El-Fallah O., Kellay K., Mashreghi J., Ransford T.'s A Primer on the Dirichlet Space PDF

By El-Fallah O., Kellay K., Mashreghi J., Ransford T.

ISBN-10: 1107047528

ISBN-13: 9781107047525

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En−1 }. If x ∈ E and t < n , then B(x, t) meets Enj for only one j, so μ(B(x, t)) ≤ 1/2n . Consequently K( IK (μ) ≤ K( 1 ) + n≥1 − K( n ) ≤ 2n n+1 ) n≥0 K(en ) , 2n the final inequality because K( n+1 ) − K( n ) is either zero or K(en ) − K( n ). By definition, 1/cK (E) ≤ IK (μ). 3). 3 1. Let E be the Cantor set as constructed above. Prove that there exists a Borel probability measure μ on E such that μ(Enj ) = 1/2n for all j, n. Under what circumstances is μ unique? 4 Logarithmic capacity Though the notion of capacity has been developed in some generality, we shall be interested mostly in the following special case.

3 (iii), we have n cK (F) ≤ n cK (Fk ) ≤ k=1 cK (Uk ) ≤ k=1 cK (Uk ). k≥1 As this holds for each such F, we obtain cK (∪k Uk ) ≤ k cK (Uk ), proving the result in this case. For the general case, we may suppose that c∗K (Ek ) < ∞ for all k, otherwise there is nothing to prove. 2 Equilibrium measures 19 that Uk ⊃ Ek and c(Uk ) < c∗ (Ek ) + /2k . Then ∪k Uk is an open set containing ∪k Ek and, by what we have already proved, cK (∪k Uk ) ≤ k Thus c∗K (∪k Ek ) ≤ ∗ k cK (E k )+ c∗K (Ek ) + . cK (Uk ) ≤ k .

Fix α with 0 < α < 1. 3. (i) Show that there exists a constant Bα > 0 such that, for all ζ1 , ζ2 ∈ T, 1 π A Bα (|w|2 − 1)−α dA(w) ≤ . |w − ζ1 ||w − ζ2 | |ζ1 − ζ2 |α [Hint: By symmetry, it is enough to estimate the integral on the set A := A ∩ {w : |w − ζ1 | ≤ |w − ζ2 |}. 3. Show that there exists a constant Aα > 0 such that, for all g ∈ Lα2 (A), cα (Cg > t) ≤ Aα g 2 /t2 Lα2 (A) (t > 0). (iii) Deduce that, if f ∈ Dα , then it has a non-tangential limit at each point of T outside a set of outer cα -capacity zero.

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A Primer on the Dirichlet Space by El-Fallah O., Kellay K., Mashreghi J., Ransford T.


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