Integral representations of Beppo Levi functions and the existence of limits at infinity (Q914891)

Let \(\omega\) : \([0,\infty)\to (0,\infty)\) be monotone and such that, for some positive A, \(\omega\) (r)/A\(\leq \omega (2r)\leq A\omega (r)\) when \(r>0\). Given \(p>0\) (with q its Hölder conjugate) and positive integers m and n, suppose that the smallest \(\ell\) such that \[ \int^{\infty}_{1}r^{-1+q(m-n/p-\ell -1)}\{\omega (r)\}^{-q/p} dr<\infty \] satisfies \(\ell \geq m-n\). The author proves that if u is a Beppo-Levi function in \(BL_ m(L^ p_{loc}({\mathbb{R}}^ n))\) (that is, \(u\in L^ p_{loc}({\mathbb{R}}^ n)\) and all partial derivatives of u of order m are in \(L^ p({\mathbb{R}}^ n))\) such that \[ \sum_{| \lambda | =m}\int | D^{\lambda}u(x)|^ p \omega (| x|) dx<\infty, \] then there exists a polyharmonic polynomial P with \(\Delta^ mP\equiv 0\) in \({\mathbb{R}}^ n\) such that, almost everywhere in \({\mathbb{R}}^ n\), u-P equals the sum of potentials in \({\mathbb{R}}^ n\) of kernels made out of partial derivatives of Riesz kernels with, as weight functions, the partial derivatives \(D^{\lambda}u\) of u with order \(| \lambda | =m\). Applications are given to extensions of the author's previous results on the existence of radial limits of Beppo-Levi functions.