By Marius Mitrea

The e-book discusses the extensions of simple Fourier research suggestions to the Clifford algebra framework.

Topics lined: building of Clifford-valued wavelets, Calderon-Zygmund concept for Clifford valued singular critical operators on Lipschitz hyper-surfaces, Hardy areas of Clifford monogenic features on Lipschitz domain names. effects are utilized to power concept 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 parts of wavelet thought, Harmonic and Clifford research.

It also will curiosity the experts thinking about the functions of the Clifford algebra equipment to Mathematical Physics.

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**Additional resources for Clifford Wavelets, Singular Integrals, and Hardy Spaces**

**Example text**

E. g. the one associated to the Haar system. One cannot conclude here because of the lack of regularity of the Haar system, but a suitable perturbation of this special case will do. w HAAR CLIFFORD WAVELETS First let us introduce some notation. , m}, v E Z m, with side-length l(Q) := 2 -k, and set 9c := k~Z~-k. Each dyadic cube Q e ~" has 2m "children" {QJ}2~ 1 :-- {Q' e ~-k+l ; Q' c Q). For Q cube and A positive constant, AQ will stand for the cube having the same center as Q and side-length The CMRA of )~l(Q).

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. 23) and the J o h n Nirenberg inequality, IQI-' SQ ~ fR-, ~ IA,(b0)(-)l'dIbl'

Similarly, we define 7)(m)bl, i = 1, 2. Consider T a continuous morphism of right-Clifford-modules from bl:D(m) into (:D(m)b2)* (the duality sign refers to the left-Clifford-module structure of 7)(re)b2), and say that T is associated with a standard kernel if for some Clifford algebra valued continuous function K(x, y) defined for x # y, x, y E R n, and for some number 5, 0<5_