Frame Decomposition of Mixed-Smoothness Besov and Triebel-Lizorkin Spaces on Product Metric Spaces Associated to Operators

The purpose of this study is to develop a discrete frame decomposition of the Besov and Triebel-Lizorkin spaces on the product X₁× X₂ X 1 × X 2 of doubling metric measure spaces X₁ X 1 , X₂ X 2 associated with non-negative self-adjoint operators L₁ L 1 , L₂ L 2 , whose heat kernels have Gaussian localization. To achieve this, we first establish a pair of frames with sub-exponential spatial localization and compact spectral support. Some advances of independent interest in the theory related to product spaces, including the lower bounds of the Lᵖ L p norms of kernels, of cut-off functions are also obtained.