By J. Garnett

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The next theorem is essentially proved in [76], but it may have been noticed even earlier. 5: compact set Assume that E and f(oo) is locally integrable, analytic off a f = O. c. is of bounded variation. V( 1') When this is the case the is attained for any grid C f· VI dR IJ. Ids < '" R for which -44- for all R E R. Moreover Ifil Cu) V(fj U) for a,ny open set U. 5 is analogous to a familiar result on functions of one variable: the function f on the rea,l line is of bounded variation if and only if there is a finite measure f(b) - f(a) = In this case replaced V~(f) ~ f(b) - f(a) f by -1/7T dV(x) '" v( [a,b) a and df/dx v.

Rhen can only vanish when E of positive E lies on a sufficiently E has zero length. 2 above and the sharpest result here is due to Ivanov [52]. Its intricate proof below is not used later and may be Skipped. Let o 5. 11. 1: (Ivanov) • Let r be a 2- r q:{s} = [, (s) : is real, jer(s) j ::: 1/2. 1) u 12 ~s) - cp(sO) . Is - s 0 I 0 where M depends only on that if is a subset of E r. ds

Cal Ijnes. 2 : ( Royden [70 J). e. 4. ReD 9,. In particular, if D. ,J R whose closures lie in some admissible grid analytic on r 1F'(z)lds < QQ, dR p Then F 1< is in Xp and for CoorD), R IE: ~ with small, we have by Fubini's theorem J"dR (F * X )(z)dz =JrdR'J1F(z P Ox p Jr (OclScl1)dz F(w)dwdSdn, z ° 2l(R-O because almost every translate now implies R - is analytiC on F ... 3: sat isfy Let f f( (0) = O. C € Let g, F of As D. is in 11,. ( z ) so that F = g almost everywhere. L~oc be analytiC off a compact set E and Il be a measure on E.