A system of conjugate functions on parabolic Bloch spaces. (English) Zbl 1429.35200

The authors consider solutions of the parabolic operator \[ L^{(\alpha)}u = \frac{\partial u}{\partial t} - (\Delta_x)^{\alpha} = 0, \] where \(\Delta_x\) is the spatial Laplacian.
This is the heat equation when \(\alpha = 1\) and Laplace’s equation when \(\alpha = \frac12\). For Laplace’s equation, the notion of harmonic conjugates of the solution are well understood, and in [the authors, ibid. 65, No. 2, 487–520 (2013; Zbl 1327.35405)], the authors introduced \(\alpha\)-parabolic conjugate functions, with analogous properties for certain values of \(\alpha\). However, in general one is considering \(0 < \alpha \leq 1\), and the \(\alpha\)-parabolic conjugates exist for \(0 < \alpha \leq \frac12\), but not always for other values of \(\alpha\). To remedy this, they introduce \(L^{(\alpha)}\)-conjugates which exist for \(0 < \alpha \leq 1\) and have the expected properties.
\(H\) is the upper half-space \(\mathbb{R}^{n+1}_+ = \{ (x, t)| x \in \mathbb{R}^n, t>0 \}\), and the solutions are taken in the sense of distributions. The \(C^1(H)\) solutions are in \(\mathcal{B}_{\alpha}(\sigma)\) if the norm, \( || u ||_{\mathcal{B}_{\alpha}(\sigma)}\), with \(m(\alpha) = \min \{1, 1/2 \alpha\}, \sigma > - m(\alpha)\), is finite, where \[ || u ||_{\mathcal{B}_{\alpha}(\sigma)} := \sup_{(x, t) \in H} t^{\sigma} \left\{ t^{\frac{1}{2 \alpha}} |\nabla_x u(x, t)| + t |\partial_t u(x,t)| \right\} < \infty. \] \(\tilde{\mathcal{B}}_{\alpha}(\sigma)\) is the set of all \( u \in \mathcal{B}_{\alpha}(\sigma)\) with \(u(0,1) = 0\), and is the parabolic Bloch space.
In E. M. Stein and G. Weiss [Acta Math. 103, 25–62 (1960; Zbl 0097.28501)], an \(n\)-tuple \((v_1, \ldots, v_n)\) is a harmonic conjugate of \(u\), if \[ \partial_j v_k = \partial_k v_j, 1 \leq j,k \leq n \] \[ \partial_j u = -\mathcal{D}_t v_j, 1 \leq j \leq n, \] \[ \mathcal{D}_t u = \sum_{j=1}^n \partial_j v_j. \] where \(\mathcal{D}\) is a fractional derivative defined by a fractional integral. The \(\alpha\)-parabolic conjugates were defined by the same first two sets of equations, but with the third equation replaced by \(\sum_{j=1}^n \partial_j v_j = u - \lim_{t \to \infty} u(0,t) \) in the case of the heat equation (\(\alpha = 1\)) and equal to \(\mathcal{D}^{(1/\alpha) - 1}_t u\) if \(0 < \alpha < 1\). Finally, the \(L^{(\alpha)}\) conjugates are defined by the first equation, but with the last two equations replaced by \[ \partial_j u = -\mathcal{D}_t^{1/2 \alpha} v_j, 1 \leq j \leq n, \] \[ \mathcal{D}_t^{1/2 \alpha} u = \sum_{j=1}^n \partial_j v_j. \] Their main result is that if \(0 < \alpha \leq 1, \sigma > - m(\alpha)\), if \(u \in \tilde{\mathcal{B}}_{\alpha}(\sigma)\), there exist a unique \(L^{(\alpha)}\) conjugate \((v_1, \ldots, v_n)\) and a constant \(C\) such that \[ C^{-1} || u ||_{\mathcal{B}_{\alpha}(\sigma)} \leq \sum_{j=1}^n || v_j ||_{\mathcal{B}_{\alpha}(\sigma)} \leq C|| u ||_{\mathcal{B}_{\alpha}(\sigma)}, \] and that conversely if there are functions \((v_1, \ldots, v_n) \in \tilde{\mathcal{B}}_{\alpha}(\sigma)\) satisfying the first set of equations defining conjugates, there is a unique \(u \in \tilde{\mathcal{B}}_{\alpha}(\sigma)\) such that \((v_1, \ldots, v_n)\) is the \(L^{(\alpha)} \) conjugate of \(u\) with the estimate \[ C^{-1} \sum_{j=1}^n || v_j ||_{\mathcal{B}_{\alpha}(\sigma)} \leq || u ||_{\mathcal{B}_{\alpha}(\sigma)} \leq C \sum_{j=1}^n || v_j ||_{\mathcal{B}_{\alpha}(\sigma)}. \] Their final main result gives an isomorphism result for \( \tilde{\mathcal{B}}_{\alpha}(\sigma) \). If \(0 < \alpha \leq 1, \sigma_1, \sigma_2 > - m(\alpha)\), then \(\tilde{\mathcal{B}}_{\alpha}(\sigma_1) \cong \tilde{\mathcal{B}}_{\alpha}(\sigma_2) \) if \(\mathcal{D}_t^{-\sigma_1 + \kappa} u = \mathcal{D}_t^{-\sigma_2 + \kappa} v\) for \(u \in \tilde{\mathcal{B}}_{\alpha}(\sigma_1), v \in \tilde{\mathcal{B}}_{\alpha}(\sigma_2)\) for some \(\kappa > \max\{0, \sigma_1, \sigma_2\}\). Moreover, there is a constant \(C\) such that \[ C^{-1} ||v ||_{\mathcal{B}_{\alpha}(\sigma_2)} \leq ||u||_{ \mathcal{B}_{\alpha}(\sigma_1)} \leq C ||v||_{ \mathcal{B}_{\alpha}(\sigma_2)}. \] This is similar to a result they had given for the parabolic Bergman space in [the authors, Potential Anal. 40, No. 4, 525–537 (2014; Zbl 1293.35129)].


35R11 Fractional partial differential equations
26A33 Fractional derivatives and integrals
30H30 Bloch spaces
42B99 Harmonic analysis in several variables
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