The Bochner-Martinelli Integral and Its Applications by Alexander M. Kytmanov

0+ J 8D + (-l)P+q +! p(z) /\ Un-p,n-q-l(Z,O ,(C) /\ ! p(z)/\Un- p,n- q-2(z,O]. p has type (n, n - 1)). It CHAPTER 1. 10) for f E C(aD x D_< \ D<) (since the coefficients of the forms under the integral sign in the last formula are of this type).

Each r~alline parallel to the normal to T z and passing through a point intersects 8D at one point, which we denote by ((u); U E B' 2. Isin,61 2: 1/2 when U E B', where,6 is the angle between the normal to 8D at z and the line passing through z and (( u); 3. I((u) - zl < E for U E B'; 38 CHAPTER 1. THE BOCHNER-MARTINELLI INTEGRAL 4. w(I((u) - zl) < E for u E B'j 5. I((u) - ul < Elul for u E B'j 6. lu - z±1 :s: 21((u) - z±1 for u E B'. 1, and the radius d of the ball B' can be chosen to be independent of z.

Therefore T and Ware bounded in /2 2 (5) with IITII = IIWII = 1. The spectra of T and Ware discrete (in contrast to the spectrum of M), with the single limit point 0 for T and 1/2 for W. We remark that if we denote by Wu twice the singular double-layer potential: x E 5, then, by the jump formula for this potential, Wuf Wuf =~ Un f on 5, that is, r f(y)(l-Ix2(x,y) + IYI2) du = Tf. 2 = 2Wf - = 3 they are given in [192, chap. 5, §1]). 7). The computation of M f for an arbitrary function fin /2 2 (5) can be reduced to the calculation of a one-dimensional integral.

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