Multivariate spectral multipliers for tensor product orthogonal expansions
For \(n = 1, 2, \dotsc, d\), consider the eigenfunction expansion in \(L^2(A_n, \nu_n)\) of a positive self-adjoint operator \(\mathcal{L}_n\) with eigenvalues \(0 < \lambda_0^n < \lambda_1^n < \dots\), i.e., \[ \mathcal{L}_n f = \sum_{r=0}^\infty\lambda_r^n\langle f, e_r^n \rangle e_r^n, \] where \(\{e_r^n\}_{r=0}^\infty\) is an orthonormal basis consisting of eigenfunctions of \(\mathcal L_n\). Let \(A = A_1 \times A_2 \times \dotsb \times A_d\), \(\nu = \nu_1 \times \nu_2 \times \dotsb \times \nu_d\). If \(m\) is a bounded function of \(d\) variables, define \[ m(\mathcal{L})f = m(\mathcal{L}_1, \dotsc, \mathcal{L}_d ) f = \sum_{k \in \mathbb N^d} m(\lambda_{k_1}^1, \dotsc, \lambda_{k_d}^d) \langle f, e_k \rangle e_k, \] where \(k=(k_1,\dotsc,k_d)\) is a multi-index and the \(e_k=e^1_{k_1}\times\dotsb\times e^d_{k_d}\) form an orthonormal basis and \(f \in L^2(A, \nu).\) This paper is concerned with \(L^p\)-boundedness of the operator \(m(\mathcal{L})\) under certain assumptions on the norm growth of the imaginary powers of \(\mathcal{L}_n\). The author develops a multidimensional analogue of \textit{S. Meda}'s Mellin transform approach [Proc. Am. Math. Soc. 110, No. 3, 639--647 (1990; Zbl 0760.42007)] to prove the main results.
scientific article; zbMATH DE number 6103711
scientific article