January 16, 2017 1 1. Villani f : R R f 2 f 0 x, y R t [0, 1] f((1 t)x + ty) (1 t)f(x) + tf(y) f 2 f 0 x, y R t [0, 1] f((1 t)x + ty) (1 t)f(x) + tf(y) (simple) (general) (stable) f((1 t)x + ty) (1 t)f(x) + tf(y) 3 f((1 t)x + ty) (1 t)f(x) + tf(y) (useful) (simple, general, stable, useful Villani 2. 1
3. f, g C f g C f g S 4. (Φ (L) ) 1 (Φ (L) ) 1 (x) = 2 n Φ (L) (2 1 x) 5. 6. L p, 1 p < S (i) (ii) f N 7. B s 0 p 0 q 0 = F s 0 p 0 q 0 a k f = a k γ (2k ) k=1 f B s0 p 0q 0 \ F s0 p 0q 0 8. f Fp s0 0q 0 \ Bp s0 0q 0 (3.75) Ψ 9. (3.77) (3.78) (3.80) {h (j) } j N0 Ψ (F s pq ) q > 1 (3.78) q = 1 {φ j (D)F (j) } j N Lp (l 1 ) = φ j (D)F (j) j=1 L p = sup φ j (D)F (j) G G L p j=1 L 1 = sup sup φ j (D)F (j) (x)a j (x)g(x) dx G L p a j L, a j L R n j=1 2
φ j (D) {φ j (D)F (j) } j N L p (l 1 ) = sup sup G L p a j L, a j L R n j=1 F (j) (x)φ j (D)[a j G](x) dx {φ j (D)F (j) } j N Lp (l 1 ) sup sup G L p a j L, a j L sup sup G L p a j L, a j L {F (j) } j N L p (l 1 ) 10. R n j=1 R n j=1 F (j) (x)m[a j G](x) dx F (j) (x)mg(x) dx S 0 Z S 11. [f], Φ([f]) 12. (3.101) f, τ φ j (D)f [f] 13. L = 0 Φ(f) = φ j (D)f S 14. j= L n + L 15. c 0 500 l(q l,2 j ) 2l(Q k,2 j 1) 3
16. non-smooth 17. L br L b {R j } j=1 L b Rj L b, R j L b = lim L b Rj b L j 18. (1 ψ(d))f(x) h(x) dx 19. x P sup 2 j (c(p ) y) n η f(y) y R n η,q inf M (η) [χ 10nP f](x) x P + 2 jn( 1 q 1 η ) l(p ) n η sup k Z n ( 20. f(y) q dy (l(p )k+p ) φ j = ψ j (φ j 1 + φ j + φ j+1 ) ψ S 21. ) 1 q 4
F s q Ḃ s f S B(2 j ) f(x) sup m Z n 22. ( 1 Q jm Q jm f(y) q dy T q ({F j } j A ) G q ({G j } j A ) q = 23. O λ 24. O O = {F > λ} ) 1 q µ{(y, t) R n+1 + : F (y, t) > λ} 3 n µ Carleson O 25. u( ; t) M s pq f M s pq 26. M 0 21 M 0 p1 27. supp(ψ) B(2 r ) r < r supp(ψ) B(2 r ) r = ρ 28.. 5
R 29. Λ νm 2 ν(s n p ) inf M ( η0 2 ) [φν (D)f](y) y Q νm i. ν 1 y Q νm 1 ( φν m ) (D)f M ( η 0 2 ) [φν (1 + 2 ν y 2 ν m ) 2n/η0 2 ν (D)f](y) ( Λ νm = 2 (s n p )ν m ) φ ν (D)f 2 ν = (1 + 2 ν y 2 ν m ) 2n/η 0 2 (s n p )ν 1 ( m ) φ (1 + 2 ν y 2 ν m ) 2n/η ν (D)f 0 2 ν (1 + n) 2n/η0 2 (s n p )ν 1 (1 + 2 ν y 2 ν m ) 2n/η0 φν (D)f 2 (s n p )ν M ( η 0 2 ) [φν (D)f](y) y Λ νm 2 ν(s n p ) inf y Q νm M ( η0 2 ) [φν (D)f](y) ii. ν = 0 30. z C 1 ρ 1, ρ 2, ρ 3, ρ 4 ( m 2 ν ) ρ 1 (l) = s q q l s l, ρ 2 (l) = p p l q q l, ρ 3 (l) = 1 p p l, ρ 4 (l) = q q l, l = 0, 1 z C 1 ρ 1, ρ 2, ρ 3, ρ 4 ρ 1 (l) = s q q l s l, ρ 2 (l) = p p l q q l, ρ 3 (l) = 1 p p l, ρ 4 (l) = q q l, l = 0, 1 6
31. (5.126) f F s pq f = λ νmaνm ν N 0 m Z n λ = {λ νm} ν N0 λ fpq f F s pq, α a νm 2 ν(s n/p)+ α ν a νm = 2 ν(s n/p) a νm, λ νm = 2 ν(s n/p) λ νm f = ν N 0 m Z n λ νm a νm { } λ fpq = 2 νn/p λ νmχ Qνm m Z ν=0 { } = 2 νs λ νm χ Qνm m Z ν=0 L p (l q ) L p (l q ) 32. (Key theorems in function spaces) 33. B M (R n ) B N (R n ) 34. (5.150) I ψ B 1 (R n )+ ψ 1 B 1 (R n ) B 1 (R n ) B N (R n ) 7
35. (5.158) (5.158) f S S B11 n λβ νm, (βqu) νm f = λ β νm(βqu) νm (1) m Z n β N n 0 ν N 0 B n 11 Bn 11 B 0 1 BUC (5.158) f S 36. φ(x) = (x, x n ω(x )) ω D(R n 1 ) 37. sup j N 0 (m j+1 m j ) < 0 lim j m j = 38. y β η α e iy η dy dη = (2π) n ( i) α α!δ αβ R 2n 39. (6.119) Θ 40. f g, f, g, S(R n ) S(R 2n ) T 41. T [φ(τ )] L 2 42. T [(1 κ B )φ(τ )] K 8
43. A 44. l 1 Ψ l ( y)φ k ( y) = Φ k ( y) 2 1. 2. β [(1 + ξ 2 ) σ (ψ j+3 ψ j 1 )] L 1 2 j(2σ+n β ) M > 0 F 1 [(1 + ξ 2 ) σ (ψ j+3 ψ j 1 )](x) 2 j(2σ+n M) x M F 1 [(1 + ξ 2 ) σ (ψ j+3 ψ j 1 )](x) dx R n ( ) n+1 2 j(2σ+n) 1 min 1, R 2 j dx x n 2 2jσ 9
3. ( 1) j+1 j=1 ξ j ( 1) j+1 j=1 ξ 2j 4. (1 ) s 1 2 (1 ) s 1 2 5. (1.34) W m 2 = W m 2 := 6. m m 7. Ff(ξ) = F 1 f( ξ) Ff = F 1 f( ) 8. 9. 10 U S φ S U S φ U 10. 8 ε 1, ε 2,..., ε L > 0 10
ε 1, ε 2,..., ε L (0, 1) 11. 8 j = 1, 2,..., k j = 1, 2,..., L 12. 7 N 1, N 2,..., N L N α 1, α 2,..., α L N n n 0, β 1, β 2,..., β L N 0 13. 7 (2.8) L {ψ S : p αl,β l (ψ φ) < ε l } U l=1 14. 7 j = 1, 2,..., k j = 1, 2,..., L 15. 7 α j + β j α j + β j + 1 16. φ (N;α) (x) := e N x α φ(x) φ (N;α) := e N α φ 17. φ y (x) := φ(x y) φ y := φ( y) 11
18. sup p N (φ t ) t [0,1] n sup p N (φ y ) y [0,1] n φ y = φ( y) 19. (2.13) 2 4 N 1 2 N 1 20. (2.19) τ S 21. α f S φ S 22. ψ η 23. (2.30) φ S 24. Θ j (ξ) := Θ(2 j ξ) Θ j := Θ(2 j ) 12
25. ψ(ξ) := η( ξ ), φ(ξ) := η( ξ ) η(2 ξ ) ψ := η( ), φ := η( ) η(2 ) 26. 27. 28. (2.38) p N (φ) = p N (φ) = α N 0 n α N = α N 0 n α N α N 0 n α N = α N 0 n α N sup x N α φ(x) x R n sup α φ(x) (1 + x 2 ) N x R n ( ) sup x N α φ(x) x R n ( ) sup α φ(x) (1 + x 2 ) N x R n 29. 30. 13
dx dξ 31. j = 1, 2,..., n 32. ix ξ ix ξ 33. E(x) := exp ( 1 2 x 2), E t (x) := E(tx) E := exp ( 1 2 2), E t := E(t ) 34. FE t (ξ) = 1 ( ) ξ t n E (2π) n 2 t FE t = 1 ( ) t n E (2π) n 2 t 35. Fφ(x)Fψ(x) dx Fφ(ξ)Fψ(ξ) dξ 36. (2π) n 2 (2π) n 2 37. F (x) := m Z n f(x 2πm) 14
F := m Z n f( 2πm) 38. 39. (2.90) 40. Φ (L) (x) := vol(s n 1 ) 1 κ (L+1) ( x ), Ψ (L) (x) := vol(s n 1 ) 1 ψ (L) ( x ) Φ (L) := vol(s n 1 ) 1 κ (L+1) ( ), Ψ (L) := vol(s n 1 ) 1 ψ (L) ( ) 41. γ L γ = L 42. α L α = L 43. L p S 0 < p < 1 L p S 44. sup λ B λ < X = R B λ = ( λ, λ), λ (0, ) 15
45. (2.112) f(x) dx f(y) dy 46. max(1, p) min(1, p) 47. n n (p 1 ) p n n (p 1 ) p 48. 0 (p < 1 ) 0 (p > 1 ) 49. min(1, p) η = 2 min(1, p) η := 2 50. f p... f p h n p... 51. (2.158) (x z) ( z) 16
52. 53. f(x) dx f(y) dy 54.... f(x) M dyadic f(x)...... f(x) M dyadic f(x)... 55. 2 R jn f(x) dx n 2 jn R n f(y) dy 56. T g(x) dx R n T g(x) dx = R n 57. λ λ > 0 58. K(x) := F 1 [ψ(2 j )m](x) K := j= j= F 1 [ψ(2 j )m] 17
59. x R n \ {0} 60. m j (ξ) = ψ(2 j ξ)m(ξ) m j = ψ j m 61. ψ(2 j ) ψ(2 j ξ) 62. m(d)f j (x) m(d)f(x) m(d)f j (x) m(d)f(x) 63. 0 Ψ χ B(4)\B(3) χ B(3.9)\B(3.1) Ψ 2 χ B(4)\B(3) 64. χ B(2) Ψ χ B(3) χ B(2.9) Ψ χ B(3) 65. f = f := 66. φ j (D)f = 2 jn a j F 1 [φ j ] 18
φ j (D)f = 2 jn a j F 1 [φ j Ψ j+1 ] 67. min(p, q) < 1 68. ψ j+2 (2 j+2 ) (ψ j+2 (2 j+2 ) ψ j 3 (2 j+2 )) 69. ψ j+2 (2 j+2 ) (ψ j+2 (2 j+2 ) ψ j 3 (2 j+2 )) 70. ψ j+2 (2 j+2 x) (ψ j+2 (2 j+2 x) ψ j 3 (2 j+2 x)) 71. F (x) p 0 p 1 f F s 0 p 0 F (x) p 0 p 1 f 1 p 0 p 1 F s 0 p 0 72. f a := a j F 1 [κ j ] f a := j=1 2 j(n+s n/p) a j F 1 [κ j ] j=1 73. 19
(3.1) 74. (3.51) F s pq (3.1) F s pq 75. (3.51) N 3 76. A s pq B s n/p A s pq B 0 1 BUC 77. (3.1) 78. (3.51) N 2 79. (3.52) supp(f N G N ) B(2 N+4 ) supp(f(f N G N )) B(2 N+4 ) 80. 81. BUC 20
C c C c 82. (3.56) κ η 83. (3.56) χ B(2.1)\B(1.9) κ χ B(2.2)\B(1.8) χ Q(2.1)\Q(1.9) η 1 χ Q(2.2)\Q(1.8) 84. F 1 η(2 k ) F 1 [η(2 k )] 85. e i2k x 1 F 1 [η( 2 l e 1 )] F 1 [η( 2 l e 1 )]( 2 l e 1 ) 86. F 1 (α (k) ) α (k) 87. F 1 (β (k) ) β (k) 88. (3.61) (3.62) (3.61) (3.62) 21
89. (3.61), (3.62) F 1 (γ (k) ) γ (k) 90. F 1 (δ (k) ) δ (k) 91. (3.64) χ B(1) ψ χ B(3/2) χ Q(11/10) ψ χ Q(3/2) 92. η ρ η 2 93. (3.68) (η) (ρ) η 2 94. F 1 [δ (k) ] δ (k) 95. F (A s pq) Ψ (A s pq) 96. 22
φ j (D)F (j) 2 js φ j (D)F (j) 97. (3.77) F (F s pq ) Ψ (F s pq ) 98. (3.79) L p (l q ) L p (l q ) 99. (3.80) φ j (D)h (j) 2 js φ j (D)h (j) 100. (3.80) g F s pq {h (j) } j N0 Lp (l q ) Ψ (F s pq ) < g F s p q {h(j) } j N0 L p (l q ) Ψ (F s pq ) < 101. (3.80) F (0) = ψ(d)h (0), F (j) = φ j (D)h (j) F (0) := ψ(d)η, F (j) := 2 js φ j (D)η 102. h (j) φ j (D)η 2 js h (j) φ j (D)η 103. φ j (D)ηh (j) 2 js φ j (D)h (j) 23
104. τ j (x) = τ(2 j x) τ j = τ(2 j ) 105. S 0 S 0 S 106. (3.97) lim J J j= J F 1 [ 2s Ff] F 1 [ 2s φ j Ff] 107. f A s pq Φ S 0 p N (f) f Ȧs pq 108. F s pq Ḃs p min(p,q) p N (Φ) Φ Ȧs pq F s pq Ḃs p max(p,q) 109. 2 j 2(n+1) 2 j 2(n+1) 24
110. 2 (s+2n+2)j f Ȧs pq n+1 τ 1 2 (n+1)j f Ḃ n 1 n+1 τ 1 111. f Ȧs pq p n+1 (τ) (τ S 0 ) f Ḃ n 1 p 2n+2(τ) (τ S 0 ) 112. s > 0 s R 113. s > 0 s R 114. (3.104) (3.104) L > s n p. 115. (3.105) 116. (3.105) A s pq Ḃs σp max(1,p) A s pq Ḃs n/p 117. (3.105) (3.105) 25
f Ḃs, L > s 118. (3.108) (3.108) α φ j (D)f, τ φ j (D)f 119. (3.109) (3.109) R n α φ j (D)f, τ 2 α j φ j (D)f 120. (3.109) (3.109) 1 j= α φ j (D)f, τ 121. (3.110) (3.110) L s n p, q 1 122. (3.112) (3.112) s < n p 123. j 0 j 0 124. ( ) 1 s < n min p, 1 1 j= 26 2 (n+ α )j dx 2 j x (n+1) 2 α j φ j (D)f 2 ( α s)j f Ḃs s,l f Ḃs
s < n p 125. ( ) 1 s = n min p, 1, q 1 s = n p, q 1 126. (3.122) (ω ) 2 ω 2 127. 128. n (1) A min(a, A + s) > (2) min(1, p) n A min(a, A + s) > min(1, p, q) 129. η 0 < η < min(1, p, q), Aη > n 130. (3.141) L = N A + 1 L = [A + 1] 131. (3.142) 27
2(j l)n+jn 2 j x A 2(j l)a+jn 2 j x A 132. (3.143) 2 (j l)n+jn 2 (j l)a+jn 133. (3.145) 2 (j l)n+jn 2 (j l)a+jn 134. (3.146) 2 (j l)n+jn 2 (j l)a+jn 135. 2 (j l)n+(l m)n+ln 2 (j l)a+(l m)a+ln 136. 2 (j m)n+ln 2 (j m)a+ln 137. 2 (j m)n+mn 2 (j m)a+mn 138. 2 (j m)n+mn 2 (j m)a+mn 28
139. (3.147) 2 (j m)n+mn 2 (j m)a+mn 140. N 1 Nη > n 141. (3.148) 2 (j m)nη+mn 2 (j m)aη+mn 142. (3.149) 2 (j m)(nη n)+mn 2 (j m)(aη n)+mn 143. (3.151) 2 (j l)(nη n)+jn 2 (j l)(a f η n)+jn 144. 2 (m l)nη 2 (m l)aη 145. 2 (m j)(nη n)+mn 2 (m j)(aη n)+mn 146. 29
2 (m j)(nη n)+mn 2 (m j)(aη n)+mn 147. 2 (m j)(nη n)+mn 2 (m j)(aη n)+mn 148. (3.152) 2 (m l)((n+s)η n)+mn 2 (m l)((a+s)η n)+mn 149. 150. (1) L > σ p + s (2) L > σ pq + s 151. (3.160) 0 / supp(η) 0 / supp(fη) 152. sup y R n y N α [ y 2N Fφ(y)] f 1 sup y R n α [ y 2N Fφ(y)] 153. P M 1 P M 30
154. p, q > 1 155. x y 156. ( L Φ) f(x) ( L Φ) l f(x) l=0 157. 158. (4.2) L p 159. L 2 [0, 1] L 2 [0, 1) 160. l p L p [0, 1) 161. 31
4.1.3 4 (3) (4.2) a j = 0 l 2 162. L p [0, 1] L p [0, 1) 163. (4.14) N r j (t)φ j (D)f j=1 N r j (t)φ j (D)f L p j=1 164. fg fg 165. fg fg 166. sup fg fg g S, g p =1 sup fg p f F 0 p2 g S, g p =1 167. 32
s > 0, 1 < p, 0 < q s > 0, 1 p, 0 < q 168. 1 < p, 0 < q, s > 0 1 p, 0 < q, s > 0 169. 170. (4.21) φ j (D)f φ k (D)f 171. knr kn 172. 2 jnr 2 jn 173. 2 jnr 2 jn 174. (4.51) sup{l N : 1000Q l 1000Q j } <. j N sup {l N : 1000Q l 1000Q j } <. j N 33
175. Q(x) Q j 176. (4.54) ( x ψ (j) c(qj ) (x) := ψ l(q j ) ( ψ (j) c(qj ) := ψ l(q j ) ), Ψ(x) := ), Ψ := 177. l=1 l=1 sup {l N : supp(φ (l) ) 1000Q l } <. j N sup {l N : supp(φ (j) ) 1000Q l } <. j N 178. m(d)f(x) = lim R m(d)ψ(r 1 D)f(x) m(d)f(x) = m(d)ψ(r 1 D)f(x) 179. κ F N, ψ (l) (x), φ (j) (x) := ψ(j) (x) Ψ(x) ψ (l), φ (j) := ψ(j) Ψ ψ(r 1 D)f H p = M[ψ(R 1 D)f] p = sup κ j ψ(r 1 D)f. j Z ψ(r 1 D)f H p sup ψ(2 j D)ψ(R 1 D)f. j Z 180. ψ(r 1 D)f H p sup κ F N, j Z 34 κ j f p p p = Mf p = f H p.
ψ(r 1 D)f H p Mf p = f H p. 181. φ (j) (x) := φ (j) ( x) φ (j) := φ (j) ( ) 182. { m(d) f } 1 > λ λ 2 m(d) f(x) 2 dx R ( ) n e t 1 F (x) F, exp x 2 (x R n ) (4πt) n 4t F S (R n ) M 0 F (x) sup e t F (x) j Z { M 0 [m(d) f] } 1 > λ λ 2 m(d) f(x) 2 dx R n 183. { m(d) f } 1 > λ λ 2 f(x) 2 dx 1 R λ n { M 0 [m(d) f] } 1 > λ λ 2 f(x) 2 dx R n 1 λ 2 Mf(x) 2 dx + Ω R n \Ω 184. (4.62) Ψ (j) x (y) φ (j) (x y) ( F 1 m(d)(y) 1 I (j) R n \Ω Mf(x) dx + Ω ) F 1 m(d)(x z)φ (j) (z) dz R n Ψ (j) x ( F 1 m(d)(x z j + y) 1 I (j) (y) ) F 1 m(x z)φ (j) (z) dz R n φ (j) (z j y) 35
185. (4.63) Ψ (j) x f(x) Ψ (j) x f(z j ) 186. Ψ (j) x (y) = 1 ( [ F 1 m(d)(y) F 1 m(d)(x z) ] ) φ (j) (z) dz I (j) R n φ (j) (x y) Ψ (j) x (y) = 1 ( [ F 1 m(x z I (j) j + y) F 1 m(x z) ] ) φ (j) (z) dz R n φ (j) (z j y) 187. ( 1 [ F 1 m(d)(y) F 1 m(d)(z) ] ) φ (j) (x z) dz I (j) R n φ (j) (x y) ( 1 [ F 1 m(d)(x z I (j) j + y) F 1 m(d)(z) ] ) φ (j) (x z) dz R n φ (j) (z j y) 188. (4.64) α F 1 m(d)(w) α F 1 m(w) 189. Ψ (j) x Ψ (j) x f(x) f(z j ) l(q j ) n+1 Mf(x) x c(q j ) n+1 l(q j ) n+1 x c(q j ) n+1 Mf(z j) 36
190. Mf(x) p dx R n Mf(x) p dx R n 191. { m(d)[f f] > λ } Ω + { m(d)[f f] > λ } Ω Ω + 1 m(d)[f λ f](x) dx Ω R n \Ω { M 0 [m(d)[f f]] > λ } Ω + { M 0 [m(d)[f f]] > λ } Ω c Ω + 1 M 0 [m(d)[f λ f](x)] dx R n \Ω Ω M 192. { m(d)f > 2λ } { m(d) f) > λ } + { m(d)[f f] > λ } 1 λ 2 f(x) 2 dx + { Mf > λ } {Mf λ} 1 λ 2 Mf(x) 2 dx + { Mf > λ } {Mf λ} { M 0 [m(d)f] > 2λ } { M 0 [m(d) f)] > λ } + { M 0 [m(d)[f f]] > λ } 1 λ 2 f(x) 2 dx + { Mf > λ } {Mf λ} 1 λ 2 Mf(x) 2 dx + { Mf > λ } {Mf λ} 37
M 193. m(d)f p p = M 0 [m(d)f] p p = p 0 p λ p 1 { m(d)f > λ } dλ = p 2 p λ p 1 { m(d)f > 2λ } dλ 0 λ p 3 min{mf(x) 2, λ 2 } dλ dx R n 0 Mf(x) p dx f p H p R n 0 λ p 1 { M 0 [m(d)f] > λ } dλ = p 2 p λ p 1 { M 0 [m(d)f] > 2λ } dλ 0 λ p 3 min{mf(x) 2, λ 2 } dλ dx R n 0 Mf(x) p dx f p H p R n M 194. x R n \ 2Q 195. Q 1 2 1 p Q 1 p 1 2 196. (4.84) ( ) x ψ (j) c(qj ) (x) := ψ, φ (j) ψ (j) (x) := l(q j ) χ R n \Ω + k N ψ(k) 38
( ) ψ (j) c(qj ) := ψ, φ (j) ψ (j) := l(q j ) χ Rn \Ω + k N ψ(k) 197. (4.90) α, β N 0 n 198. (4.94) p (j) (x) = f, e(j,k) φ(j) e (j,k) (x) I (j) k K j p (j) = f, e(j,k) φ(j) e (j,k) I (j) k K j 199. p (j) (x) p (j) 200. (4.95) x R n 201. 100Q j 100000Q j 202. Φ j,t (x) = Φ j,t (x) := 203. (4.109) M[e (j,k) φ (j) ] l(q j ) n, f, e (j,k) φ (j) Mf(x 0 ) 39
M[e (j,k) φ (j) ] 1, f, e (j,k) φ (j) l(q j ) n Mf(x 0 ) 204. (4.118) 1 ( ) t n φ b (j) (x 0 ) t Mf(z l(q j ) n+l+1 Mf(z j ) j) (l(q j ) + x 0 c(q j ) ) n+l+1 1 ( ) t n φ b (j) (x 0 ) t l(q j ) n+l+1 Mf(z j ) (l(q j ) + x 0 c(q j ) ) n+l+1 λl(q j ) n+l+1 (l(q j ) + x 0 c(q j ) ) n+l+1 205. (x) 206. 207. (4.121) f φ (l) f, 1 ( ) t n φ t l J j,0 f φ (l) f, 1 ( ) x t n φ t l J j,0 ( ) sup x N x zj + t x x R n α φ 1 φ (l) (z j t x) t l J j,0 ( ) sup x N x zj + t v v R n α φ 1 φ (l) (z j t v) t l J j,0 208. 40
b (j) b 209. D = {f X p : Mf 1, f < } D = {f H p : Mf 1 < } Mf 1 < f L 1 210. H p Y p Y p H p 211. (4.133) f H p f Y p f Y p f H p 212. (4.135) g 2 J 1 g 2 J 1 213. (4.141) ( ) A j,k = 1 φ (l) l,2 φ (k) j 1 l,2 f p (k) j 1 2 φ (k) j 1 2 + j 1 A j,k = + l=1 ( 1 l=1 l=1 φ (l) l,2 j 1 p (l) 2 j φ (l) 2 j φ (k) 2 j 1 + ) l=1 214. (p, ) (1, ) φ (k) l,2 j 1 f p (k) 2 j 1 φ (k) 2 j 1 p (j,k,l) φ (l) 2 j l=1 p (l) 2 j φ (l) 2 j φ (k) 2 j 1 41
215. 1 ( ψ ) sup ψ F N, 0<t 1 t n f t p 1 ( ) sup ψ t n f t p ψ F N, 0<t 1 216. κ(td)f κ(td)f(x) 217. (4.167) p 218. (4.167) p 219. V m (x) V m (x) := χ m+[0,1] n(x)ψ(d)f(x) V m V m := χ m+[0,1] nψ(d)f 220. 221. dx 42
222. Q λ} λ} 223. λ > 0 224. θ θ > 0 225. ψ S ψ S 226. (2π) n 2 (2π) n 2 227. R n 228. j 2 j Q 2 h BMO = j 2 j h BMO j 2 jn Q 2 h BMO = j 2 jn h BMO 229. 43
φ j (D)f φ j (D)f(x) 230. ( ) 1 q f(x) q dx (l(p )k+p ) ( ) 1 q f(y) q dy (l(p )k+p ) 231. f(y) f(y) min w B(x,δ) min w B(y,δ) f(w) + 2δ f(w) + δ 232. sup x y n η y R n sup x y n η y R n 233. sup w B(x,δ) sup w B(y,δ) f(w), x y δ 1 f(w), δ 1 min f(w) + 2δ sup x y n η w B(x,δ) y R n min f(w) + δ sup x y n η w B(y,δ) y R n sup f(w) w B(x,δ) sup f(w) w B(y,δ) sup x y n η min f(w) + 2δ sup x w n η f(w) y R n w B(x,δ) w R n sup x y n η min f(w) + δ sup x w n η f(w) y R n w B(y,δ) w R n 234. P 1 235. 44
( x y n ( x y n f(w) η dw B(y,δ) 236. ( x y n ( x y n 237. ) 1 η χ 10nP (w) f(w) η dw B(y,δ) f(x + w) η dw B(y x,δ) ) 1 η ) 1 η χ 10nP (x + w) f(x + w) η dw B(y x,δ) P 1 238. ( x y n ( x y n 239. ( 1 (1 + x y ) n ( 1 (1 + x y ) n f(x + w) η dw B(1+ x y ) ) 1 η ) 1 η χ 10nP (x + w) f(x + w) η dw B(1+ x y ) B(1+ x y ) B(1+ x y ) 240. min f(w) w B(y,δ) f(x + w) η dw ) 1 η ) 1 η χ 10nP (x + w) f(x + w) η dw ) 1 η 45
inf f(w) w B(y,δ) 241. k k Z n \ [ 1, 1] n 242. k Z n k Z n \ [ 1, 1] n 243. k = 0 Q 0m P \ P m m 244. 245. Q jm T q (G j )(x) q dx 2 inf{g q (G)(x) q : x Q jm } Q jm T q (G j )(x) q dx 2 inf{g q (G j )(x) q : x Q jm } 246. (4.219) 2 1/q G q ({F k } k A )} 2 1/q G q ({F k } k A )(x)} 247. 46
2 T q ({F j } j= )(x) R n 2 T q ({F j } j= )(x) R n m Z n j=j(x) m Z n j=j(x) Qj(x)m G j (y) q Qjm G j (y) q dy Q j(x)m dy Q jm 1 q 1 q dx dy 2 T q ({F j } j A )(x) G j (y) R n m Z n j=j(x) Qjm q dy Q jm 2 T q ({F j } j A )(x)t q ({G j } j A [j(x), ) )(x) dx R n 2 1+1/q R n T q ({F j } j A )(x)g q ({G j } j A )(x) dx 1 q dx 248. F s 1q F q s F s 1q F s q F1q s F q s F 1q s F q s 249. (4.222) j= j= φ j φ j 2 250. B(y,2 j ) B(x,2 n j ) 251. 47
L F 1q s L ( F 1q) s 252. S S 0 253. 1 2 js φ j (D)f(x) q dx P 1 P P j= log 2 l(p ) P j= log 2 l(p ) 2 js φ j (D)f(x) q dx 254. m P 2 js φ j (D)f q m P j= log 2 l(p ) j= log 2 l(p ) 255. a j C (Int(P )) 2 js φ j (D)f q j= log 2 l(p ) P j= log 2 l(p ) 1 q 1/q a j (x) q = χ 2P (x) a j (x) q dx = P 48
256. (4.225) j= log 2 l(p ) k N P j= log 2 l(p ) φ j (D)a j (x) q χ 2P (x) φ j (D)a j (x) q dx P q = 1 1 < q < φ j (D)a j (x) Ma j (x) k N 257. x 2 k+1 P \ 2 k P x 2 k+1 P \ 2 k P, L > n 258. φ j (D)a j (x) 2jn a j (y)f 1 φ(2 j (x y)) dy P P φ j (D)a j (x) 2 jn a j (y)f 1 φ(2 j (x y)) dy 259. (2 j l(p ) 1 ) n (2 k+j l(p )) L 2 jn (2 k+j l(p )) L 260. (4.226) P P P a j (y) dy φ j (D)a j (x) q 2 kql P q 1 φ j (D)a j (x) q 2 kql P q a j (y) dy P P a j (y) q dy a j (y) q dy 49
261. (4.227), (4.228) 1/q 262. B (0, r(b)) Int(conv((c(B), r(b)) B {0})) conv 263. 264. B (0, r(b)) Int(conv((c(B), r(b)) B {0})) 265. N µ B j N O = B j, B j j=1 3 n µ Carleson, B j O = j=1 j=1 B j 266. Ô M j=1 ˆB j m j=1 ˆ 3B j 50
N Ô M ˆB j 3Bˆ ι(j) j=1 j=1 267. B j B ι(j) 268. = 269. 270. sup κ(td)f(y) κ(td)f(x) Mf(x) (y,t) Γ(x) sup κ(td)f(y) Mf(x) (y,t) Γ(x) 271. exp(it D 2 )f exp( it D 2 )f 272. 2iξ k 2itξ k 273. (4.238) 51
τ( k) 1 k Z n 274. 3Q 0m 3Q νm 275. κ > κ n n min(1, p, q) 276. = (5.19) κ > n 277. (5.35) m νm (x) m νm 278. λ jm = 0 λ jm 0 (c) 1 279. ν N 0 ν=1 280. (5.61) 52
281. L - L - 282. (5.67) = c = 283. (5.67) (5.68) ρ 284. (5.67) 5.1.24 a > 1 5.1.24 5.1.24 a > 1 a 285. N 1 286. (5.68) (a N) ( ) 2n N η 0 287. 53
288. Λ νm 2 ν(s n p ) inf M ( η0 2 ) [φν (D)f](y) y Q νm ν = 0 289. < s 0 < s < s 1 <. < s 1 < s < s 0 <. 290. f S f S 0 291. (5.104), (5.107) 292. G F (z) = e K(z θ) F (z) G F (z) = e δz2 δθ 2 +K(z θ) F (z) 293. = inf F e δ inf F Λ 0m inf y Q 0m M ( η0 2 ) [τ(d)f](y) 294. (5.111), (5.112) 295. M 1 θ M 1 θ inf F 54
296. (5.126) 2 νs 2 νs 297. (5.126) Λ ν (x) Λ ν 298. l = 0, 1 F (l + i t) F s l p l q l F (l + it) F s l p l q l f [F s 0 p 0 q 0,F s 1 p 1 q 1 ] θ l = 0, 1 F (l + i t) F s l p l q l F (l + it) s F l f p l q F s l pq 299. φ(x) = ψ(x) ψ(2x) φ(ξ) = ψ(ξ) ψ(2ξ) 300. ψ(d)g p0 + 2 js φ j (D)g L p 0 (lq ) ψ(d)g p2 + 2 js φ j (D)g L p 2 (lq ). 301. i I i = 1, 2,, I. 302. ψ 55
λ β,i ν, m := (βqu) i ν, m := λ β,i ν, m := (βqu) i ν, m := { λ β ν,θi ν( m) m Mi ν 0 { (βqu) ν,θ ν i ( m) ψ m M ν i 0. { λ β ν,θi ν( m) 0 { (βqu) ν,θ ν i ( m) ψ m ι ν i (M ν i ) m ι ν i (M i ν) 0. 303. I ψ B1 (R n ) + ψ 1 B1 (R n ) I ψ,ψ 1 1 304. f i,β F s pq (R n ) ψ {λ β,i νm } ν N 0, m Z n f pq (R n ) ψ λ β fpq (R n ) f i,β F s pq (R n ) ψ 2 (r+ε) β {λ β,i νm } ν N 0, m Z n f pq(r n ) ψ 2 (r+ε) β λ β fpq(r n ) 305. δ = ρ r ε 306. λ β = λ β := 307. (5.159) v j := β N n 1 0 ν N 0 m Z n 1 β λ νm (2L + j)!2 ν(2l+j) L [((β, 2L + j)qu) ν(m,0)] 56
v j := β N n 1 0 ν N 0 m Z n 1 β λ νm (2L + j)!2 ν(2l+j) ((β, 2L + j)qu) ν(m,0) 308. (5.162) 3 δ jl (2L + j)! δ jl (2L + j)! β N n 1 0 β N n 1 0 λ β ν N 0 m Z n 1 λ β ν N 0 m Z n 1 309. (5.167) E 1 (x) = λ ν1 mχ ν1 m(x) m Z n E 1 λ ν1 mχ ν1 m m Z n 310. Ẽ1(x) = m Z n λ ν1 mκ(2 ν1 x m) Ẽ1 m Z n λ ν1 mκ(2 ν1 m) 311. κ(2 ν 2 x m) κ(2 ν 2 m) 312. κ(2 ν l x m) κ(2 ν l m) 313. κ νm Tr R n 1 2L+j x n νm Tr R n[ 2L+l x n χ [ 1,1] κ χ [ 2,2] κ : R R. (x 2L j n (β qu) νm ) (x 2L+j n (β qu) νm )] 57
314. (5.174) R m0 (x) = ρ(x 10am am 0 ) m Z n R m0 := ρ( 10am am 0 ) m Z n 315. (J σ g 1 ) R n = (J σ g 2 ) R n (J σ g 1 ) R n + = (J σ g 2 ) R n + 316. (6.16) δ 0l 1 317. M (n + 1)N 318. (6.19) (6.21) (βqu) νm ((βqu) νm ) 319. (6.23), (6.27) lim ϵ 0 Tr R n + Ext N [f(, n + ϵ) R n + ] lim ϵ 0 Tr R next N [f(, n + ϵ) R n + ] 320. (6.24) ( 1 δ s (n 1) q )+ 1 δ > 0 58
321. f(, n + ϵ) = Ext N f(, n + ϵ) f(, n + ϵ) = Ext N [f(, n + ϵ) R n + ] 322. ξ m δ α +ρ β ξ m+δ β ρ α 323. α, β N 0 n 324. j 1 a k=0 a k j 1 a(x, ξ) a k (x, ξ) k=0 325. 2 γ 2 γ k 326. γ ξ [φ(2 k ξ)] x β α γ ξ [a k (x, ξ)] ξ mj+δ β ρ α 2 γ ξ m k m j+ρ γ χ B(2)\B(1) (2 k ξ) ξ m j+δ β ρ α 2 γ +m k m j +ρ γ χ B(2)\B(1) (2 k ξ) 2 m k m j ξ mj+δ β ρ α χ B(2)\B(1) (2 k ξ) 59
ξ (mj+δ β ρ α ) γ ξ [φ(2 k ξ)] x β α γ ξ [a k (x, ξ)] 2 γ k ξ m j+m k +ρ γ χ B(2)\B(1) (2 k ξ) 2 (m k m j +(ρ 1) γ )k χ B(2)\B(1) (2 k ξ) 2 (m k m j )k. 327. a S m0 ρδ 328. min(µ j (α, β), m j ) > k min(µ j (α, β), m j ρ β + δ α ) > k 329. m j > k m j ρ β + δ α > k 330. ξ m ξ m j ρ β +δ α 331. A(x, ξ) γ 332. 60
= = = i α γ α,β t α γ ( α γ ) R 2n y β γ e iy η x α+β γ ξ α [a(x + t y, ξ + t η)] dy dη = ( ) (it) α + β γ t γ α γ γ α,β γ min(α,β) γ min(α,β) x α+β γ R 2n C α,β,γ t α γ α+β γ ξ a(x + t y, ξ + t η)]e iy η dy dη y β γ e iy η x α+β γ ξ α [a(x + t y, ξ + t η)] dy dη R 2n C α,β,γ t α+β γ t γ ( ) α γ x α+β γ R 2n α+β γ ξ [a(x + t y, ξ + t η)]e iy η dy dη. 333. α+β+γ x α+β γ x α+β+γ ξ a α+β γ ξ a 334. a(x, η + ξ) y 2L (1 η ) L y 2L (1 η) L a(x, η + ξ) 335. ( ) α β 61
336. α α β β ( α α α α,β β ) ( ) β β c α,β,α β α x β ξ R 2n α R 2n x β ξ a(x, η + ξ) α α x a(x, η+ξ) α α x dy dη β β ξ b(y + x, ξ) e iy η dy dη β β ξ b(y+x, ξ) e iy η. 337. α x β ξ a Sm 1+δ α ρ β α x β ξ a Sm 1 δ α +ρ β 338. x α α β β ξ a S m 1+δ α α ρ β β x α α β β ξ a S m 1 δ α α +ρ β β 339. ξ m1+m2 (ρ δ)( α + β )+2δL ξ m 1+m 2 +(ρ δ)( α + β )+2δL 340. R 2n R 2n 341. dt 62
342. N = [s + 1] + N 0 N = [s + 1] + + [σ pq + 1] N 0 (d) s + 2N > σ pq + 1 343. 2 2jM α +m+δl β 2 (2M α +m+δl β )j 344. 2 m+δl 2 (m+δl)j 345. Ff N (x) Ff N (ξ) 346. 347. exp( x n 1 + ξ 2 iξ y ) exp( x n 1 + ξ 2 iξ y ) 348. N+1 j=1 N+1 j=1 Ef 0 (x, jx) λ j Ef 0 (x, jx n ) 63
349. 350. φ S(R n ) 351. φ 2 φ 1 352. T a f 2 a BMO f 2 1 2 φ j (D)a 2 ψ j 4 (D)f 2 a BMO f 1 j= 1 353. T a f T a f(x) 354. T a 1, A T a 1, A 355. Ta,J 1, A T a,j 1, A 64
356. T k = lim r 0 T k,r T k f = lim r 0 T k,r f 357. 002 L. Jorgen J. Löfström 358. 006 D. Edmund D. E. Edmunds 359. B. X. Wang and C. Huang 360. Characterization of the Besov-Lipschitz and Triebel-Lizorkin spaces. The case q < 1 361. 362. Journal d Analyse Mathématique 363. Some observations of Besov and Lizorkin-Triebel spaces Some observations on Besov and Lizorkin-Triebel spaces 364. Pseudo-differential Pseudodifferential 365. 65