A conjugate system and tangential derivative norms on parabolic Bergman spaces (Q989429)

Let \(H=\{(x,t): x\in \mathbb R^n, t>0 \}\) be the upper half space of \(\mathbb R^{n+1}\). For \(0<\alpha \leq 1\), the parabolic operator \(L^{(\alpha)}\) is defined by \(L^{(\alpha)}= \frac \partial {\partial t} +(-\triangle_x)^\alpha \), where \(\triangle_x\) is the Laplacian on the \(x\)-space \(\mathbb R^n\). A real valued continuous function \(u\) on \(H\) is said to be a \(L^{(\alpha)}\)-harmonic if \(u\) satisfies \(L^{(\alpha)}u=0\) in the sense of distributions. For \(\lambda > -1\) and \(1\leq p < \infty\), the \(\alpha\)-parabolic Bergman space \(\mathbf b^p_\alpha(\lambda)\) is the set of \(L^{(\alpha)}\)-harmonic \(u\) on \(H\) with \(\|u\|_{L^p(\lambda)}:= (\int_H|u(x,t)|^pt^\lambda \,dV(x,t))^{1/p} < \infty,\) where \(dV\) is the Lebesgue volume measure on \(H\) and \(L^p(\lambda) := L^p(H,t^\lambda dV)\). For a function \(u \in \mathbf b^p_\alpha(\lambda)\), we shall say that a vector valued function \(\mathbf V = (v_1, \dots ,v_n)\) on \(H\) is an \(\alpha\)-parabolic conjugate function of \(u\) if \(v_j\in C^2(H)\) and \(\mathbf V\) satisfies the equations \[ \begin{aligned} \nabla_xu=\partial_t \mathbf V, \quad \nabla_xv_j&=\partial_j\mathbf V \qquad (1\leq j \leq n),\\ (-\partial_t)^{\frac 1 \alpha -1} u&= \nabla_x \cdot \mathbf V, \end{aligned} \] The main results of the paper under review are the following theorems of the existence and uniqueness of the \(\alpha\)-parabolic conjugate function and the tangential estimates of \(\alpha\)-parabolic harmonic functions. Theorem. Let \(0< \alpha \leq 1, 1\leq p <\infty, \lambda >-1\), and \(u \in \mathbf b^p_\alpha(\lambda)\). If \(\alpha,p\) and \(\lambda\) satisfy the condition \(\eta=p(\frac 1{2\alpha} -1)+\lambda >-1\), then there exists a unique \(\alpha\)-parabolic conjugate function \(\mathbf V=(v_1,\dots ,v_n)\) of \(u\) such that \(v_j\in \mathbf b^p_\alpha(\eta)\). Also there exists a constant \(C=C(n,p,\alpha,\lambda)>0\) independent of \(u\) such that \[ C^{-1}\|u\|_{L^p(\lambda)}\leq \| |\mathbf V| \|_{L^p(\eta)}\leq C\|u\|_{L^p(\lambda)} . \] Theorem. Let \(0< \alpha \leq 1, 1\leq p <\infty, \lambda >-1\), and \(u \in \mathbf b^p_\alpha(\lambda)\). Then for each \(m\in \mathbb N_0\), there exists a constant \(C=C(n,p,\alpha,\lambda,m)>0\) independent of \(u\) such that \[ C^{-1}\|u\|_{L^p(\lambda)} \leq \sum_{|\gamma|=m}\|t^{\frac m{2\alpha}}\partial^\gamma_x u\|_{L^p(\lambda)}\leq C\|u\|_{L^p(\lambda)}. \] The present work is a generalization of the previous work of \textit{W. Ramey} and \textit{H. Yi} [Trans. Am. Math. Soc. 348, No. 2, 633--660 (1996; Zbl 0848.31004)], where they consider the case \(\alpha = \frac 12, \lambda = 0\). Furthermore, the authors also prove more relations between \(u\) and its \(\alpha\)-parabolic conjugate \(\mathbf V\) in the following theorems. Theorem. Let \(0< \alpha \leq 1, 1\leq p <\infty, \lambda >-1\), and \(u \in \mathbf b^p_\alpha(\lambda)\). Then a vector valued function \(\mathbf V=(v_1,\dots ,v_n)\) on \(H\) is an \(\alpha\)-parabolic conjugate function of \(u\) if and only if there exists a function \(g\in C^2(H)\) such that \(g(x,\cdot) \in \mathcal FC^{\frac 1\alpha}\) for each \(x\in \mathbb R^n\) and \[ ((-\partial_t)^{\frac 1\alpha}+\triangle_x)g=0\;\text{ on } H\quad \text{and}\quad \nabla_{(x,t)} g =(v_1,\dots ,v_n,u). \] Theorem. Let \(0< \alpha \leq 1, 1\leq p <\infty, \eta >-1\), Suppose that a vector valued function \(\mathbf V=(v_1,\dots ,v_n)\) on \(H\) satisfies \(v_j\in \mathbf b^P_\alpha(\eta)\) and \(\nabla_xv_j=\partial_j\mathbf V\) for all \(1\leq j \leq n\). If \(\alpha,p\) and \(\eta\) satisfy the condition \(\lambda=p(\frac 1{2\alpha} -1)+\eta >-1\), then there exists a unique function \(u\) on \(H\) such that \(u \in \mathbf b^p_\alpha(\lambda)\) and \(\mathbf V\) is an \(\alpha\)-parabolic conjugate function of \(u\). Also, there exists a constant \(C=C(n,p,\alpha,\eta)>0\) independent of \(\mathbf V\) such that \[ C^{-1}\||\mathbf V |\|_{L^p(\eta)} \leq \|u\|_{L^p(\lambda)} \leq C\||\mathbf V |\|_{L^p(\eta)}. \]