Orthogonal systems and semigroups
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- Ματθαίος Παπαστεφάνου
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1 Orthogonal systems and semigroups José Luis Torrea Orthonet Logroño, 23 Febrero 213 () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
2 Jorge Betancor, Raquel Crescimbeni, Eleonor Harboure Teresa Martinez, Ricardo Testoni, Peter Sjögren Orthogonal systems and semigroups Juan Carlos Fariña, José Luis Torrea Oscar Ciaurri Luz Roncal, Juan Luis Varona, Juan Carlos Fariña Beatriz Viviani, Silvia Harzstein, Orthonet Lourdes Rodriguez-Mesa F. J. Martín-Reyes, Logroño, 23 Febrero 213 A. De La Torre, Gustavo Garrigós Alejandro Sanabria, Zhang Chao, Ma Tao Pablo Stinga, Beatriz Viviani, Roberto Macías Mario Pérez, Bruno Bongioanni, Sundaram Thangavelu Carlos Segovia, Cristian Gutierrez, () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
3 Fourier sums Poisson sums f (θ) = k a k e ikθ () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
4 Fourier sums Poisson sums f (θ) = k a k e ikθ P r f (θ) = k r k a k e ikθ = 1 + k> r k a k e ikθ + k> r k a k e ikθ }{{} = 1 + a k z k + r k a k z k = U(z) (z=re iθ ) k> k> () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
5 Fourier sums Poisson sums f (θ) = k a k e ikθ P r f (θ) = k r k a k e ikθ = 1 + k> r k a k e ikθ + k> r k a k e ikθ }{{} = 1 + a k z k + r k a k z k = U(z) (z=re iθ ) k> k> Harmonic. ( r 2 2 r + r r + 2 θ) Pr f (θ) = z z U = () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
6 Fourier sums Poisson sums f (θ) = k a k e ikθ P r f (θ) = k r k a k e ikθ = 1 + k> r k a k e ikθ + k> r k a k e ikθ }{{} = 1 + a k z k + r k a k z k = U(z) (z=re iθ ) k> k> Harmonic. ( r 2 2 r + r r + 2 θ) Pr f (θ) = z z U = (r r ) = r r with respect to dµ(r) = dr r en [, 1]. ( θ ) = θ with respect to a dθ. Decomposition: [ ] r 2 r 2 + r r + θ 2 = (r r ) (r r ) + ( θ ) ( θ ) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
7 Conjugate harmonic function (Fourier series) f (θ) = k a k e ikθ () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
8 Conjugate harmonic function (Fourier series) f (θ) = k a k e ikθ Q r f (θ) = i k = i k> sign k r k a k e ikθ r k a k e ikθ + i k> r k a k e ikθ }{{} = i a k z k + i r k a k z k = V (z) (z=re iθ ) k> k> () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
9 Conjugate harmonic function (Fourier series) f (θ) = k a k e ikθ Q r f (θ) = i k = i k> sign k r k a k e ikθ r k a k e ikθ + i k> r k a k e ikθ }{{} = i a k z k + i r k a k z k = V (z) (z=re iθ ) k> k> Harmonic. ( r 2 2 r + r r + 2 θ) Qr f (θ) =. z z V =. () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
10 Cauchy Riemann equations ( Fourier series) P r f (θ) = k r k a k e ikθ = 1 + k> a k z k + k> r k a k z k = U(z) Q r f (θ) = i sign k r k a k e ikθ = i a k z k + i r k a k z k = V (z) k k> k> () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
11 Cauchy Riemann equations ( Fourier series) P r f (θ) = k r k a k e ikθ = 1 + k> a k z k + k> r k a k z k = U(z) Q r f (θ) = i sign k r k a k e ikθ = i a k z k + i r k a k z k = V (z) k k> k> F (z) = U(z) + iv (z) = k> a k z k is holomorfic: z F = () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
12 Cauchy Riemann equations ( Fourier series) P r f (θ) = k r k a k e ikθ = 1 + k> a k z k + k> r k a k z k = U(z) Q r f (θ) = i sign k r k a k e ikθ = i a k z k + i r k a k z k = V (z) k k> k> F (z) = U(z) + iv (z) = k> a k z k is holomorfic: z F = Cauchy-Riemann (Fourier series) θ (P r f )(θ) = r r (Q r f )(θ) ( ) r r (P r f )(θ) = θ (Q r f )(θ) i.e. (r r ) (P r f )(θ) = ( θ ) (Q r f )(θ) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
13 On R f (x) = f (ξ)e ixξ dξ R () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
14 On R f (x) = R f (ξ)e ixξ dξ P t f (x) = R = }{{} (z=t ix) e t ξ f (ξ) e iξ x dξ e zξ f (ξ)dξ + e zξ f ( ξ)dξ = U(z) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
15 On R f (x) = R f (ξ)e ixξ dξ P t f (x) = R = }{{} (z=t ix) e t ξ f (ξ) e iξ x dξ e zξ f (ξ)dξ + e zξ f ( ξ)dξ = U(z) Harmonic. ( 2 t + 2 x ) Pt f (x) = z z U = () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
16 On R f (x) = R f (ξ)e ixξ dξ P t f (x) = R = }{{} (z=t ix) e t ξ f (ξ) e iξ x dξ e zξ f (ξ)dξ + e zξ f ( ξ)dξ = U(z) Harmonic. ( 2 t + 2 x ) Pt f (x) = z z U = ( x ) = x with respect to dx ( t ) = t with respect to dt Decomposition. [ ] t 2 + x 2 = ( t ) ( t ) + ( x ) ( x ) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
17 Conjugate harmonic function (line) f (x) = f (ξ)e ixξ dξ R () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
18 Conjugate harmonic function (line) f (x) = R f (ξ)e ixξ dξ Q t f (x) = i R }{{} = i (z=t ix) sign ξ e t ξ f (ξ) e iξ x dξ e zξ f (ξ)dξ + i e zξ f ( ξ)dξ = V (z) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
19 Conjugate harmonic function (line) f (x) = R f (ξ)e ixξ dξ Q t f (x) = i R }{{} = i (z=t ix) sign ξ e t ξ f (ξ) e iξ x dξ e zξ f (ξ)dξ + i e zξ f ( ξ)dξ = V (z) Harmonic. ( 2 t + 2 x ) Qt f (x) = z z V = () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
20 Cauchy Riemann equations (line) P t f (x) = e t ξ f (ξ) e iξ x dξ = R Q t f (x) = i sign ξ e t ξ f (ξ) e iξ x dξ = i R e zξ f (ξ)dξ + e zξ f ( ξ)dξ = U(z) e zξ f (ξ)dξ + i e zξ f ( ξ)dξ = V (z) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
21 Cauchy Riemann equations (line) P t f (x) = e t ξ f (ξ) e iξ x dξ = R Q t f (x) = i sign ξ e t ξ f (ξ) e iξ x dξ = i R e zξ f (ξ)dξ + e zξ f ( ξ)dξ = U(z) e zξ f (ξ)dξ + i e zξ f ( ξ)dξ = V (z) F (z) = U(z) + iv (z) is holomorphic: z F = () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
22 Cauchy Riemann equations (line) P t f (x) = e t ξ f (ξ) e iξ x dξ = R Q t f (x) = i sign ξ e t ξ f (ξ) e iξ x dξ = i R e zξ f (ξ)dξ + e zξ f ( ξ)dξ = U(z) e zξ f (ξ)dξ + i e zξ f ( ξ)dξ = V (z) F (z) = U(z) + iv (z) is holomorphic: z F = Cauchy Riemann (line) x P t f (x) = t Q t f (x) ( ) t P t f (x) = x Q t f (x) i.e. ( t ) P t f (x) = ( x ) Q t f (x) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
23 Orthogonal polynomials (Hermite) Hermite Polynomials on R: H n (x) = ( 1) n e x 2 /2 d n 2 dx n e x /2. () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
24 Orthogonal polynomials (Hermite) Hermite Polynomials on R: H n (x) = ( 1) n e x 2 /2 d n 2 dx n e x /2. f (x) = n a n H n (x) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
25 Orthogonal polynomials (Hermite) Hermite Polynomials on R: H n (x) = ( 1) n e x 2 /2 d n 2 dx n e x /2. f (x) = n a n H n (x) Poisson sums : P r f (x) = r n a n H n (x)... (r = e t ) n= () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
26 Orthogonal polynomials (Hermite) Hermite Polynomials on R: H n (x) = ( 1) n e x 2 /2 d n 2 dx n e x /2. f (x) = n a n H n (x) Poisson sums : P r f (x) = r n a n H n (x)... (r = e t ) n= There is a kernel : [ P t f (x) = e tn a n H n (x) = e tn n= = R n= [ ] e tn H n (y) H n (x) f (y)e y 2 dy = n= R ] f (y)h n (y)e y 2 dy H n (x) R K t (x, y)f (y)e y 2 dy... () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
27 Literature B. Muckenhoupt, Poisson integrals for Hermite and Laguerre expansions, Trans. Amer. Math. Soc. 118 (1965), B. Muckenhoupt, Hermite conjugate expansions, Trans. Amer. Math. Soc. 139 (1969), B. Muckenhoupt and E. M. Stein Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965), () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
28 Literature () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
29 Understanding Muckenhoupt Two alternatives following Muckenhoupt P t f (x) = n e t2n a n H n (x) versus P t f (x) = n e t 2n a n H n (x). () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
30 Understanding Muckenhoupt Two alternatives following Muckenhoupt P t f (x) = n e t2n a n H n (x) versus P t f (x) = n e t 2n a n H n (x). The second satisfies ( 2 t + x 2 ) 2x x (Pt f )(x) = ( t 2 ) + ( x 2x) x (Pt f )(x) ] = [( t ) ( t ) + ( x ) x (P t f )(x) = () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
31 Understanding Muckenhoupt Two alternatives following Muckenhoupt P t f (x) = n e t2n a n H n (x) versus P t f (x) = n e t 2n a n H n (x). The second satisfies ( 2 t + x 2 ) 2x x (Pt f )(x) = ( t 2 ) + ( x 2x) x (Pt f )(x) ] = [( t ) ( t ) + ( x ) x (P t f )(x) = ( x ) = ( x 2x) Adjoint with respect to measure dγ(x) = e x 2 dx () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
32 Q t f (x) = n Understanding Muckenhoupt Moreover 2n e t 2n a n H n 1 (x) satisfies ( 2 t + x ( x 2x) ) (Q t f )(x) = [ ( t )( t ) + x ( x ) ] (Q t f )(x) = () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
33 Understanding Muckenhoupt Moreover 2n e t 2n a n H n 1 (x) Q t f (x) = n satisfies ( 2 t + x ( x 2x) ) (Q t f )(x) = [ ( t )( t ) + x ( x ) ] (Q t f )(x) = Cauchy Riemann equations (Hermite) x P t f (x) = t Q t f (x) ( ) t P t f (x) = ( x 2x)Q t f (x), i.e. ( t ) P t f (x) = ( x ) Q t f (x) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
34 Common path? In all the cases, lim Q tf (x) = t () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
35 Common path? In all the cases, Hence Q t f (x) = t s Q s f (x) ds = lim Q tf (x) = t t x P s f (x) ds = x P s f (x) ds t () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
36 Common path? In all the cases, Hence Q t f (x) = t s Q s f (x) ds = lim Q tf (x) = t lim Q tf (x) drives to an important operator: t t x P s f (x) ds = x P s f (x) ds (1) lim i sign k r k a k e ikθ = i sign k a k e ikθ (Conjugate function). t k k (2) lim i sign ξe t ξ f (ξ)e iξx dξ = Ĥf (ξ)e iξx dξ (Hilbert transform). t R R t () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
37 Common path? In all the cases, Hence Q t f (x) = t s Q s f (x) ds = lim Q tf (x) = t lim Q tf (x) drives to an important operator: t t x P s f (x) ds = x P s f (x) ds (1) lim i sign k r k a k e ikθ = i sign k a k e ikθ (Conjugate function). t k k (2) lim sign ξe t ξ f (ξ)e iξx dξ = Ĥf (ξ)e iξx dξ (Hilbert transform). i t R (3) lim t t s Q s f (x) ds = x P s f (x) ds R t () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
38 Common path? In all the cases, Hence Q t f (x) = t s Q s f (x) ds = lim Q tf (x) = t lim Q tf (x) drives to an important operator: t t x P s f (x) ds = x P s f (x) ds (1) lim i sign k r k a k e ikθ = i sign k a k e ikθ (Conjugate function). t k k (2) lim sign ξe t ξ f (ξ)e iξx dξ = Ĥf (ξ)e iξx dξ (Hilbert transform). i t R (3) lim t t s Q s f (x) ds = x P s f (x) ds R x P s f (x) ds? () Orthogonal systems and semigroups Logroño, 23 Febrero / 29 t
39 Diffusion semigroups E. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley theory, Princeton, 197. (M, dµ) measure space. {T t } t> : L 2 L 2 : T t1+t 2 = T t1 T t2. T = Id. lim t T t f = f in L 2. T t f p f p, (1 p ). Contraction. T t selfadjoint in L 2. T t f si f. Positivity. T t 1 = 1. Markov. () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
40 First example of semigroup. Classical heat equation The example of diffusion semigroup in L 2 (R): 1 T t f (x) = e x y 2 (4πt) 1/2 4t f (y) dy R () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
41 First example of semigroup. Classical heat equation The example of diffusion semigroup in L 2 (R): 1 T t f (x) = e x y 2 (4πt) 1/2 4t f (y) dy For good functions f we have: t (T t f (x)) = 2 x (T t f (x)) = x (T t f (x)) R In symbols T t f (x) = e t f (x) heat semigroup () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
42 First example of semigroup. Classical heat equation The example of diffusion semigroup in L 2 (R): 1 T t f (x) = e x y 2 (4πt) 1/2 4t f (y) dy For good functions f we have: t (T t f (x)) = 2 x (T t f (x)) = x (T t f (x)) R In symbols Interesting remark T t f (x) = e t f (x) heat semigroup T t f (ξ) = e t ξ 2 f (ξ) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
43 Second example of diffusion semigroup. Orthogonal polynomials Ilustration. L with eigenfunctions {φ k } k and eigenvalues {λ k } k e tl φ k (x) = e tλ k φ k (x), e tl( ) c k φ k (x) = e tλ k c k φ k (x) k k () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
44 Second example of diffusion semigroup. Orthogonal polynomials Ilustration. L with eigenfunctions {φ k } k and eigenvalues {λ k } k e tl φ k (x) = e tλ k φ k (x), e tl( ) c k φ k (x) = e tλ k c k φ k (x) k k We have the heat equation for L: t (e tl f (x)) = Le tl f (x) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
45 Second example of diffusion semigroup. Orthogonal polynomials Ilustration. L with eigenfunctions {φ k } k and eigenvalues {λ k } k e tl φ k (x) = e tλ k φ k (x), e tl( ) c k φ k (x) = e tλ k c k φ k (x) k k We have the heat equation for L: t (e tl f (x)) = Le tl f (x) Fourier series case. e ikθ = k 2 e ikθ. e t( )( a k e ikθ) = k k e t k 2 a k e ikθ () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
46 Second example of diffusion semigroup. Orthogonal polynomials Ilustration. L with eigenfunctions {φ k } k and eigenvalues {λ k } k e tl φ k (x) = e tλ k φ k (x), e tl( ) c k φ k (x) = e tλ k c k φ k (x) k k We have the heat equation for L: t (e tl f (x)) = Le tl f (x) Fourier series case. e ikθ = k 2 e ikθ. e t( )( a k e ikθ) = k k e t k 2 a k e ikθ Hermite polynomials case. L = x 2 + 2x x, LH n = 2nH n. Then e tl( ) a n H n (x) = e t2n a n H n (x) n (remember Muckenhoupt) n () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
47 Third example of Poisson semigroup Formula for Gamma function: e t λ = t 2 π e t2 4s s 3/2 e sλ ds, λ >. () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
48 Third example of Poisson semigroup Formula for Gamma function: e t λ = t 2 π e t2 4s s 3/2 e sλ ds, λ >. If L is positive, e t L f = t 2 π e t2 4s s 3/2 e sl f ds. Harmonic ( 2 t L ) (e t L f ) = e t L f : Poisson semigroup () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
49 Third example of Poisson semigroup Formula for Gamma function: e t λ = t 2 π e t2 4s s 3/2 e sλ ds, λ >. If L is positive, e t L f = t 2 π e t2 4s s 3/2 e sl f ds. Harmonic ( 2 t L ) (e t L f ) = e t L f : Poisson semigroup If L can be decompose L = ( x ) ( x ), we get the conjugate harmonic function: Q t f = t s Q s f ds = t x P s f ds () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
50 Determination of kernels If we know e tl then we know e t L. () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
51 Determination of kernels If we know e tl then we know e t L. t 2 π e t2 4s s 3/2 e sl (x, y) ds = t 2 e t2 /4s π = }{{} ( t2 + x y 2 4s =u) = t π s 3/2 1 t π(t 2 + x y 2 ). e x y 2 4s 4πs ds 1 (t 2 + x y 2 ) e u du () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
52 Determination of kernels If we know e tl then we know e t L. t 2 π e t2 4s s 3/2 e sl (x, y) ds = t 2 e t2 /4s π = }{{} ( t2 + x y 2 4s =u) = t π s 3/2 1 t π(t 2 + x y 2 ). e x y 2 4s 4πs ds 1 (t 2 + x y 2 ) e u du Conclusion. If we know the kernel of the heat semigroup, we know the kernel of the Poisson semigroup. () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
53 Riesz Transforms Gamma function firmula: λ α = 1 Γ(α) t α e λt dt, λ, α >. t () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
54 Riesz Transforms Gamma function firmula: λ α = 1 Γ(α) t α e Given φ k, eigenfunction of L with eigenvalue λ k, we have λt dt, λ, α >. t L α φ k = 1 λ α φ k = 1 k Γ(α) t α e λ k t φ k dt t = 1 Γ(α) t α e tl φ k dt t Hence, L α f = 1 Γ(α) t α e tl f dt t () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
55 Riesz Transforms Gamma function firmula: λ α = 1 Γ(α) t α e Given φ k, eigenfunction of L with eigenvalue λ k, we have λt dt, λ, α >. t L α φ k = 1 λ α φ k = 1 k Γ(α) t α e λ k t φ k dt t = 1 Γ(α) t α e tl φ k dt t Hence, L α f = 1 Γ(α) t α e tl f dt t L 1/2 f = ( L) 1 f = 1 Γ(1) Going back to our slide (?) t 1 e t L f dt t = e t L f dt lim Q tf = lim t t t s Q s f (x)ds = x P s f (x)ds = x L 1/2 f () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
56 Stein knew everything!.
57 Stein knew everything!. Look into the red book. 197! () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
58 Stein knew it... () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
59 Boundedness in L 2 of Riesz transforms L = x x () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
60 Boundedness in L 2 of Riesz transforms L = x x Eigenvalue case: ψ k = x (L) 1/2 φ k ( ) ψ k ψ l dµ = x (L) 1/2 φ k )( x (L) 1/2 φ l dµ ( )( ( ) = x x (L) 1/2 φ k (L) 1/2 φ l )dµ = L(L) 1/2 φ k )((L) 1/2 φ l dµ ( ) = (L) 1/2 φ k )((L) 1/2 φ l dµ = λ 1/2 k λ 1/2 l φ k φ l dµ () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
61 Boundedness in L 2 of Riesz transforms L = x x Eigenvalue case: ψ k = x (L) 1/2 φ k ( ) ψ k ψ l dµ = x (L) 1/2 φ k )( x (L) 1/2 φ l dµ ( )( ( ) = x x (L) 1/2 φ k (L) 1/2 φ l )dµ = L(L) 1/2 φ k )((L) 1/2 φ l dµ ( ) = (L) 1/2 φ k )((L) 1/2 φ l dµ = λ 1/2 k λ 1/2 l φ k φ l dµ Spectral theorem: x (L) 1/2 f, x (L) 1/2 f = x x (L) 1/2 f, (L) 1/2 f = (L) 1/2 f, (L) 1/2 f = f, f () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
62 Boundedness in L p of Riesz transforms General procedure in Harmonic Analysis. Once we know the L 2 -boundedness, the kernel is used to get L p boundedness. () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
63 Boundedness in L p of Riesz transforms General procedure in Harmonic Analysis. Once we know the L 2 -boundedness, the kernel is used to get L p boundedness. x L 1/2 f (x) = x e tl 1/2 dt f (x) t t = x e tl 1/2 dt f (x) t t = x e tl 1/2 dt (x, y)f (y) dy t R t ( n ) = x e tl 1/2 dt (x, y) t f (y) dy R n t = K(x, y)f (y) dy. R n () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
64 Possible Applications a priori estimates a priori estimates L = i ( i ) i Lu = f f L p = i j u L p () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
65 Possible Applications a priori estimates a priori estimates L = i ( i ) i Lu = f f L p = i j u L p i j u = i j L 1 f L p () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
66 Application Sobolev Spaces Assume f L p L 1/2 f L p. Equivalently L 1/2 g L p g L p. () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
67 Application Sobolev Spaces Assume f L p L 1/2 f L p. Equivalently Then L 1/2 g L p g L p. {f L p : L 1/2 f L p } = {f L p : f L p } () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
68 Application Sobolev Spaces Assume f L p L 1/2 f L p. Equivalently Then L 1/2 g L p g L p. {f L p : L 1/2 f L p } = {f L p : f L p } Sobolev spaces W k,p L = {f L p : k f L p } = {f L p : L k/2 f L p } () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
69 Muchas gracias! () Orthogonal systems and semigroups Logroño, 23 Febrero / 29
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