Clifford Wavelets, Singular Integrals, and Hardy Spaces by Marius Mitrea

By Marius Mitrea

The ebook discusses the extensions of uncomplicated Fourier research ideas to the Clifford algebra framework.
Topics coated: building of Clifford-valued wavelets, Calderon-Zygmund idea for Clifford valued singular essential operators on Lipschitz hyper-surfaces, Hardy areas of Clifford monogenic services on Lipschitz domain names. effects are utilized to capability thought and elliptic boundary worth difficulties on non-smooth domain names. The ebook is self-contained to a wide volume and well-suited for graduate scholars and researchers within the parts of wavelet thought, Harmonic and Clifford research.
It also will curiosity the experts involved in the functions of the Clifford algebra equipment to Mathematical Physics.

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Assume now that x E QJ for some j . i=/X_Yl>a+~X_Yl

As in the general case, /l k --: Ek+l - Ek. Moreover, since 2m--1 aks = ( f , hQ)hQ, ' ' Z QEDCk i = 1 we have that +oc ~,~ IAkfl 2dz = Ilfll~, SC L2(Rm)(r~). 19). Note that E L f = Ek(fb)Ek(b) -1. 24) < IAk(fb)l + IEk(ib)l]Ek+l(b) -1 - Ek(b)-ll < IA#<(/b)l + IEk(fb)llAk(b)l, since IE~(b)l ~ 1. 24) show that it is enough to prove that +c~ L IEk('fb)121Ak(b)12dx< The proof of this is more or less standard. 25) We shall follow Christ [Ch] with very minor alterations. We make the following definition.

With the above notations, for all j, k, e we have P r o o f . The constant function 1 belongs to the L2(R '~ e-~l~ldx)(n)-closure of Vk and, consequently, everything follows from B ( X L, Vk) = B(Vk, X~) = O. 9 Exercise. g. [St]). ~ ]IBI]bH~ ~ IlCllbH~. In particular, {O~,j,k}~,j, L k and {O~j,k}~,j,a are unconditional basis for bHl(R~)(n). Remarks. L (~R t 1 | C(,~) for all e,j,k. (1) Since X L, Xff C Vk+l, we have that OE,j,k, e,j,k E Vg+ (2) Using the exponential decay of the O's as before, we can get higher order vanishing moments for O's provided the initial multiresolution analysis is 29 suitably chosen.

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