On the existence of limits along lines of Beppo Levi functions (Q1083590)

Let \(m\in {\mathbb{N}}\) and \(p>1\), let S denote the unit sphere in \({\mathbb{R}}^ n\), and \(B_{m,p}(E)\) denote the Bessel capacity of index (m,p) of a set \(E\subset {\mathbb{R}}^ n\). The author establishes an integral representation theorem for Beppo Levi functions u of order m attached to \(L^ p({\mathbb{R}}^ n)\), and uses this to investigate the existence of radial and perpendicular limits of such functions. Let \(A(r)=r^{(n- mp)/p}\) if m-n/p is not a non-negative integer; otherwise let \(A(r)=r^{(n-mp)/p} (\log r)^{-1/p'}\), where \(1/p+1/p'=1.\) Theorem. Let u be as above. (i) If \(mp\leq n\), then there exist a polynomial P of degree at most m-1 and a set \(E\subset S\), for which \(B_{m,p}(E)=0\), such that \(A(r)\{u(r\theta) - P(r\theta)\} \to 0\) as \(r\to \infty\) for every \(\theta\in S\setminus E\). (ii) If \(mp>n\), then there is a polynomial P of degree at most m-1 such that \(A(| x|)\{u(x)-P(x)\}\to 0\) as \(| x| \to \infty.\) The behaviour of \(u(x_ 1,...,x_ n)\) as \(x_ n\to +\infty\) is also considered. A number of examples are presented to demonstrate the sharpness of the stated results.