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Atomic and molecular decompositions of anisotropic Besov spaces. (English) Zbl 1079.42016

In this work the author introduces and studies Besov spaces in R n associated with an expansive dilation A. The starting point is a basic representation formula for tempered distributions

f= q𝒬 f,ϕ Q ψ Q ,where𝒬={A -j ([0,1] n +k):jZ,kZ n },

and ϕ Q and ψ Q are translates and dilates of functions ϕ and ψ that satisfy the Calderón conditions

suppϕ ^,suppψ ^[-π,π] n {0}

and

jZ ϕ ^((A * ) j ξ) ¯ψ ^((A * ) j ξ)=1forallξR n {0}·

This study extends the isotropic Paley–Littlewood methods of dyadic ϕ–transforms of M. Frazier and B. Jawerth [Indiana Univ. Math J. 34, 777–799 (1985; Zbl 0551.46018); J. Funct. Anal. 93, No. 1, 34–170 (1990; Zbl 0716.46031)] to nonisotropic settings.


MSC:
42B35Function spaces arising in harmonic analysis
42B25Maximal functions, Littlewood-Paley theory
42C40Wavelets and other special systems
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47B37Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47B38Operators on function spaces (general)
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