Notes March 9, 27 1 Fourier transform and L p spaces For a function in f L 1 (R n ) define the Fourier transform ˆf(ξ) = f(x)e 2πi x,ξ dx. Properties R n 1. f g = ˆfĝ 2. δλ (f)(ξ) = ˆf(λξ), where δ λ f(x) = λ n f(x/λ). 3. τh (f)(ξ) = ˆf(ξ)e 2πi h,ξ, where τ h (f)(x) = f(x + h). 4. j f(ξ) = 2πiξ j ˆf(ξ). 5. 2πix j f(ξ) = ξ j 6. fg = ˆf ĝ. ˆf(ξ) Plancherel s formula fĝdx = R n fḡdx = R n R n ˆfgdx R n ˆf ĝdx, f L2 = ˆf L2 1
Inversion formula f(x) = ˆf(ξ)e 2πi x,ξ dξ Schwartz class and distributions φ is in the Schwartz class S, if all of the seminorms R n p α,β (φ) = sup x x α D β φ(x) = sup x α 1 1... x αn n β 1 1... n βn φ(x) x are finite, the distributions are bounded (in the topology of S) linear functionals on S. Operations on distributions 1. α h, φ := ( 1) α h, α φ 2. ĥ, φ := h, ˆφ 3. hg, φ := h, gφ, as long as g C with polynomial bounds. 4. h g, φ := h, g φ, where g(x) := g( x). Homogeneous Distributions ( x a )(ξ) = πa n/2 Γ((n a)/2) ξ a n. Γ(a/2) The fundamental solution of the Laplace s operator G(X) (i.e. G = δ) is G(X) = { Γ(n/2) Explicit solution to some notable PDE s The solution to the Helmholtz equation x 2 n 2(2 n)π n/2 n 3, 1 2π n = 2 u = f is given by u(x) = { Γ(n/2) 2(n 2)π n/2 f(y) dy n 3 x y n 2 (2π) 1 f(y) ln( x y )dy n = 2 2
The solution to the heat equation u t = u; u(, x) = f(x), x R n is given by 1 u(t, x) = e x y 2 /4πt f(y)dy. (4tπ) n/2 R n The solution to the Schrödinger equation is given by iu t = u; u(, x) = f(x), x R n u(t, x) = Approximation of identity 1 e i x y 2 /4πt f(y)dy. (4itπ) n/2 R n Lemma 1. Suppose Φ is sufficiently decaying and Φdx = 1, then for all 1 p <. lim Φ t f f t L p = 2 Interpolation Complex Interpolation Theorem 1. (Riesz-Thorin interpolation theorem ) Let 1 p, p 1, q, q 1. Suppose that for some measure spaces (Y, ν), (X, µ), and a linear operator T T f L q (Y,ν) C f L p (X,µ), T f L q 1 (Y,ν) C 1 f L p 1 (X,µ), i.e. T : L p (X, µ) L q (Y, ν), i.e. T : L p 1 (X, µ) L q 1 (Y, ν). Then for all θ (, 1) and (p, q) : 1/p = (1 θ)/p + θ/p 1 ; 1/q = (1 θ)/q + θ/q 1, there exists a constant C where C C (1 θ) C θ 1. T f L q (Y,ν) C f L p (X,µ), i.e. T : Lp (X, µ) L q (Y, ν) 3
Real Interpolation Theorem 2. (Marcinkiewicz interpolation theorem ) Let 1 p, p 1. Suppose that for a subinear operator T T f L p, C f L p, i.e. T : L p L p,, T f L p 1, C 1 f L p 1, i.e. T : L p 1 L p1,. Then for all θ (, 1) and p : 1/p = (1 θ)/p + θ/p 1, there exists a constant C = C(C, C 1, θ), but independent of f, so that T f L p C f L p, i.e. T : L p L p. Hausdorf-Young inequality For all 2 p, ˆf L p C p f L p, where 1/p + 1/p = 1. Young s inequality For all 1 p, q, r and 1 + 1/q = 1/p + 1/r, f g L q f L p g L r. Moreover, if in addition p > 1, 1 < q, r <, then Weak L p spaces f g L q f L p, g L r. A function f is in the space L p,, if sup λ 1/p {x : f(x) λ} <. λ> 3 Sobolev embedding theorems Define the Sobolev spaces Ẇ p,s = {f : s f L p (R n ) < } W p,s = Ẇ p,s L p 4
Lemma 2. (Non-sharp version) For all 1 p < q with n(1/p 1/q) < s, f q C q,p f W p,s. Lemma 3. (Sharp version) For all 1 < p < q < with s = n(1/p 1/q), f q C q,p f Ẇ p,s. 4 Hardy-Littlewood maximal function and singular integrals The (uncentered) Hardy-Littlewood maximal function is defined by 1 Mf(x) = f(y) dy. Q sup Q x,q cube We have that in pointwise sense M cubes f M balls f M dyadic balls M dyadic cubes. Theorem 3. The Hardy-Littlewood maximal function Mf is bounded on L p, 1 < p and is of weak type (1, 1), that is Q Mf p C p f p sup λ {x : Mf(x) λ} C f 1. λ Lemma 4. Let φ be such that there exists an integrable, radial decreasing majorant ψ. Then sup f φ ε (x) ψ L 1Mf(x). ε Definition 1. We say that an operator given by T f(x) = K(x, y)f(y)dy for all x / supp(f) is Calderón-Zygmund if they satisfy 1. T : L 2 L 2 5
2. sup y,z x y 2 y z condition. K(x, y) K(x, z) dx C, which is known as the Hörmander s Condition 1 above is equivalent to ˆK L, when K(x, y) = K(x y). The Hörmander s condition is a consequence of the easier to verify gradient condition, when K(x, y) = K(x y). which is K(x) C x n 1. Theorem 4. The Calderón-Zygmund singular integrals are bounded operators on L p and are of weak type (1, 1), that is Examples T f p C p f p sup λ {x : T f(x) λ} C f 1. λ The Hilbert transform is 1 Hf(x) = lim ε π x y >ε f(y) x y dy It is also given by Ĥf(ξ) = isgn(ξ) ˆf(ξ), whence H 2 = Id. The Riesz transforms are defined by R j f(x) = Γ((n + 1)/2) π (n+1)/2 lim ε x y >ε with R j f(ξ) = i ξ j ξ ˆf(ξ), whence R 2 j = Id. Corollary 1. We have for all 1 < p <, that x j y j f(y)dy x y n+1 f L p (R 1 ) Hf L p (R 1 ) n f L p (R n ) R j f L p (R n ) 6 j=1
Also, for every 1 < p < and all integers s, s f L p (R n ) n α: α =s α f L p (R n ) In particular, any of these two expressions can be taken for a norm in Ẇ p,s. Corollary 2. For the Helmholtz equation, u = f, one has sup j k u L p (R n ) C n f L p (R n ). j,k=1,...,n Vector valued Calderón-Zygmund operators Theorem 5. Suppose T f(x) = K(x y)f(y)dy, where for every x, K(x) is a bounded operator between two Banach spaces A and B, that is K(x) L(A, B). If T satisfies 1. T : L 2 (A) L 2 (B), 2. K(x) L(A,B) C x n 1. Then T : L p (A) L p (B) and is of weak type (1, 1), that is ( ) 1/p ( T f(x) p B dx C p ) 1/p f(x) p A dx sup λ {x : T f(x) B λ} C f(x) A dx. λ Corollary 3. Let T : L 2 L 2 and satisfies the Hörmaner s condition. Then for every 1 r, ( ) 1/r ( ) 1/r T f j r L p (R n ) C n f j r L p (R n ) j j 7
5 Littlewood-Paley theory Define a function Φ C (R n ), so that Φ is even and supp Φ {ξ : ξ 2} and Φ(ξ) = 1, ξ 1. Take ψ(ξ) := Φ(ξ) Φ(2ξ). Note k Z ψ(2 k ξ) = 1. Define P k f(ξ) = ψ(2 k ξ) ˆf(ξ) P k f(x) = 2 kn ˇψ(2 k (x y))f(y)dy R n P k = m k P m etc. Proposition 1. For all 1 p : c 1 sup P k f p f p k k P k f p. For all 1 p : P k f p 2 k P k f p. 5.1 Bernstein inequality & Sobolev embedding Proposition 2. (Bernstein) Let A R n be any measurable set, P A f(ξ) := χ A (ξ) ˆf(ξ). Let 1 q 2 p. Then P A f L p A 1/q 1/p f L q. For Littlewood-Paley pieces P k f this can be generalized to 1 q p P k f L p (R n ) C n 2 kn(1/q 1/p) P k f L q (R n ). 5.2 Littlewood-Paley theory - Square functions Theorem 6. For all 1 < p <, one has ( ) 1/2 P k f 2 f p. p k Z 8
The dual inequality for an arbitrary sequence {f k } k (maybe not Littlewood-Paley pieces!) ( ) 1/2p P k f k C p fk 2. p k 5.3 Hörmander-Mikhlin multiplier criteria Theorem 7. Let T m be a multiplier type operator acting by T m f(ξ) = m(ξ) ˆf(ξ). Suppose that j m(ξ) C j ξ j, j =,...,. Then T : L p L p for all 1 < p <. 6 Sobolev spaces and product estimates 6.1 Sobolev Spaces Theorem 8. s f L p ( k 2 2sk P k f 2 ) 1/2 L p. 6.2 Littlewood-Paley projections on products Lemma 5. For every two functions f, g supp fg supp ˆf + suppĝ. As a consequence, P k (fg) = P k [P <k+5 fp k 3< <k+3 g] + P k [P k 1< <k+1 fp k 3 g] + +P k [ P l 2 l+2 fp l g]. l k+3 6.3 Product estimates Theorem 9. Let 1 < p, q 1, q 2 <, 1 < r 1, r 2, 1/p = 1/q 1 + 1/r 1 = 1/q 2 + 1/r 2, s >. Then s (fg) L p s f L q 1 g L r 1 + s g L q 2 f L r 2. 9
If s is an integer, then one can replace s with s above. Corollary 4. If s > n/p, the space W s,p (R n ) is a Banach algebra, that is whenever f, g W s,p (R n ), then fg W s,p (R n ) and fg W s,p f W s,p g W s,p 7 Semigroups and Dynamics For the linear homogeneous equation u + Au = u(x, ) = u (x) one has a solution x(t) = S(t)u = e ta u in appropriate sense. Under suitable assumptions on A : D(A) X X (which is in general unbounded) one has the semigroup properties 1. S(t) B(X) and S(t + s) = S(t)S(s) 2. S() = Id 3. S(t)x S(t )x as t t for every fixed x. Given the semigroup S(t), one may recover the generator by D(A) := S(t)u u {u : lim t t Au := S(t)u u lim t t For the nonhomogeneous equation u + Au = F (x, t) u(x, ) = u (x) one solves by the Duhamel s formula u(t) = S(t)u + t exists} S(t s)f (s, )ds. 1
7.1 Examples 1. Heat semigroup. Generated by A =, D(A) = H 2 (R n ) L 2 (R n ). S(t)f = e t f Ŝ(t)f(ξ) = e 4π2 t ξ 2 ˆf(ξ). Using the kernel representation S(t)f(x) = 1 (4tπ) n/2 R n e x y 2 /4πt f(y)dy. S(t) is a semigroup of contractions on all L p (R n ), 1 p. 2. Schrödinger semigroup. Generated by A = i, D(A) = H 2 (R n ) L 2 (R n ). S(t)f = e it f Ŝ(t)f(ξ) = e 4π2 it ξ 2 ˆf(ξ). Semigroup of unitary operators on L 2 (R n ), but not a semigroup on any L p (R n ), p 2. 3. Wave equation. For u tt u =, take v := u t to write in the form ( ) ( ) ( ) u 1 u = v v t ( ) 1 For the semigroup generated by A =, ( ) ( f e ta cos(t ) sin(t )/ = g sin(t ) cos(t ) ( f u(t) = S(t) g ) = cos(t )f + sin(t ) g. The semigroup S(t) : L 2 (R n ) Ḣ 1 (R n ) L 2 (R n ) Ḣ 1 (R n ). For the inhomogeneous equation u tt u = F (x, t), one has the Duhamel s formula u(t, x) = cos(t )f + sin(t ) g + 11 t ) sin((t s) ) F (s, )ds.
7.2 A Local solvability theorem Theorem 1. Assume that there exist T >, Banach spaces X T, N T that and s >, so X C([, T ], H s ), i.e. C([,T ],H s ) C X T, S(t)u XT C u H s, t S(t s)f (s, )ds Co(t) F NT, where o(t) as t. X T G(u) NT C u α X T for some α > 1. G(u) G(v) NT C( u α 1 X T + v α 1 X T ) u v XT for some α > 1. Then, there exists a time t = t ( u H s) < T, so that in the interval t t, there is a unique mild solution to the Cauchy problem u = Au + G(u) u() = u. That is, there exists an unique solution to the integral equation t u(t, x) = S(t)u + S(t s)f (u(s, ), s, )ds. Moreover, the solution has Lipschitz dependence on initial data u v XT C u v H s. 7.3 Global solvability for small data Theorem 11. Assume that there exist Banach spaces X, N and s >, so that X C([, ), H s ), i.e. C([, ),H s ) C X, S(t)u X C u H s, 12
t S(t s)f (s, )ds F (u) N C u α X X C F N. for some α > 1. F (u) F (v) N C( u α 1 X + v α 1 X ) u v X for some α > 1. Then, there exists an ε >, so that whenever u H s solution to the integral equation ε, there is a unique global t u(t, x) = S(t)u + S(t s)f (u(s, ), s, )ds. Moreover, the solution has Lipschitz dependence on initial data, that is u v X C u v H s. 7.4 Solving nonlinear equations by fixed point methods For the nonlinear equation u + Au = F (u, x, t) u(x, ) = u (x) produce a fixed point for the map Λ : X X in appropriate space X. Λ(u) = S(t)u + t S(t s)f (u(s, ), s, )ds. *********************************************** 8 Strichartz estimates Definition 2. We say that a pair of exponents (q, r) is σ admissible, if q, r 2 : 1/q + σ/r σ/2. If equality holds, we say that such pair is sharp σ admissible. 13
Theorem 12. (Keel-Tao) Suppose that for the one parameter family of mappings U(t) and for every Schwartz function f and some σ > U(t)f L 2 C f L 2 (energy estimate) or U(t)U(s) f L C t s σ f L 1 untruncated decay estimate U(t)U(s) f L C(1 + t s ) σ f L 1 truncated decay estimate Then under the untruncated decay estimate assumption and for every sharp σ admissible pairs (q, r), ( q, r), we have the Strichartz estimates U(t)f L q C f t Lr x L (1) 2 U(s) F (s, )ds F L q (2) L 2 t L r x t U(t)U(s) F (s, )ds F (3) L q t L r x L q L r Under the truncated decay estimate assumption the estimates above hold for every σ admissible pairs (q, r), ( q, r). 8.1 Strichartz estimates for the Schrödinger equation Theorem 13. For the Schrödinger equation iu t + u = F, propagator U(t) = e it for which one has the energy and decay estimates u(x, ) = u, one has the e it f L 2 = f L 2 (4) e it f L (R n ) Ct n/2 f L 1 (R n ) (5) 14
As a consequence of the abstract theorem for every (q, r), ( q, r) : q, r, q, r 2, 1/q + n/(2r) = n/4, 1/ q + n/(2 r) = n/4, e it f L q C f t Lr x L 2 (6) e is F (s, )ds F L q (7) L 2 t L r x t e i(t s) F (s, )ds F (8) L q t L r x L q L r 8.2 Strichartz estimates for the wave equation Theorem 14. For the wave equation u tt u = F, u(x, ) = f, u t (x, ) = g and for every α R and all (q, r) : q, r 2, 1/q + (n 1)/(2r) (n 1)/4 one has 1/q+n/r+α u(x, t) L q L r C( f Ḣn/2+α + 1/ q +n/ r +α 2 F (9) L q L r 9 Applications 9.1 Semilinear Schrödinger equations For the semilinear Schrödinger equation iu t + u + u p 2 u = (1) take p = 2(n + 2)/n. This is the L 2 scale invaraint semilinear Schrödinger equation. Theorem 15. For (1), one has global well-posedness for small data in L 2 and local well posedness in H ε whenever ε >. More generally, let iu t + u + g(u) = u(x, ) = u (x) 15
Theorem 16. Let n 3, (q, r) be Schrödinger admissible in R n and r [2, 2n/(n 2)). Let M = max( u L 2, v L 2). g(u) g(v) L r K(M)( u α L r + v α L r ) u v L r Then if α + 2 < q and for every u L 2, there exists a time T max and a unique solution u C([, T max ), L 2 (R n )) L q ([, T max ), L r (R n ))). If T max <, then lim t Tmax u(t, ) L 2 = (blowup alternative) For every u, v, the corresponding solutions satisfy u v C([,Tmax),L 2 (R n )) L q ([,T max),l r (R n ))) C u v L 2 Lipschitz continuity with respect to initial data If in addition Im g(φ) φdx = for all Schwartz functions φ, then the solutions are global (i.e. T max = ) and there is the energy conservation law u(t, ) L 2 = u L 2. 9.2 Wave maps This is a system of equations in the form φ + a φ = a a 2 + (( φ) 2 ) u(x, ) = f(x) u t (x, ) = g(x) Theorem 17. In dimension n 4, the wave map system is globally well-posed for small data (f, g) Ḣn/2 Ḣn/2 1. References [1] Javier Duoandikoetxea,F ourier analysis. American Mathematical Society, Providence, RI, 21 16
[2] Loukas Grafakos, C lassical and modern Fourier analysis,, Prentice Hall, 23. [3] T. Tao, Notes from UCLA class. 17