Tangential characterzations of Hardy and mixed-norm spaces of harmonic functions on the real hyperbolic ball
Let \(B^n\) (\(S^{n-1}\)) be the unit ball (resp. unit sphere) in \(\mathbb R^n,\) \(n\geq 3,\) with the volume measure \(m,\) and let \(\Delta_h\) denote the Laplace-Beltrami operator for the Poincare metric on \(B^n.\) A function \(u\) on \(B^n\) is \(h\)-harmonic (or \(u\in h\)) if \(\Delta_hu=0\) on \(B^n.\) Let \(\Gamma_\alpha(\xi)=\{x\in B^n:| x-\xi| <\alpha(1-| x| )\},\) \(\xi\in S^{n-1},\alpha>1\) be the non-tangential approach region, and define the area function \[ S_\alpha u(\xi)=\left(\int_{\Gamma_\alpha(\xi)}| \nabla_hu(x)| ^2dm_h(x)\right)^{1\slash 2}, \] where \(dm_h=C_ndm\slash (1-| x| ^2)^n,\) and \(\nabla_hu\) is the hyperbolic gradient. Let \(N=\sum_jx_j\partial_{x_j}\) be the normal derivation operator and \(T_{ij}=x_i\partial_{x_j}-x_j\partial_{x_i}.\) Define two modified area functions \[ S_\alpha^Nu(\xi)=\left(\int_{\Gamma_\alpha(\xi)}(1-| x| ^2)^{2-n}| Nu(x)| ^2dm(x)\right)^{1\slash 2} \] and \[ S_\alpha^Tu(\xi)=\left(\int_{\Gamma_\alpha(\xi)}(1-| x| ^2)^{2-n}\sum_{i<j}| T_{ij}u(x)| ^2dm(x)\right)^{1\slash 2}. \] The first main result is the following theorem. Let \(0<p<\infty.\) Then for an \(h\)-harmonic function \(u\) in \(B^n\) the following statements are equivalent: (i) \(S_\alpha u\in L^p(S^{n-1}),\) (ii) \(S_\alpha^N u\in L^p(S^{n-1}),\) (iii) \(S_\alpha^T u\in L^p(S^{n-1}).\) Let \(h^{p,q,\alpha}=h\cap L^{p,q,\alpha},\) \(p>0,\) \(q>0,\) \(\alpha>0,\) where \(L^{p,q,a}\) is the space of Borel functions \(u\) on \(B^n\) for which \(\| u\| _{p,q,\alpha}^q=\int_0^1(1-r)^{q\alpha-1}M_p(r,u)^qdr<\infty,\) where \(M_p(r,u)^p=\int_{S^{n-1}}| u(ry)| ^pd\sigma(y),\) and \(\sigma\) is the usual rotation invariant measure. The second theorem is the following. Let \(u\in h,\) \(p>0,\) \(q>0,\) \(\alpha>0.\) Then the following statements are equivalent: (i) for all \(i,j,\) \(T_{ij}u\in h^{p,q,\alpha},\) (ii) \(Nu\in h^{p,q,\alpha},\) (iii) \(| \nabla u| \in h^{p,q,\alpha}.\)
scientific article; zbMATH DE number 5150229