Download Approximation by Solutions of Partial Differential Equations by D. H. Armitage (auth.), B. Fuglede, M. Goldstein, W. PDF

0 implies II O~ - gwIlLq,w'(D) --+ 0, where, as usual, w' = Wl/(I-p). Extend 9 to be identically equal to zero on DC, and fix f > O. Let E = {z E DC such that IC(gw)(z)1 > flIT} and let gn = ~(gw - &;:;).

P, M+ (E) is the set of (positive) Borel measures supported on E, IlJilh is the total variation of J1 and * denotes the operation of convolution. Finally, the weighted Riesz capacity is defined to be: Ri,p(E):= inf{jU(z))Pw(z)d>'2(z): f ~ 0 and I~I * f ~ Ion E} REMARK 2. It should be noted that for certain weights and certain values of p, these capacities are trivial (see Section 4: Capacities Revisited). REMARK 3. In the definition of i;, we can assume that the function f vanishes at 00 (and thus is analytic at 00).

_ WEIGHTED LP APPROXIMATION BY HOLOMORPHIC FUNCTIONS 43 9. The Annihilator of AP,W(D) Let D be a bounded open subset of the complex plane. We would now like to describe the annihilator of the space AP,W(D). Recall that AP,W(D) is the space of those functions in Lf~:(D) which are analytic on D. l if 9 E Lf~:(D) and JD f(z)g(z)w(z)d>'2(z) = 0 for all f E AP,W(D). PROPOSITION 2. 3]) Let Dee be bounded and open and let 9 E Lf~:(D). e. ,G)-capacity. Proof (see [12]): (4) ~ (3). w- l )wd>'2 = 0, so (4) implies (3).

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