We deal with homogeneous Besov and Triebel-Lizorkin spaces in the setting of a doubling metric measure space in the presence of a nonnegative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. The class of almost diagonal operators on the associated sequence spaces is developed and it is shown that this class is an algebra. The boundedness of almost diagonal operators is utilized for establishing smooth molecular and atomic decompositions for the above homogeneous Besov and Triebel-Lizorkin spaces. Spectral multipliers for these spaces are established as well.
ABSTRACT. Homogeneous mixed-norm Triebel-Lizorkin spaces are introduced and studied with the use of a discrete wavelet transformation, the so-called ϕ-transform. This extends the classical ϕ-transform approach introduced by Frazier and Jawerth to the setting of mixed-norm spaces. Moreover, the theory of the ϕ-transform is enhanced through a precise definition of the synthesis operator, in terms of a Pettis integral, and a number of rigorous results for this operator. Especially its terms can always be summed in any order, without changing the resulting distribution.
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