Growth properties for modified Poisson integrals in a half space. (Q1880112)

Let \(K_{\lambda} = \| x\| ^{-\lambda}\), \(x\in {\mathbb R}^n,\) \[ K_{\lambda, m}=\begin{cases} K_{\lambda}(x-y), &\| y\| <1 \\ K_{\lambda}(x-y)-P_{m-1}(x,y),& \| y\| > 1 , \end{cases} \] where \(P_{m-1}(x, y)\) is Taylor's polynomial of degree \(m-1\) for the function \(K_{\lambda}(x-y)\) in the variable \(x\) at the expansion point \(x_0=0.\) The authors investigate the integral \[ K_{\lambda,m} f(x)=\int_{{\mathbb R}^{n-1}} K_{\lambda,m}(x,y)f(y)\,dy \] in the half space \({\mathbb R}_n^+ =\{ x\in {\mathbb R}^n\), \(x_n>0 \}\) under the condition \( \int_{{\mathbb R}^{n-1}}(1+ \| y\| )^{-\gamma} | f(y)| ^p\,dy< \infty\), \(p\geq 1.\) They find conditions (theorem 1) for \(\mu\) under which the following identities are fulfilled \[ \begin{aligned} \lim_{x \to \infty} x_n^{\lambda}\| x\| ^\mu K_{\lambda, m}f(x) &=0,\\ \lim_{x \to \infty} x_n^{\lambda}\| x\| ^\mu (\log \| x\| )^ {\frac{1}{q}}K_{\lambda, m}f(x) &=0, \quad \frac{1}{p} + \frac{1}{q} = 1.\end{aligned} \] Then (theorem 2) the authors estimate the size of the exceptional set \(E\) such that \[ \lim_{\substack{ x \to \infty\\ x \notin {\mathbb R}_n^+ \setminus E}} x_n^{-\beta}\| x\| ^\mu K_{\lambda, m}f(x) =0. \] In theorem 3 the authors investigate boundary properties of the integrals. There are many previous investigations of such type of questions.