\text{loc}}(G)\), where \(D^{\mu} u\) is the distributional derivative of \(u\) of order \(\mu\). In Theorem 2.1 it is proved that if \(u\in BL_m(L^p_{\text{loc}}(\mathbb{R}^n))\) satisfies \(\int_{\mathbb{R}^n}|\nabla_m u(y)|(1+|y|)^{m-n-l-1}dy <\infty\) for some integer \(l\geq -1\), then there exists a polynomial \(P\) such that \[ u(x) = \sum_{|\mu|=m}a_{\mu}\int K_{\mu,l}(x,y)D^{\mu}u(y)dy + P(x) \] a.e. on \(\mathbb{R}^n\), where the \(K_{\mu,l}\) are modified kernel functions derived from \(K_{\mu}(x) = x^{\mu}\big/|x|^n\). As applications of the above integral representation, the author studies continuity and differentiability properties of Beppo Levi functions \(u\) satisfying \(\int \Psi_p(|\nabla_mu(x)|)dx<\infty \), where \(\Psi_p(r) = r^p\psi(r)\), with \(\psi\) a positive non-decreasing function on \((0,\infty)\). In the second part of the paper the author studies the existence of limits at infinity for Beppo Levi functions. The author also considers the existence of fine-type limits at infinity with respect to Bessel capacities, which yields the radial limit result at infinity.
scientific article; zbMATH DE number 1045346