lecture7_001

1 proposition 12 let t s 0 be real numbers if u is l

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Unformatted text preview: er version of (4) holds for all t, not just when t/m is a power of 2. 1 Proposition 1.2 Let t > s > 0 be real numbers. If u is L harmonic on Bt then � �α � � t 2 u2 . u≥ 2s Bs Bt Proof Take 2m s ≤ t ≤ 2m+1 s. Since u2 is positive we have � � u. u≤ B 2m s Bt (5) (6) We can estimate this bound by (4), so 2mα Also 2 Plugging this into (7) gives � Bt mα � u≤ Bs � u. Bt (7) =2 (m+1)α −α s ≥2 −α � �α t . s u≥ as required. 2 � t 2s �α � Bs u2 (8) We can do exactly the same calculation for the Dirichlet energy to give Proposition 1.3 Let t > s > 0 be real numbers and let β = on Bt then � Bt log(1+d) log 2 . If u is L harmonic |�u | ≥ There is a nice corollary to this 2 � t 2s �β � Bs |�u|2 . (9) Corollary 1.4 Take u : Rn → R an L harmonic function. If � u2 < ∞ Rn then u = 0 on Rn . 2 Proof Suppose u is not identically zero, say u(x0 ) �= 0. Then � u2 = � > 0, B1 (x0 ) so � Bt (x0 ) u= which goes to infinity as t gets large. 2 � t 2s �α �, Now we will prove one of the two inequalities at the s...
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