By Marius Mitrea

ISBN-10: 0387578846

ISBN-13: 9780387578842

The publication discusses the extensions of easy Fourier research options to the Clifford algebra framework.Topics coated: development of Clifford-valued wavelets, Calderon-Zygmund thought for Clifford valued singular imperative operators on Lipschitz hyper-surfaces, Hardy areas of Clifford monogenic capabilities on Lipschitz domain names. effects are utilized to power thought and elliptic boundary worth difficulties on non-smooth domain names. The ebook is self-contained to a wide quantity and well-suited for graduate scholars and researchers within the components of wavelet thought, Harmonic and Clifford Analysis.It also will curiosity the experts fascinated with the purposes of the Clifford algebra equipment to Mathematical Physics.

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For every b in BMO(Rm)(n), {]Ak(b)]2}lr is Car]eson, with norm not exceeding (a multiple of)]lbil2Mo . 18. Suppose w = {wk}k is Carleson. Then, for any 1 < p < o% uniformly for f E Lv(~m)(n). 14 is therefore complete. 9 For the reader's convenience, we shall include the proofs of these lemmas. 17. We need to estimate for an arbitrary, fixed, dyadic cube Q. Since Ak annihilates constants, we may suppose that bQ, the integral mean of b over Q, is in fact zero. Write b E BMO(~m)(n) as b = b0 + boo with bo = XQb, and boo = XR~\Qb.

0 § The three Haar Clifford wavelets living in the same dyadic cube for m = 2. The main result of this section is the following. 14. , 2 m - 1; (4) fRm e~,dz)b(=)e~,,~,(=) dz = ~Q,Q,~,~,, for ali Q, Q', i, i', (5) { eLQ,i}Q,i is a left-Riesz basis for L2(Nm)(n) and {e~,i}Q,i is a right-Riesz basis for L2(~m)(n). 33 P r o o f . The only thing that we still have to check is (5). , 2 m - 1, is a left-Riesz basis for X L uniformly in k E Z. Restricting our attention to one dyadic cube Q E ~k and using the explicit expressions of the | we readily see that XQ2 is spanned by XQ~ and 04,1 in the set of C(n)-valued functions on R'* with its natural structure as a left Clifford module.

However, it is natural to try to prove this without relying on the rotation method, working directly on the surface and the first results in this direction are due to Coifman, Murray and McIntosh ([Mu], [Mc]). To explain our approach, we first need to introduce some notation. We shall work in the Euclidean space IRn+l canonically embedded in the Clifford algebra ]R(n). e. for some M > 0 one has [g(x) - g(Y)l <- Mix - Yl, for all x, y 9 IRn, and denote by E its graph: r, := { x = (g(~), 5) ; ~ 9 IRn} g ~t~+l ~ R(~).

### Clifford Wavelets, Singular Intervals and Hardy Spaces by Marius Mitrea

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