By Bilusyak N. I., Ptashnyk B. I.

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Extra resources for A Boundary-Value Problem for Weakly Nonlinear Hyperbolic Equations with Data on the Entire Boundary of a Domain

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17) @~,i take on values in the Clifford group of N(~). 1. 0 ยง 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.

6 we have xE0,,/21-) = L L XR"\2[O,1/2]n)E ~,~

Similar estimates are also valid for C~. We shall postpone the proof of this for a moment and first derive some of its consequences. First recall the following version of Schur's test. 2. Assume that the rows and the columns of an infinite matrix A = (aij)ij satisfy Wi-1Z [aij[wj <~ C < +oo for each i, J and wT1 Z [alj[wi < C < +oo for each j, i for some constant C > 0 and some positive numbers (wj)j. Then A defines a bounded operator on g2(N) with norm _< C. 1 immediately yields the following.

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