I. Relationship with previous work

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Amie Mitchell
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9 i t

10 j J t = {0, 1,...J t } j t (p jt, x jt, ξ jt ) p jt R + x jt R k k ξ jt R ξ t T j = 0 t (z i, ζ i, G i ), ζ i z i R m G i G i (p j, x j ) i j U(z i, ζ i, x j, p j, ξ j ; G i ) = u(ζ i, x j, p j, ξ j ; x r, p r )dg i (x r, p r ) u(s i, ζ i, x j, p j, ξ j ; x r, p r ) r r i j u(z i, ζ i, x j, p j, ξ j ; x r, p r, ξ r ) = υ(z i, ζ i, x j, p j, ξ j ) + η(x j, p j ; x r, p r ) ξ it x ij

11 υ(z i, ζ i, x j, p j, ξ j ) j η(x j, p j ; x r, p r ) (x r, p r ) η(x j, p j ; x r, p r ) = 0 r η(x j, p j ; x r, p r ) = η p (p j ; p r ) + η x (x j ; x r )

12 n p (p j ; p r ) = µ p (p r p j ) µ p : R R µ p (x) x µ p (0) = 0 µ p (x) y > x > 0 µ p (y) + µ p ( y) < µ p (x) + µ p ( x) η x (x j ; x r ) = µ x ( d(x j, x r )) d(, ) : R k R k R µ x ( )

13 Y i = (z i, ζ i, G i ) P Y A j (x j, p j, ξ j ) = {Y : U(z i, ζ i, x j, p j, ξ j ; G i ) > U(z i, ζ i, x n, p n, ξ n ; G i ) n J}, j J t σ j (p, x, ξ) = P Y (dy) A j (x j,p j,ξ j ) (x j, p j, ξ j ) supp(g) (x r, p r, ξ r ) G i (x r, p r ) = 1 i G i = G a.e.

14 U(z i, ζ i, x j, p j, ξ j ; x r, p r ) = υ(z i, ζ i, x j, p j, ξ j ) + µ x ( d(x j, x r )) +µ p (p r p j ) a b r x 1 x 2 x 1 a x 1 b b x 2 a r a b a b r r r a b r x 2 b a r a b (ζ i, z i ) Ã(x j, p j, ξ j ) = {Y i : υ(z i, ζ i, x j, p j, ξ j ) > υ(z i, ζ i, x n, p n, ξ n ) n J}

15 Ã(x j, p j, ξ j ) j σ(x j, p j, ξ j ) = Ã(x j,p j,ξ j ) P Y (dy) σ(x j, p j, ξ j ) (ζ i, z i ) σ j (p, x, ξ) x n p n n / {0, j} j n (p, x, ξ) x r < x r d(x r, x j ) < d(x r, x j ) σ j > 0 x r < x r d(x r, x j ) < d(x r, x j ) σ j < 0

16 a b r 1 r 2 x 1 x 2 x 1 a > r 1 > b > r 2 x 2 x 1 x 2 x 1 r 1 r 2 x 2 a r 2 r 1 b

17 r x 1 x r a r 1 r 2 r 2 r 1 r 1 x 1 a a a 1 4 r 1 x x 1 b b r 2 r 1 R N R (p, x, ξ) x rn < x r n d(x r n, x j ) < d(x rn, x j ) σ j > 0 R (p, x, ξ) x r < x r

19 t j J t \{0} i u (z i, ζ i, x jt, p jt, ξ jt ; θ) = k x jkt βik α i p jt +ξ jt +ϵ ijt λ(p jt p rt )I {p jt > p rt } γ x jt x rt 2 β ik = β k + z ir βkr o + βu k ν ik r α i = ᾱ + z ir αr o + α u ν ip r t, u(z i, ζ i, x 0t, p 0t, ξ 0t ) = ϵ i0t ν i = (ν i1,..., ν ik, ν ip ) ϵ it = (ϵ i0t, ϵ i1t,..., ϵ ijt ) 2 L 2 Norm λ γ t i

20 ν i ϵ it ϵ it σ jt= ( exp ) x jtk βi αp jt +ξ jt λ(p jt p rt)i{p jt >p rt} γ x jt x rt 2 k 1+ Jt ( exp n=1 k x ntk βi αp nt+ξ nt λ(p nt p rt)i{p nt>p rt} γ x nt x rt 2 )df (ν)d F (z) F (z) F (ν) M F (ν) θ = ( β, α, λ, γ) ξ w jt ξ E[ξ jt w jt, x jt ] = 0 C E[ξ jt.h(w jt, x jt )] = 0 h s jt j t s t = (s jt ) J j=0 σ j (p, x, ξ; θ) = S j

21 θ,ξ g(ξ) Ωg(ξ) σ(p, x, ξ; θ) = S g(ξ) = 1 T T t=1 J j=1 ξ jth(z jt, x jt ) Ω T = 30 J = 10 K = 3 p jt ξ jt x 1 j x 2 j x 3 j 0 N 0, ξ jt i.i.d. N(0, 1), p jt = 0.5ξ jt + e jt + x 1j + x 2j e jt N(0, 1) j t w jt D = 6 w jtd = u (e jt k=1 x kjt) u Uniform (0, 1) z 2 jtd z3 jtd x2 jk x3 jk 6 d=1 z jtd 3 k=1 x jk z jtd x j1 z jtd x j2 i β = ( ) β u = ( ) ᾱ = 3 α u =.44 λ = 0 γ = 0

22 t τ [1,..., J t ] τ δ τ = x τ β αp τ I{τ r} ϕp τ I{τ > r} γd(x τ, x r ) + λp r I{τ > r} + ξ r ϕ = α + λ ς = [τ 1,.., τ Jt ] τ 1 > 1 τ 2 < J t Ψ r H 0 : r α ϕ = 0 γ 0 SupΨ = supψ τ τ ς λ = 5 γ = 4

23 ᾱ α u β 1 β1 u β 2 β2 u β 3 β3 u λ γ Ψ H o

24 ᾱ α u β 1 β1 u β 2 β2 u β 3 β3 u λ γ j

25 d(x j, x r ) = 1.76 d(x j, x r ) = 0 σ j

31 (λ) (γ)

32 ᾱ α u µ = 0 γ = 0

33 (λ) (µ) Ψ

35 λ = γ = 0

39 (p, ξ) A j A j = {w W : U(w,, x j ; x r ) > U(w, x n ; x r ) n J} A j = {w W : υ(w, x j ) + µ( d(x j, x r )) > υ(w, x n ) + µ( d(x n, x r )) n J} Ãj Ã j = {w W : υ(w, x j ) > υ(w, x n ) n J} x r x r d(x j, x r) < d(x j, x r ) P W (dw) Ã j Ã j P W (dw) Ã j J, x r x r w Max n J {υ(w, x n )} = υ(w, x m) > υ(w, x m ) = Max {υ(w, x n)} n J w A j = {υ(w, x j ) + µ( d(x j, x r)) > υ(w, x m) + µ( d(x m, x r))}

40 w µ( d(x j, x r)) µ( d(x m, x r)) > υ(w, x m) υ(w, x j ) υ(w, x m) > υ(w, x m ) w µ( d(x j, x r)) µ( d(x m, x r)) > υ(w, x m ) υ(w, x j ) d(x j, x r) > d(x j, x r ) µ( d(x j, x r )) µ( d(x m, x r)) > υ(w, x m ) υ(w, x j ) d(x m, x r ) > d(x m, x r) w µ( d(x j, x r )) µ( d(x m, x r )) > υ(w, x m ) υ(w, x j ) w A j A j A j σ j = A j P W (dw) > P W (dw) = σ j A j n C r [0, 1] r σ j σ j = ( n r=1 A jn P W (dw) ) M(C r ) M( ) A jr = {w W : U(w,, x j ; x r ) > U(w, x n ; x r ) n J} r ( ) r {1,..., n} M(C r ) > M(C r ) σ jr = P Aj W (dw) r x r x r d(x j, x r ) < d(x j, x r ) σ j > σ r j r

41 σ j r σ jr r {1,.., n}\{r } M(C r )(σ j r σ jr ) > ) M(C n ) (σ σ jn jn σ j > σ j M(C r ) n n as n M(C r ) 0 σ j σ j σ j > σ j n σ j σ j n R

Carleson measures and elliptic boundary value problems Abstract. Mathematics Subject Classiﬁcation (2010). Keywords. 1.

Carleson measures and elliptic boundary value problems Abstract. Mathematics Subject Classiﬁcation (2010). Keywords. 1. 2 3 4 5 6 7 8 9 0 A 2 3 4 5 6 7 8 9 20 2 22 23 24 25 26 27 28 29 30 3 32 33 34 BMO BMO R n R n+ + BMO Q R n l(q) T Q = {(x, t) R n+ + : x I, 0 < t < l(q)} Q T Q dµ R n+ + C Q R n µ(t (Q)) < C Q Q 35 36