M. Riesz' kernels as boundary values of conjugate Poisson kernels (Q539847)

For every \(1\leq p\leq \infty\) and \(\mu \in \mathbb{R}\) the weighted spaces of distributions \(\mathcal{D}'_{L^p,\mu}:=(1+|x|^2)^{-\mu /2}\mathcal{D}'_{L^p}\), and \(\mathcal{D}'_{L^p,\mu ,-1}:=(1+|x|^2)^{-\mu /2}\log (2+|x|^2)\mathcal{D}'_{L^p}\) are considered. The authors first establish that the \(\mathcal{S}'\)-convolution \(*\) is continuous from \(\mathcal{D}'_{L^1,\mu}(\mathbb{R}^n)\times \mathcal{D}'_{L^\infty,\nu}(\mathbb{R}^n)\) into \(\mathcal{D}'_{L^1,\mu}(\mathbb{R}^n)\), when \(-\nu\leq \mu<\nu -n\), \(\nu >n\). Also, for certain kernels \(K\) and \(-n-1\leq \mu <1\), they obtain the convergence in \(\mathcal{D}'_{L^1,\mu}\) of \(T*K_t\), as \(t\rightarrow 0^+\), to \(T\in \mathcal{D}'_{L^1,\mu}\). (Here, as usual, \(K_t(x)=t^{-n}K(x/t)\), \(x\in \mathbb{R}^n\), \(t>0\)). Thus, a result obtained by \textit{J. Alvarez, M. Guzmán-Partida} and \textit{S. Pérez-Esteva} [Math. Nachr. 280, 1443--1466 (2007; Zbl 1138.46032)] is generalized. As consequence, it is shown that \(U(t)=T*P_t\), \(t>0\), is a solution of the Dirichlet problem for \(\Delta _{n+1}\) in the upper half-space, for boundary value \(T\) in \(\mathcal{D}'_{L^1,\mu}(\mathbb{R}^n)\), \(-n-1\leq \mu <1\). The convolution mapping associated to the generalized Hilbert kernel \(N_0\) is also analyzed. The authors obtain that it is continuous from \(\mathcal{D}'_{L^1,\mu}(\mathbb{R}^n)\) into \((\mathcal{D}'_{L^1,\mu ,-1}(\mathbb{R}^n))^n\). Then, the conjugate Poisson kernels \(Q_t=N_0*P_t\), \(t>0\), furnish a continuous convolution mapping from \(\mathcal{D}'_{L^1,\mu}(\mathbb{R}^n)\) into \((\mathcal{D}'_{L^1,\mu ,-1}(\mathbb{R}^n))^n\), when \(-n\leq \mu <0\), and it holds that \(\lim_{t\downarrow 0}Q_t*T=N_0*T\), for \(T\in (\mathcal{D}'_{L^1,\mu ,-1}(\mathbb{R}^n))^n\).