Radial averaging of domains, estimates for Dirichlet integrals and applications

Abstract

RADIAL AVERAGING OF DOMAINS, ESTIMATES FOR DIRICHLET INTEGRALS AND APPLICATIONS by Moshe Marcus Let & = {D,,...,D } be a family of domains in the plane, containing the origin. We define a radial averaging transformation R. on S by which we obtain a starlike domain D . When & is such that the domains D,,...^ are obtained from a fixed domain D by rotation or reflexion, ft. becomes j\ a radial symmetrization. One of the results we present is an inequality relating the conformal radius of D to the conformal radii of D, ,...,D at the origin. This result includes, as particular cases, the radial symmetrization results of Szego [11] (for starlike domains), Marcus [7] (for general domains) and Aharonov and Kirwan [1]. The inequality for the conformal radii is obtained via an inequality for conformal capacities, which seems to be of independent interest. A number of applications in the theory of functions are discussed. Here we introduce a definition of a class of functions {f}, analytic in the unit disk | §| 0 the set CL (f^) is open (relative to M) and that for 0 0.) We now keep j fixed. Let P1 = (x- y-Jefi . Kt(f.) and x x x a ^ D j f . ( P ) = A , . Denote by Kc(P,) the open d i s k of r a d i u s 6, c e n t e r e d at I>1. If 0 X ("X A ) 4r6 (x -x ) Since this holds for every j, we obtain (under the same assumptions) (1-9) **(x ,A ) ^l*(x ^ ) + [S (x.-x ) ] 1 / . From (1.9) it follows that: (1.10) |f*(P)-f*(P' ) | (i = 1,2), in na, b,(f*) such that: (1.11) |PX-P2 | = 6 k6. Suppose f*(P1) A2 and A + k5 £*(x2,A2). On the other hand inequality (1.9) holds for these values of X T > X 2 > ^ T * ^ 9 ' H e n c e we obtain: 2 > l + t 6 -( i2 ) 2 ] 1 / 2 -* I1"2 > ' which is a contradiction to (1.11). Definition 1.3. Let feB(M) and denote: = {(x5y) |0 < f(x,y) < 1} PI M; YA(f) = {(x,y) |f(x,y) = A} 0 M, 0 < A < 1. A Suppose that f€C(Q(f)). Let PQ = (xQ,y0) be an interior point of Cl(f) and f (PQ) = AQ. We shall say that PQ is a regular point of f, if Sf/5y j4 0 at all the points of the set Y-x PI {x = xQ} and if this set is contained in the interior of fi(f). Otherwise we shall say that PQ is a critical point and AQ a critical value of f on x = x . Lemma 1.2. Let feB(M) n C(n(f)). Suppose that AQ, (0 < 7\Q < 1) , is not a critical value of f on x = x , (0 < xQ < 1) . Then l(x,~k;±) eC in a neighborhood of (xQ,A ) . Proof. Since feB(M), y-v intersects the line x = xn at a 0 finite number of points {p. , . . . ,P.} . Let P. = (x ,y.) and suppose that y, < y2 <...< y,. Then the sequence has alternating signs. Let y. = y.(x,A) be the inverse function (with respect to *j «J y) of A = f(x,y), in a neighborhood of p.. Then for