(1) ENCOUNTER with MATHEMATICS 2001 10 26
(2) 1807 J. B. J. Fourier 1 2π f(x) f(x) = n= c n (f) = 1 2π c n (f)e inx (1) π π f(t)e int dt Fourier 2 R f(x) f(x) = F[f](ξ)= 1 2π F [f](ξ)e ixξ dξ f(t)e iξx dx f(x) = 1 2π F [f](ξ) = F [f](ξ)e ixξ dξ f(t)e iξx dx
N 1 c n (f)e inx f(x) (N ) 1 2π n= N R R e iξx F [f](ξ)dξ f(x) (R ) (3) Fourier etc.
(4) 1. Hardy-Littlewood 2. Calderón-Zygmund 3. Littlewood-Paley 1. 2. 3. Plancherel L p ( )
(5) 1 Hardy-Littlewood 3 f L 1 loc (Rd ) 1 Mf(x)= Q sup x Q Q: f(y) dy Q Hardy-Littlewood 4 (Hardy-Littlewood ) f L 1 {x : Mf(x) >λ} C 1 λ f L 1 (1,1) {x:tf(x)>λ} C λ f L 1
(6) 5 (Marcinkiewicz ) (X, µ) (Y,ν) :σ 1 <p 1 T :L 1 (X, µ)+l p 1(X, µ) {Y }( ) T : L p (1,1) = T : L p (X, µ) L p (Y,ν)( ) 1 <p<p 1 Mf L f L Hardy-Littlewood Marcinkiewicz Mf L p C p f L p (1 <p< ) 6 ( ) f(x) d dx x lim h 0 1 2h a f(t)dt = f(x) x+h x h f(t)dt = f(x)
(7) Lebesgue : 7 (Lebesgue ) f L 1 loc (Rd ) 1 lim r 0 Q(x, r) lim r 0 1 Q(x, r) f(y)dy = f(x)a.e. x R d Q(x,r) Q(x,r) f(y) f(x) dy =0 a.e. x R d (2) Q(x, r) x r (2) f(x) Lebesgue
(8-i) 7 f L 1 p 1 T (p)(x)= lim sup f(y) f(x) dy r 0 Q(x, r) Q(x,r) {x : T (f)(x) > 0} = 0 (3) λ>0: {x : T (f)(x) >λ} = 0 (4) Lebesgue ε>0 ε >0, g C 0 : f g L 1 <ε f = h + g (h := f g) T (g) =0 T (f) =T (h + g) T (h)+t(g) =T (h)
(8-ii) T (h)(x) Mh(x) + h(x) { {T (h) >λ} Mh> λ { + 2} h > λ 2} Hardy-Littlewood : {Mh> λ} C 1 λ h L 1 < C λ ε Chebyshev : { h > λ } 2 2 λ h L 1 {x : T (f)(x) >λ} < C λ ε Hardy-Littlewood Hardy-Littlewood
(9) Hardy-Littlewood Π + = {(x, t) :x R, t > 0} Π + = R f L p (R) P [f](x, t) = 1 π t x t 2 + t 2f(y)dy, (x, t) Π + (Poisson ) Π + Hardy-Littlewood x Π + α Stoltz Γ α (x) ={(y, t): x y <αt}, x Π +, α > 0 (x) x Π + u(x, t) N α (u)(x)= sup (y,t) Γ α (x) u(y, t) N α (P [f])(x) CMf(x), x Π + (5)
(10) 8 x 0 Π + f(x) Lebesgue, lim P [f](x, t) =f(x 0 )(6) (x,t) (x 0,0), (x,t) Γ α (x 0 ) 2 (5) x y <αt t π P [f](y, t) y z t y z 2 + t 2 f(z) dz t + k=0 2 k t< y z 2 k+1 t y z 2 + t 2 f(z) dz 1 f(z) dz t y z t 1 + f(z) dz (2 2k +1)t k=0 y z 2 k+1 t 1 f(z) dz t x z (1+α)t 1 + f(z) dz (2 2k +1)t k=0 x z (α+2 k+1 )t 1 Mf(x) + 2kMf(x) Mf(x) k=0
(11) (6) x 0 :Lebesgue ε >0 δ >0:δ r>0= 1 f(y) f(x 0 ) dy < ε Q(x 0,r) Q(x 0,r), g(x) = f(x) f(x 0 ) χ Q(x0,δ) Mg(x 0 ) Cε. y <αt P [f](x 0 y, t) f(x 0 ) = 1 P t (z y) {f(x 0 z) f(x 0 )} dz := (I) π z 2 +t 2 z y 2 + y 2 +t 2 z y 2 +(1+α 2 )t 2 P t (z y) C α P t (y) (I) P t (z) f(x 0 z) f(x 0 ) dz + z <δ z δ P t (z)g(x 0 z)dz z <δ + P t (z) f(x 0 z) f(x 0 ) dz =:(II)+(III) z δ (II) P [g](x 0,t) Mg(x 0 ) <ε (III) 0 (t 0)
(12) 2 Calderón-Zygmund ( ) Tf(x) = K(x, y)f(y)dy (7) K(x, x) = : Hilbert K(x, y) = 1 x y, x,y R Hilbert L p = Fourier L p
(13) Fourier S R f(x) = 1 2π R R Plancherel e ixξ F [f](ξ)dξ (Fourier ) f L 2 (R): lim R S Rf f L 2 = 0 (8) L p 1 1 Hf(x)= lim ε>0 π x y >εx y f(y)dy M a f(x) =e iax f(x) S R f(x) = i 2 (M ahm a M a HM a ) Hilbert L p = (8) Hilbert L p M. Riesz Calderón&Zygmund
(14) f L p (R)(1 p< ) (x, y) Π + P y [f](x) = 1 π Q y [f](x) = 1 π y (x u) 2 + y 2f(u)du (Poisson ) x u (x u) 2 + y 2f(u)du ( Poisson ) u(x, y)=p y [f](x), ũ(x, y)=q y [f](x):π + z = x + iy f(z):= u(x, y)+iũ(x, y) {x + iy : x R, y > 0} ũ(x, y) : u(x, y) u(x, y) f(x) ũ(x, y) Hf(x) (y 0)a.e. x (y 0)a.e. x
(15) 1. 2. Hilbert
(16) 9 ( ) = { (x, x) :x R d} K(x, y) δ>0 K(x, y) :R d R d \ 1 K(x, y) C x y d K(x, y) K(x, z) C y z δ x y d+δ, y z < 1 2 x y K(x, y) K(w, y) C x w δ x y d+δ, x w < 1 2 x y T : S(R d ) S (R d )!K S (R d R d ): Tϕ,ψ = K, ψ ϕ, ϕ,ψ S(R d ) ψ ϕ(x, y) =ψ(x)ϕ(y) K : T
(17) 10 T : S(R d ) S (R d ) T L 2 Calderón-Zygmund
(18) : 1. T L p 2. f L p (R d ) Cauchy lim K(x, y)f(y)dy ε 0 x y >ε 3. T : Littlewood-Paley Lipschitz Wavelet ( Haar wavelet ) : Calderón-Zygmund
(19) L p (Calderón-Zygmund ) 1. L 2 2. (1,1) 3. Marcinkiewicz 1 <p<2 4. duality argument 2 <p< L 2 : Cotlar-Stein Shur T1 Tb 2 : Calderón-Zygmund =. T ( T ( ), T( )
(20) 2 R d 2 k Z m =(m 1,,m d ) Z d [ m1 Q k,m = 2, m ) [ 1 +1 md k 2 k 2, m ) d +1 k 2 k ( k 2 ) D k = { Q k.m : m Z d} D= k Z D k, 11 (1) R d = Q D k Q k Z (2) Q k,m Q k,m = (m m ) (3) k l Q D k Q D l = Q Q Q Q = (4) k l Q D k,!q D l : Q Q (5) Q D k, Q D k 1 Q 2 d Q
(21) 2 : Calderón-Zygmund Littlewood-Paley 2 12 (Calderón-Zygmund ) f L 1 (R d ) f 0 λ >0 2 Q j (j =1, 2, ) ( 2 ) g(x) b j (x) (j =1, 2, ): (1) f(x) =g(x) + b j (x) (2)0 g(x) Cλ a.e.x g L 1 C f L 1 (3) {x : b j (x) 0} = Q j (4) b j (x)dx =0 (5) b j L 1 C Q j (6) Qj C λ f L 1
(22) 13 (Calderón-Zygmund ) f L 1 (R d ) f 0 λ>0 2 Q j (j =1, 2, ) (1) f(x) λ a.e. x/ Q j (2) Q j 1 λ f L 1 (3) λ< 1 f(x)dx (:= m j ) 2 d λ Q j Q j Calderón-Zygmund 13 g(x) =f(x)χ R d \Ω λ + m j χ Qj (x) b j (x) ={f(x) m j } χ Qj (x) j 13 : E k f(x) = Q D k f L 1 loc ( ) 1 f(y)dy χ Q (x) Q Q x R d M d f(x)= sup k E k f(x) 2
(23) E k : D k σ F k Doob Hardy-Littlewood Martingale Lebesgue 14 (1) C>0 {x : M d f(x) >λ} C λ f L 1, f L 1 (2) f L 1 lim E kf(x) =f(x)a.e. x k + lim E kf(x) =0 k x R d λ>0 Ω λ = {x : M d f(x) >λ}. x/ Ω λ Lebesgue f(x)= lim k E k f(x) M d f(x) λ Ω λ =0 g = f b j =0
(24) Ω λ 0 : A k = {x : E k f(x) >λ,e j f(x) λ (j <k)} Ω λ = A k, A k A k = (k k ) k Z E k f E j f (j <k): Q D k Ω λ = Q j, Q j : 2 Hardy-Littlewood Qj = Ω λ C λ f L 1 Q j : Q j Q j λ< 1 f(x)dx Q j Q j 2 1 Q j f(x)dx λ Q j 1 f(x)dx = Q j Qj Q j 1 Q j Q j f(x)dx Q j 2d Q j f(x)dx 2 d λ Q j
(25) Haar 15 (X, ) Banach {x i : i N} X x X,! {a i } N i=1 C : lim N x a i x i =0 i=1 {x i : i N} { x σ(i) : i N } X Banach Hardy H 1 Maurey (1980)( ) Carleson, Wojtaszczyk ( ) Y. Meyer H 1, Besov Triebel-Lizorkin etc. Donoho (1993, 1996)
(26) Donoho sparsity critical index X =[0, 1] or [0, 1] 2 {φ i } : L 2 (X) θ i (f) = f, φ i f θ i φ i i=1 θ (i) : θ wl p = sup i 1 i 1/p θ (i) ( ) 1 p, δ > 0 {i : θ i >δ} θ wl p Θ δ p (Θ)= inf {p :Θ wl p } p sparse optimal (Donoho)
(27) 16 (Haar ) h(x) =χ [ 0, 1 2) (x) χ [ 1 2,1 ) (x) j, k Z : {h j,k } : Haar h j,k (x) =2 j/2 h(2 j x k) Haar : L 2 (R) L p (R) : Calderón-Zygmund β = {β j,k }, β j,k 1 0 T β f(x) = K β (x, y) = j Z k Z K β (x, y)f(y)dy L p (1 <p< ). h j,k (x)h j,k (y)
(28) T β f(x)g(x)dx h j,k (x)g(x)dx h j,k (y)f(y)dy j Z k Z = h j,k,g h j,k,f j Z k Z h j,k,g 2 1/2 j,k Z j,k Z = g L 2 f L 2 h j,k,f 2 1/2 (1,1) : f L 1 (R) f 0 λ>0 CZ : f = g + b, b = bj { { Tf >λ} Tg > λ { + 2} Tb > λ 2}
[Good part ] { Tg > λ } 4 2 λ 2 C λ [Bad part ] Tg 2 dx C g 2 dx λ 2 g dx C f L 1 (29) b i x / Q i Tb i (x) = h j,k (x) h j,k (y)b i (y)dy =0 j,k Q : h j,k 2 i) Q Q i = ii) Q Q i iii) Q i Q i): h j,k b i =0 ii): h j,k (x) =0 iii): b i cancellation hj,k b i =0 (h j,k Q i = ) { Tb > λ 2} Qi C λ f L 1
(30) Lipschitz Cauchy a(x, ξ) : S m : a(x, ξ) C (R d R d ) β x α ξ a(x, ξ) C α,β (1 + ξ ) m α ( (x, ξ) R d R d) T a f(x) = 1 a(x, ξ)f[f](ξ)dξ (2π) d R d a(x, ξ) S 0 Calderón-Zygmund. S 0 1,1 CZO Wavelet Bergman ( )
(31) 3 Littlewood-Paley {φ j } : L 2 β j R ( β j 1) f = cj φ j h = βj c j φ j Plancherel h L 2 f L 2 h L p C f L p p 2 (9) {φ j } : L p (9) Fourier L p (p 2)
(32) f L 1 (T ) n Fourier n s n (f; x) = c k (f)e ikx k= n g(f)(x) 2 = s 2 n+1 1(f; x) s 2 n 1(f; x) 2 = n=0 n=0 2 n k 2 n+1 1 g(f)(x) c n (f)e inx Littlewood-Paley g 2 17 (Littlewood-Paley) 1 <p< C p f L p g(f) L p C p f L p c 0 = 1 π f(x)dx =0 2π π
(33) 2 n k 2 n+1 1 β k ( β k 1) h L p C p f L p
(34) 18 ( ) ϕ j S ( R d) (j =0, 1, 2, ) (1)supp ϕ 0 { x 2} (2)supp ϕ j { 2 j 1 x 2 j} j =1, 2, (3) α ϕ j (x) c α 2 j α j =0, 1, 2, (4) ϕ j 1 j=0 Φ ( R d) j f(x) =F 1 [ϕ j F [f]] g : G(f)(x) = ( j Z j f(x) 2 ) 1/2 19 1 <p< C p f L p G(f) L p C p f L p
(35) G(f) : ( ( G s (f)(x) = 2 sj j f(x) ) ) 1/2 2, j Z <s< C p f H s p Gs (f) L p C p f H s p, 1 <p<, <s< H s p s L p Sobolev : [ ( f H s := p F 1 1+ ξ 2) s ] 2 F [f] Triebel-Lizorkin ( G s q (f)(x) = ( 2 sj j f(x) ) ) 1/q q, j Z L p <s< G s (f)(x)= sup 2 sj j f(x), <s< j Z { F s p,q (Rd )= f S (R d ): f F s p,q = G s q } < L p
(36) G s q = L p ( j Z ( 2 sj j f(x) ) q ) 1/q L p l q L p : Besov <s<, 0<p,q ( ) 1/q ( ) f B s p,q = 2 sj q j f(x) L p j Z f B s = sup 2 sj j f(x) p, L p j Z { } B s p,q (Rd )= f S (R d ): f B s p,q < B s, = Cs s>0 (Hölder Littlewood-Paley )
(37) 1 ψ S ( R d) { } 1 supp ψ x 2, 2 ( 1 ψ(x) > 0 2 x ) 2 ϕ j (x) = ψ(2 j x) ψ(2 k x) k= ϕ 0 (x) =1 ϕ j (x) j=1 (j =1, 2, ),
(38) 2 19 l 2 Hörmander- Mihlin 20 1 <p< m k,j L (R d ) f j (j, k =0, 1, 2, ) ( [ ] F 2) 1/2 1 m k,j F [f j ] k=0 j=0 L p ( ) 1/2 C p,m f k (x) 2 L p k=0 C p,m = c sup R>0 0 α [d/2] (R 2 α d R/2 x 2R j,k=0 D α m k,j (x) 2 dx ) 1/2
(39) L-P 21 0 <ε 1 {S k (x, y)} k Z : S k (x, y) C2 kd S k (x, y) =0 ( x y C2 k ) S k (x, y) S k (x,y) C2 k(d+ε) x x ε S k (x, y) S k (x, y ) C2 k(d+ε) y y ε [S k (x, y) S k (x,y)] [S k (x, y ) S k (x,y )] C2 k(d+ε) x x ε y y ε S k (x, y)dx = S k (x, y)dy =1 (x, y R d ) D k (x, y) =S k (x, y) S k 1 (x, y) (k Z) f S D k f(x) = D k (x, ),f
(40) Han, Jawerth, Taibleson, Weiss (1990): D k f S k f. h C ([0, ]) supp h [0, 2] h(x) =1, (x [0, 1/2]) T k f(x):= 2 kd h ( 2 k x y ) f(y)dy ( C 1 T k 1 C). M k f(x):= 1 T k 1(x) f(x), [ ( ) 1 1 W k f(x):= T k (x)] f(x), T k 1 S k = M k T k W k T k M k S k (x, y) :S k 1 S k (x, y) = m(x)m(y) t(x, z)t(z, y) dz w(z) m(x) =T k 1(x), t(x, y) =2 kd h(2 k x y ), [ ( ) ] 1 1 w(x) = T k (x) T k 1
(41) ( ε ) Littlewood-Paley 22 (Han, Jawerth, Taibleson, Weiss, 1990) ε <α<ε 1 p, q [ ( f 2 kα D k f ) ] 1/q q Fp.q α f B α p,q k Z [ k Z L p ] 1/q ( ) 2 kα p D k f L p Han and Sawyer (1994)
(42) 23 (Han, Lu, Yang ( ), 1999) ε <α<ε 1 p, q [ ( f F α p,q 2 kα D k f ) ] 1/q q k Z + L p f B α p,q [ k Z + ] 1/q ( ) 2 kα q D k f L p D 0 = S 0
(43) 24 (Coifman, Weiss, 1971) (X, ρ, µ) X Hausdorff ρ µ X Radon 0 <µ(b ρ (x, 2r)) Cµ (B ρ (x, r)) < B ρ (x, r) ={y : ρ(x, y) <r} {B ρ (x, r)} r>0 X 25 X 1,,X m : Hörmander R d C = (U, ρ, dx) : (Nagel, Stein, Wainger, Fefferman, Phong ) (ρ )
(44) 26 (X, g): Cartan-Hadamard < b 2 K a 2 < 0 S( ): Eberlein-O Neil ( ), ω : (on S( )) = (S( ),ρ,ω): (Anderson, Schoen; see also A [8]) CR etc.
(45) [1] J. Duoandikoetxea, Fourier Analysis, Graduate Studies in Math. vol. 29, Amer. Math. Soc. 2001. [2] J. García-Cuerva and J. L. Rubio de Fracncia, Weighted Normn Inequalities and Related Topics, North-Holland, 1985. [3] E. Hernandez and G. Weiss, A First Course on Wavelets, CRC Press, 1996. [4] E. M. Stein, Harmonic Analysis, Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, 1993. [5] 50 (1997), 29-55.
f(x)dx f(x) dx
ψ,ψ C(R) c ψ(x) (1 + x ) 1+α ψ c (x) (1 + x ) 1+α ψ(x + h) ψ(x) h β ψ (x) ψ (x + h) h β ψ(x) = ψ (x) =0 T η f = η j,k 2 j f, ψ (2 j x k) ψ(2 j s k) j,k = T η CZO CZ-norm C η l