Hyperplane means of potentials (Q1919590)

For a measurable function \(f\) on \(\mathbb{R}^n\) satisfying \(\int (1+|y |)^{ \alpha-n} |f(y) |dy<\infty\), \(0<\alpha<n\), the Riesz potential \(U_\alpha f\) of \(f\) is defined by \[ U_\alpha f(x) = \int_{\mathbb{R}^n} |x-y |^{\alpha-n} f(y)dy. \] In the paper the author investigates the limiting behavior of the \(q\)th hyperplane means of the potentials \(U_\alpha f\) over the surfaces \(D(r) = \{(x',x)\in \mathbb{R}^{n-1} \times\mathbb{R} : x_n=r\}\). For \(q>0\) set \[ S_q(u_r) = \Bigl(\int_{\mathbb{R}^{n-1}} \bigl|u_r(x') \bigr|^q dx'\Bigr)^{1/q} \] where \(u_r(x')=U_\alpha f(x',r)\). It is proved that if \(\beta<p-1\), \(1\leq p\leq q\), and \({n-\alpha p-1 \over p(n-1)} < {1\over q} < {n-\alpha + \beta\over p(n-1)}\), then for any nonnegative measurable function \(f\) on \(\mathbb{R}^n\) satisfying \(\int|f(y) |^p|y_n |^\beta dy < \infty\), we have \[ \liminf_{r\to 0} r^{(n-\alpha p + \beta)/p-(n-1)/q} S_q(u_r)=0. \] The paper also considers the limiting behavior of \(S_q(v_r)\), where \(v_r(x') = U_\alpha f(x',r)-U_\alpha f(x',0)\), and of harmonic functions \(u\) on the half space \(D\) satisfying \(\int_D |\text{grad} u(x) |^p x_n^\beta dx< \infty\).