Conjugate functions on spaces of parabolic Bloch type (Q2375997)

Let \(0<\alpha\leq 1\) and \(\sigma > \min\left\{1, 1/2\alpha \right\}\). The \(\alpha\)-parabolic Bloch type space \(\mathcal{B}_\alpha(\sigma)\) consists of those \(u\in C^1(\mathbb{R}^{n+1}_+)\) for which \[ u_t + \left(-\Delta_x \right)^\alpha u =0 \] and \[ \|u\|_{\mathcal{B}_\alpha(\sigma)} := |u(0,1)| + \sup_{(x,t)\in \mathbb{R}^{n+1}_+} t^\sigma \left(t^{1/2\alpha}|\nabla_x u(x,t)| + t|u_t(x,t)| \right). \] For \(u\in \mathcal{B}_\alpha(\sigma)\), the authors introduce an \(\alpha\)-parabolic conjugate vector function in terms of appropriate time-fractional derivatives. On \(\mathcal{B}_{1/2}(0)\), the classical harmonic Bloch space, the \(1/2\)-parabolic conjugation coincides with the standard one. For \(0<\alpha<1\) and \(\sigma> 1 - 1/\alpha\), the authors prove existence and norm equivalence of a unique \(\alpha\)-parabolic conjugate \(V=(v_1, \dots, v_n)\) such that \(v_j\in \mathcal{B}_\alpha(\eta)\), \(\eta=1/2\alpha -1 +\sigma\). An analogous result is obtained for \(\alpha=1\). As an application, \(L^\infty(\mathbb{R}^{n+1}_+)\) estimates of higher-order tangential derivatives are given in terms of \(\|u\|_{\mathcal{B}_\alpha(\sigma)}\). In the final section, inversion theorems are proven.