## 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)].

### MSC:

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

### Citations:

Zbl 1327.35405; Zbl 0097.28501; Zbl 1293.35129
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