Course No. 401-4463-62L Fourier Analysis in Function Space Theory Dozent: Tristan Rivi`ere Assistant: Alessandro Pigati Contents 1 The Fourier transform of tempered distributions 1 1.1 The Fourier transforms of L1 functions . 1 1.2 The Schwartz Space S(Rn)........................... 4 1.3 Frechet Spaces . 6 1.4 The space of tempered distributions S0(Rn).................. 12 1.5 Convolutions in S0(Rn)............................. 21 2 The Hardy-Littlewood Maximal Function 26 2.1 Definition and elementary properties. 26 2.2 Hardy-Littlewood Lp−theorem for the Maximal Function. 27 2.3 The limiting case p =1. ............................ 30 3 Quasi-normed vector spaces 32 3.1 The Metrizability of quasi-normed vector spaces . 32 3.2 The Lorentz spaces Lp;1 ............................ 36 3.3 Decreasing rearrangement . 37 3.4 The Lorentz spaces Lp;q ............................ 39 3.5 Functional inequalities for Lorentz spaces . 44 3.6 Dyadic characterization of some Lorentz spaces and another proof of Lorentz{ Sobolev embedding (optional) . 49 4 The Lp−theory of Calder´on-Zygmund convolution operators. 52 4.1 Calder´on-Zygmund decompositions. 52 4.2 An application of Calder´on-Zygmund decomposition . 54 4.3 The Marcinkiewicz Interpolation Theorem - The Lp case . 56 4.4 Calderon Zygmund Convolution Operators over Lp ............. 58 4.4.1 A \primitive" formulation . 60 4.4.2 A singular integral type formulation . 64 4.4.3 The case of homogeneous kernels . 69 4.4.4 A multiplier type formulation . 71 4.4.5 Applications: The Lp theory of the Riesz Transform and the Laplace and Bessel Operators . 74 4.4.6 The limiting case p =1......................... 77 5 The Lp−Theorem for Littlewood Paley Decompositions 83 5.1 Bernstein and Nikolsky inequalities . 83 5.2 Littlewood Paley projections . 85 p n 5.3 The spaces L (R ; `q).............................. 86 5.4 The Lp-theorem for Littlewood-Paley decompositions . 89 1 The Fourier transform of tempered distributions 1.1 The Fourier transforms of L1 functions Theorem-Definition 1.1. Let f 2 L1(Rn; C) define the Fourier transform of f as fol- lows: Z n − n −ix·ξ 8ξ 2 R fb(ξ) = (2π) 2 e f(x) dx: Rn We have that fb 2 L1(Rn) and n − 2 (1.1) kfbkL1(Rn) ≤ (2π) kfkL1(Rn): Moreover fb 2 C0(Rn) and (1.2) lim jfb(ξ)j = 0: jξj!+1 We shall also sometimes denote the Fourier transform of f by F(f). Remark 1.2. There are several possible normalizations for defining the Fourier transform of an L1 function such as for instance Z fb(ξ) := e−2iπx·ξf(x) dx: Rn None of them give a full satisfaction. The advantages of the one we chose are the following: i) f 7−! fb will define an isometry of L2 as we will see in Proposition 1.5. ii) With our normalization we have the convenient formula (see Lemma 1.11) 8k = 1 : : : n @ξk fb = −i ξk fb but the less convenient formula (see Proposition 1.32) −n g\∗ f = (2π) gbf:b Proof of Theorem 1.1. The first part of the theorem that is inequality (1.1) is straight- 0 n 1 n forward. We prove now that fb 2 C (R ). Let fk 2 C0 (R ) such that 1 n fk −! f strongly in L (R ): 1 n 1 It is clear that since fk 2 C0 (R ) the functions fbk are also C . Inequality (1.1) gives n − 2 kfb− fbkkL1(Rn) ≤ (2π) kf − fkkL1(Rn): Thus fb is the uniform limit of continuous functions and, as such, it is continuous. It remains to prove that jfj(ξ) uniformly converges to zero as jξj converge to infinity. In 1 1 n 1 n Proposition 1.9 we shall prove that (1.2) holds if f 2 C0 (R ). Let f 2 L (R ), let " > 0 1 and let ' 2 C0 such that " n (1.3) kf − 'k 1 n ≤ (2π) 2 : L (R ) 2 There exists R > 0 such that " (1.4) jξj > R =) j'(ξ)j ≤ : b 2 Combining (1.3) and (1.4) together with (1.1) applied to the difference f − ', we obtain jξj > R =) jfb(ξ)j ≤ kfb− 'bkL1 + j'b(ξ)j ≤ ": This implies (1.2) and Theorem 1.1 is proved. ∗ Exercise 1.3. Prove that for any a 2 R+ jξj2 2 1 −ajxj − 4a e\ = n e : (2a) 2 ∗ Prove that for any a 2 R+ n f[a(x) = a fb(aξ) x n where fa(x) := f( a ) for any x 2 R . It is then natural to ask among the functions which are continuous, bounded in L1 and converging uniformly to zero at infinity, which one is the Fourier transform of an L1 function. Unfortunately, there seems to be no satisfactory condition characterizing the space of Fourier transforms of L1(Rn). We have nevertheless the following theorem. Theorem 1.4. (Inverse of the Fourier transform) Let f 2 L1(Rn; C) such that fb 2 L1(Rn; C) then Z n − n ix·ξ 8x 2 R f(x) = (2π) 2 e fb(ξ) dξ: Rn Proof of Theorem 1.4. We can of course explicitly write Z Z Z − n ix·ξ − n ix·ξ ix·y (2π) 2 e fb(ξ) dξ = (2π) 2 e dξ e f(y) dy: Rn Rn Rn The problem at this stage is that we cannot a-priori reverse the order of integrations because the hypothesis for applying Fubini's theorem are not fullfilled: iξ(x−y) 1 n n (ξ; y) 7−! e f(y) 2= L (R × R ) unless f ≡ 0. 2 2 2 − " jξj The idea is to insert the Gaussian function e 4 where " is a positive number that we are going to take smaller and smaller. Introduce Z 2 2 Z −n ix·ξ − " jξj −iξ·y I"(x) := (2π) e e 4 dξ e f(y) dy: Rn Rn Now we have 2 2 − " jξj iξ(x−y) 1 n n (ξ; y) 7−! e 4 e f(y) 2 L (R × R ) and we can apply Fubini's theorem. We have in one hand Z 2 2 − n ix·ξ − " jξj I"(x) = (2π) 2 e e 4 fb(ξ) dξ: Rn We can bound the integrand uniformly as follows: 2 2 n ix·ξ − " jξj 4 8x; ξ 2 R e e fb(ξ) ≤ jfb∗ ξ)j: By assumption, the right-hand side of the inequality is integrable and we have moreover, for every x and ξ 2 2 ix·ξ − " jξj ix·ξ lim e e 4 fb(ξ) = e fb(ξ): ")0 Hence dominated convergence theorem implies that for any x 2 R Z − n ix·ξ (1.5) lim I"(x) = (2π) 2 fb(x) e dξ: ")0 Rn Applying Fubini gives also Z Z 2 2 −n −i(y−x)·ξ − " jξj I"(x) = (2π) f(y) dy e e 4 dξ n R Rn Z 2 − n − " jξj2 = (2π) 2 f(y) F e 4 (y − x) dy Rn using Exercise 1.3, we then obtain n Z 2 − n −|y−xj 2 2 2 "2 I"(x) = (2π) f(y) e n dy: Rn " One proves without much difficulties that for any Lebesgue point x 2 R for f the following holds n Z 2 − n − jy−xj 2 2 lim (2π) 2 f(y) e "2 dy = f(x): ")0 n Rn " Continuing this identity with (1.5) gives the theorem. The transformation Z 1 n − n ix·ξ f 2 L (R ) 7−! (2π) 2 e f(ξ) dξ Rn _ will be denoted f or also F −1(f). 3 Proposition 1.5. Let f and g 2 L1(Rn; C). When Z Z f(x) gb(x) dx = fb(x) g(x) dx: Rn Rn Let f 2 L1(Rn; C) such that fb 2 L1(Rn; C), then Z Z f(x) f(x) dx = fb(ξ) fb(x) dξ: Rn Rn This last identity is called Plancherel identity. Proof of Proposition 1.5. The proof of the first identity in Proposition 1.5 is a direct consequence of Fubini's theorem. The second identity can be deduced from the first one by taking g := F −1(f) and by observing that −1 F (f) = F(f) : The second identity is an invitation to extend the Fourier transform as an isometry of L2. The purpose of the present chapter is to extend the Fourier transform to an even larger class of distributions. To that aim we will first concentrate on looking at the Fourier transform in a \small" class of very smooth function with very fast decrease at infinity: the Schwartz space. 1.2 The Schwartz Space S(Rn) The Schwartz functions are C1 functions whose successive derivatives decrease faster than any polynominal at infinity. We shall use below the following notations: n α α1 αn 8α = (α; : : : ; αn) 2 N x := x1 ; : : : ; xn βn n β @ @ 8β(β; : : : ; βn) 2 @ f := ::: (f) N β1 βn @n1 @xn P and jαj := αi. Definition 1.6. The space of Schwartz functions is the following subspace of C1(Rn; C): 8 1 n 9 ' 2 C (R ; C) s.t. > > < α β = n 8p 2 N (') := sup kx @ 'k 1 n < +1 S(R ) := N p L (R ) : > jαj ≤ p > : jβj ≤ p ; The following obvious proposition holds Proposition 1.7. S(Rn) is stable under the action of derivatives and the multiplication by polynomials in C[x1; : : : ; xn]. We prove now the following proposition: 4 n Proposition 1.8.