The Leibniz rule for the Dirichlet and the Neumann Laplacian (Q6155666)

The author proves results about the behavior of products of functions in Sobolev and Besov spaces. The functions are not arbitrary, but are solutions of an elliptic equation with specified boundary values of Dirichlet or Neumann type. The paper would be strengthened if there were remarks on how these results differed from the results for products of arbitrary functions belonging to the same spaces. The author studies bilinear estimates of the form \[ \| f g\|_{\dot{H}^s_p} \leq C \| f \|_{\dot{H}^s_{p_1}} \| g \|_{L^{p_2}} + \| f \|_{L^{p_3}}\| g \|_{\dot{H}^s_{p_4}} \] where \(s >0\) and \(p, p_j \) \(( j = 1, 2, 3, 4)\) satisfy \(\frac{1}{p} = \frac{1}{p_1} + \frac{1}{p_2}= \frac{1}{p_3} + \frac{1}{p_4}\). The domain is the half space \(\mathbb R^n_+ = \{x \in \mathbb R^n : x_n > 0 \}\), and \(f , g\) satisfy boundary conditions of either Dirichlet or Neumann type. Such inequalities for the Besov spaces are also studied. The main result in the bilinear case reads as follows. Let \(A_D = - \Delta|_D\), \(A_N = - \Delta|_N\) denote the Dirichlet Laplacian (Laplacian with Dirichlet boundary conditions) and the Neumann Laplacian. The author defines the test function spaces, Sobolev spaces and Besov spaces as in his previous joint paper [\textit{T.~Iwabuchi} et al., Bull. Sci. Math. 152, 93--149 (2019; Zbl 1427.46024)] using spectral properties of \(e^{-tA_D}\), \(e^{-tA_N}\), and the test function spaces using the familiar dyadic decomposition based on a \(C^{\infty}_0\) function \(\phi_0\) with \[ \mbox{ supp } \phi_0 \subseteq \{ \lambda : 2^{-1} \leq \lambda \leq 2 \} , \quad \sum_{j \in \mathbb Z} \phi_0(2^{-j} \lambda) = 1, \mbox{ for } \lambda > 0. \] in the homogeneous case with an additional non-negative function \(\psi \in C^{\infty}_0(\mathbb R) \) such that \[ \psi(\lambda) + \sum_{j \in \mathbb N} \phi_0(2^{-j} \lambda) = 1, \mbox{ for any } \lambda \geq 0. \] in the inhomogeneous case. A typical bilinear result is given in Theorem 1.2. Suppose the \(p, p_j\) satisfy \[ 1 < p, p_1, p_4 < \infty, \quad 1 < p_2, p_3 \leq \infty, \quad \frac{1}{p} = \frac{1}{p_1} + \frac{1}{p_2} = \frac{1}{p_3}+ \frac{1}{p_4} . \] For the Dirichlet case, let \(A = A_D\), \(0 < s < 2 + \frac{1}{p}\). Then there exists a constant \(C>0\) such that for every \(f \in {\dot{H}^s_{p_1}(A_D)}\cap L^{p_2}(\mathbb R^n_+) \cap {\dot{H}^s_{p_4}(A_D)}\), \[ \| f g\|_{\dot{H}^s_p(A_D)} \leq C \| f \|_{\dot{H}^s_{p_1}(A_D)} \| g \|_{L^{p_2}} + \| f \|_{L^{p_3}}\| g \|_{\dot{H}^s_{p_4}(A_D)} . \] There is a similar result for the Neumann Laplacian and if \(s \geq 2 + \frac{1}{p}\), the inequality in the Dirichlet case does not hold. Similar results are given for Besov spaces as well as multilinear versions in the case of an odd number of products.