# Eric T. Sawyer's Function Theory: Interpolation and Corona Problems (Fields PDF

By Eric T. Sawyer

ISBN-10: 0821847341

ISBN-13: 9780821847343

Those lecture notes take the reader from Lennart Carleson's first deep effects on interpolation and corona difficulties within the unit disk to trendy analogues within the disk and ball. The emphasis is on introducing the various array of options had to assault those difficulties instead of generating an encyclopedic precis of achievements. concepts from classical research and operator idea contain duality, Blaschke product structures, merely Hilbert area arguments, bounded suggest oscillation, most sensible approximation, boundedness of the Beurling remodel, estimates on suggestions to the $\bar\partial$ equation, the Koszul advanced, use of timber, the whole decide estate, and the Toeplitz corona theorem. an in depth appendix on history fabric in practical research and serve as conception at the disk is integrated for the reader's comfort.

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Extra info for Function Theory: Interpolation and Corona Problems (Fields Institute Monographs 25)

Sample text

K=l Wk Then Ur,e,N = Iii solves a crzUr,e,N(Z) = d/-Lr,e,N(Z) with uniform LOO(R) bounds. 18) in some suitably useful way. Of course we will let r, IS -+ 0 and N -+ 00 28 2. 21) hold. At this point it is convenient to pass from the unbounded upper haIf plane to the compact disk. Then Ur,g,N is a rational function in the disk and by Green's Theorem (or the residue theorem) we have for each J E Hl(]]}), 1. -2 7r~ iTf J(z)urg' ,N(z)dz 1 () ~l -1. 45' (p. 167) in Appendix A shows that there is a sequence of these restrictions that converges weak star to a bounded function U on 'JI'.

23). (Fk) = 4~~(Fk) = 4~ (OF k) = 4~(Gk), OZ Oz oz Oz OZ and Green's Theorem in the form h (uvn - unv) = I1 (u L. 4. Other corona probleDls with u = Fk and v = log~, now yields using with 6v = 80 and k(O) = 0, l k o F(eiO)k(e iO )_ • 21r IJl ~ Jl - 21r Vn = 1 and v = 0 on 'll', together 6(F(z)k(z)) log -I 1 Idxdy Z ITJ) k'(z)G(z) log 2J rk(z) 8z (z) +; I 8G JITJ) 1~ldXdY 1 log ~dxdy + II. 8 (p. 18). 17). 11) with fR in place of f, and where [[aijlloc ::; 0 independent of R. 9) with fR in place of f and with IlgRIIHoc(ll» ::; C(8, IIfIIHoo(][j))).

11) with z(l) = x + i2£Y: L c- 1 L 4yj(yj~;)+C-1f Iz - z J ET(z(2» Zj 1 c -1 Iz - £=2 zjET(z(l»\T(z(l-l» ~S~ < C 4YJ(YJ~;) ZJ 1 ~Sh oc Y.. + Y ~ . , we set In(z) = :nn(~~)gn(z)e-a{Gn(z)-G,,(zn)}, for a positive constant Q to be chosen momentarily. 16). 4. Peter Jon_' conatructive proof of interpolation with a = "'(Z)' since if the Zj are ordered with decreasing Yj and tn for any fixed z, then = L: Yj :-:;Yn 1~:tt~1 Thus we have obtained (2 16) with constant C = e /'::,. (Z) (see [43] for a discussion of sharpness).