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

Abstract The purpose of this study is to develop a discrete frame decomposition of the Besov and Triebel-Lizorkin spaces on the product $$X_1\times X_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> of doubling metric measure spaces $$X_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> , $$X_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> associated with non-negative self-adjoint operators $$L_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> , $$L_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> , 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 $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> norms of kernels, of cut-off functions are also obtained.