By Marius Mitrea
The publication discusses the extensions of simple Fourier research ideas to the Clifford algebra framework. issues coated: building of Clifford-valued wavelets, Calderon-Zygmund thought for Clifford valued singular necessary operators on Lipschitz hyper-surfaces, Hardy areas of Clifford monogenic services on Lipschitz domain names. effects are utilized to power thought and elliptic boundary price difficulties on non-smooth domain names. The booklet is self-contained to a wide quantity and well-suited for graduate scholars and researchers within the parts of wavelet idea, Harmonic and Clifford research. it's going to additionally curiosity the experts fascinated about the purposes of the Clifford algebra equipment to Mathematical Physics.
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Additional info for Clifford Wavelets, Singular Integrals, and Hardy Spaces 1994
Sample text
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). We also set XQ for the characteristic function of Q. 11). e. a system of Haar Clifford wavelets. In fact, for this particular context, we can perform our construction in a slightly more general form. e. 14) for any dyadic cube Q in ]l~m; here IQI denotes the Euclidean volume of Q. Note that in this case, by Lebesgue's differentiation theorem, one has IIb-lllL~ ~_ 5-1.
3. The operators C L and C~ are bounded on L2(~;~n)(n), with bounds independent of & Using the Poisson-like decay of the kernel of Cfi - cL~ one can readily see that (C L - cL_~)f --~ f in L p as 6 ~ cL(cL--cL_~)f = c L + J 0, for any f E L2(]Rn)(~). ) c L f in L 2 as 6' ~ Moreover, since 0, we see that the limit lim~-+o C ~ f exists for any f E L2(llC~)(n). In fact, the next theorem (whose proof will be completed in the next chapter) shows that we can be very precise about the limit operator. , and H~f(X) :: - 2 fix - r _yl>_~ Y EY3 v_=x XI + l n ( Y ) I ( Y ) es(yl, Y- X ~, where dS is the canonical surface measure on E.
S k e t c h o f P r o o f . g. [Jo]) to this somewhat more general context, therefore we consider the sufficiency. In fact, it is enough to treat only the special situation when T(bl) = Tt(b2) = 0 (in which case, the theorem is actually due to McIntosh and Meyer [MM]). The general case, T(bl), Tt(b2) e BMO(Rn)(m) reduces to the previous one by subtracting off some paraproduct-like operators. More precisely, setting 171(f) := E k AL,21TIb ~EL, 1 ib-lt~ k ~, k 1)) k-lk 1 J) and //2(f) :-'- E t (b2))Ek_l(b L,2 2-1 f), /~kL , l(T k where AL'i ~ k and E kL'i are defined as in section w with respect to bl, i = 1,2, we would like to conclude that the operator S := T - 171 - 172t falls within the scope of the special case we have mentioned first.