On the non-tangential convergence of Poisson and modified Poisson semigroups at the smoothness points of \(L_p\)-functions (Q2003770)

Let \(\mathbb{R}^{n+1}_+=\mathbb{R}^n\times(0,\infty)=\{(x,y):x\in\mathbb{R}^n,y>0\}\) be the \((n+1)\)-dimensional half-space and let \(\Gamma_\alpha(x^0)=\{(x,y):\vert x-x^0\vert <\alpha y\}\) be the infinite cone, whose vertex is at \(x^0\). Let \(\rho\in(0,1)\) be a fixed number and the function \(\mu(r)\) be continuous on \([0,\rho]\), positive on \((0,\rho]\) and \(\mu(0)=0\). A function \(f\in L_p(\mathbb{R}^n)\) has the \(\mu\)-smoothness property at a point \(x^0\in\mathbb{R}^n\), if \[\sup\limits_{0<r\leq\rho}\frac1{r^n\mu(r)}\int_{\vert x\vert \leq r}\vert f(x^0-x)-f(x^0)\vert \,dx<\infty\,.\] Such a point \(x^0\in\mathbb{R}^n\) is called a \(\mu\)-smoothness point of \(f\). This paper investigates the rate of non-tangential convergence of Poisson and metaharmonic integrals at \(\mu\)-smoothness points of \(f\). Let \(\mu(t)\geq at\), \(0<t\leq\rho]\), for some \(a>0\), \(\mu(t)=\mu(\rho)\), for \(\rho\leq t<\infty\) and let \(\omega(t)>0\) be a locally bounded function such that \(\mu(\varepsilon t)\leq\mu(\varepsilon)\omega(t)\), \(\varepsilon\in(0,\rho)\), \(t\in(0,\rho/\varepsilon)\), \(\displaystyle\int_0^\infty\frac{\omega(t)\,dt}{1+t^2}<\infty\). It is proved that, if \(V_\varepsilon f\) (\(\varepsilon>0\)) is any of Poisson or metaharmonic integrals (semigroups) of the function \(f\in L_p(\mathbb{R}^n)\) and \(x^0\in\mathbb{R}^n\) is a \(\mu\)-smoothness point of \(f\), then the non-tangential asymptotic estimate \((V_\varepsilon f)(x)-f (x^0)=O(\mu(\varepsilon))\) holds when \((x,\varepsilon)\in\Gamma_\alpha(x^0)\) and \(\varepsilon\to 0^+\).