Polar wavelets and associated Littlewood-Paley theory

Epperson Jay ; Frazier Michael

GDML_Books, (1996), p.

Abstract

Abstract We develop an almost orthogonal wavelet-type expansion in ℝ² which is adapted to polar coordinates. We start by defining a product Fourier-Hankel transform f̂ and proving a sampling formula for f such that f̂ is compactly supported. For general f, the sampling formula and a partition of unity lead to an identity of the form $f = ∑_{μ,k,m}⟨f,φ_{μ km}⟩ψ_{μkm}$, in which each function $φ_{μkm}$ and $ψ_{μkm}$ is concentrated near a certain annular sector, has compactly supported product Fourier-Hankel transform, and is smooth away from the origin. We introduce polar function spaces $A^{α q}_{p}$, analogous to the usual Littlewood-Paley spaces. We show that $A^{02}_{p} ≈ L^{p}$, 1 < p < ∞. We prove that $f ∈ A^{α q}_{p}$ if and only if a certain size condition on the coefficients ${⟨f,φ_{μkm}⟩}_{μ,k,m}$ holds. A certain class of almost diagonal operators is shown to be bounded on $A^{αq}_{p}$, which yields a product Fourier-Hankel transform multiplier theorem. Using this, we identify a polar potential operator $P^α$ which maps $A^{βq}_{p}$ isomorphically onto $A^{α+β,q}_{p}$.

CONTENTS 1. Introduction and main results...............................................................5 2. Preliminaries.......................................................................................12 3. The sampling theorem and polar wavelet identity...............................25 4. Boundedness of almost diagonal matrices on $a^{αq}_{p}$...............27 5. Peetre’s maximal inequality.................................................................31 6. Norm characterizations.......................................................................35 7. FHT multiplier and potential operators................................................39 8. Equivalence of $L^p$ and $A^{02}_p$, 1 < p < ∞...............................42 9. Conclusion..........................................................................................49 References.............................................................................................50

Détail

Zbl 0861.42008

EUDML-ID : urn:eudml:doc:275829