Lecture notes, Harmonic Analysis minicourse Joaquim Bruna, UAB April 19, 2016 Chapter 1 Preliminaries in Real and Functional Analysis 1.1 Lebesgue Integrable functions. Let (X; µ) a measure space, µ being a σ-finite positive measure defined on a σ-algebra. Most of the time,X = Rd with dµ = dm, the Lebesgue measure (more generally a locally compact topological space, σ-compact, and dµ a Borel measure on X). Also it is to keep in mind the discrete setting, X = Zd with counting measure, and the periodic setting X = Rd=Zd. Through the map 2πix1 2πix2 2πixd (x1; x2; : : : ; xd) 7! (e ; e ; : : : ; e ); we identify Rd=Zd with Td, the d-dimensional torus, where T = fz 2 C : jzj = 1g. Using approximation by simple functions (linear combinations of indi- cator functions of measurable sets), every measurable function defined on R X taking values in [0; +1] has a well defined integral X f dµ, finite or +1. For p > 0, the space Lp(X) consists of the measurable complex-valued functions such that Z p 1 kfkp = ( jf(x)j dm(x)) p < +1: X It is thus a question of the size jfj. Functions in L1(X) are called integrable. Functions in Lp are finite a.e. We identify functions which are equal a.e. The space L1(X) consists of complex-valued functions which are bounded a.e., kfk1 being the least essential supnorm. p When X is a topological space, Lloc(X) consists of functions whose re- strictions to compact sets Y belong to Lp(Y ). 1 1.2 Convergence theorems. Monotone convergence theorem: if fn are positive and a.e. non-decreasing, R R and f(x) = limn fn(x) ≤ +1, then fdµ = limn fndµ. For general se- R R X X R P quences, lim inf fndµ ≤ lim inf fndµ. As a consequence, ( n gn)dµ = P R X X X n X gndµ for arbitrary positive functions. Dominated convergence theorem: if f(x) = limn fn(x) a.e. x , jfn(x)j ≤ g(x) R R R a.e. x with gdµ < +1, then f is integrable and fdµ = limn fndµ. X P R X P X As a consequence, if n jgnjdµ < +1, the series n gn(x) absolutely R P X P R converges a.e. and X ( n gn(x))dµ = n X gndµ. 1.3 Fubini's theorem If (X; µ); (Y; ν) are measure spaces there is a natural estructure of measure space in X × Y with the product measure µ × ν. In case X = Rd; Y = Rm, of course the product measure of Lebesgue measures is Lebesgue measure. Fubini's theorem has two parts: a) for a positive measurable function, the double integral RR f(x; y)dµ(x)dν(y) and the two iterated integrals R R X×YR R X ( Y f(x; y)dν(y))dµ(x); Y ( X f(x; y)dµ(x))dν(y) are equal, finite or not. b) If f(x; y) is integrable in X × Y , then both iterated integrals make sense and are finite ( meaning that for a.e. x; f(x; y) is integrable in Y and the function so defined a.e. in X is integrable in X, and the other way around) and equal to the double integral. Normally we use both parts in the following form: if one of the two iterated integrals of jf(x; y)j is finite, then all three integrals are finite and equal. In particular the iterated integrals are a.e. absolutely convergent. 1.4 Holder's and Minkowski's inequalities 1 1 1 Holder's inequality reads as follows: If 1 ≤ p; q; r ≤ +1; p + q = r , and p q r f 2 L ; g 2 L , then fg 2 L and kfgkr ≤ kfkpkgkq: For p = q = 2 this is known as Cauchy-Schwartz inequality. Recall that this depends on Young's inequality and that there is a version with n indexes, i.e., if 1 + 1 + ··· + 1 = 1 , then p1 p2 pn r kf1f2 : : : fnkr ≤ kf1kp1 ::: kfnkpn : In the opposite direction, if f is measurable and Z supf jfgjdµ : kgkp0 ≤ 1g < +1; X 0 1 1 p p being the conjugate exponent of p, p + p0 = 1, then f 2 L and 2 Z kfkp = supf fgdµ : kgkp0 ≤ 1g: X We call this the reverse Hlder inequality. In many occasions we already know that f 2 Lp and use this to estimate its norm. From Holder's one obtains the classical Minkowski or triangle inequality: kf + gkp ≤ kfkp + kgkp; 1 ≤ p ≤ +1, so kkp is a norm if p ≥ 1. For p < 1, p p p p since ja + bj ≤ jaj + jbj , the quantity k · kp is subadditive: X p X p k fnkp ≤ kfnkp: n n Of course this goes over to finite sums and to infinite sums as well, in the form of the so-called continuous Minkowski inequality: if we have an infinite sum of functions, say Z Z Z G(y) = F (x; y)dµ(x);G = Fxdµ(x); jGj ≤ jFxjdµ(x); X X X then Z kGkp ≤ kFykpdµ(x); 1 ≤ p X that is 1 1 Z Z p p Z Z p jF (x; y)jdµ(x) dν(y) ≤ jF (x; y)jpdν(y) dµ(x): Y X X Y One can prove this by approximating integrals by finite sums. An al- ternative proof is as follows. For p = 1 it is just Fubini's theorem and for p = +1 is trivial. To prove it for 1 < p < +1 we may assume that F ≥ 0. We have, using Fubini and Holder Z Z Z Z G(y)pdν(y) = G(y)p−1G(y)dν(y) = G(y)p−1( F (x; y)dµ(x))dν(y) = Y Y Y X Z Z = ( F (x; y)G(y)p−1dν(y))dµ(x) ≤ X Y Z Z 1 Z 1 ≤ ( F (x; y)pdν(y)) p ( G(y)pdν(y)) p0 dµ(x) = X Y Y Z 1 Z Z 1 = ( G(y)pdν(y)) p0 ( F (x; y)pdν(y)) p dµ(x): Y X Y 3 The same is true replacing the L1- norm in x above by an Lr-norm with 1 ≤ r ≤ p: 1 1 p r ! p ! r Z Z r Z Z p jF (x; y)jrdµ(x) dν(y) ≤ jF (x; y)jpdν(y) dµ(x) : Y X X Y r p In fact this follows from the above applied to jF j with p replaced by r . 1.5 The Lp(X) spaces as Banach spaces The Lp(X)-spaces, 1 ≤ p ≤ +1 equipped with the Lp norm are complete spaces, meaning that every Cauchy sequence, i.e. kfn − fmk ! 0 as n; m ! p +1, is convergent to some f 2 L (X), that is kfn − fk ! 0 as n ! +1. p If p < +1, convergence of fn to f in L implies pointwise a.e. convergence of a partial subsequence; if p = +1, we have of course uniform convergence a.e. For p = 2, Lp(X) is a Hilbert space with the scalar product defined by Z hf; gi = f(x)g(x)dµ(x): X d If X is an open set U in R , the space Cc(U) of continuous compactly supported functions is dense in Lp(U); 1 ≤ p < +1, meaning that an p arbitrary f 2 L (U) can be approximated by a sequence gn 2 Cc(U): kf − gnkp ! 0 as n ! +1. In case p = +1 the space in which Cc(U) is dense is the space C0(U) of continuous functions vanishing at infinity, meaning that for each " there is a compact set in U out of which jfj < ". In the periodic setting C(Tn) is dense in the Lp(Tn) for 1 ≤ p < +1 and is already complete with the sup-norm. Recall that the linear maps T : E ! F between linear spaces that are continuous are the bounded ones, that is, those for which kT (f)kF ≤ CkfkE for some constant C > 0, the infimum of such C being the norm of T . The density of a subspace G of E is useful to define linear continuous maps T : E ! F between Banach spaces; namely every linear continuous maps T : G ! F extends uniquely to E. This is so because T maps Cauchy sequences to Cauchy sequences, hence if f is approximated by gn 2 G, T f can be defined by the limit of the Cauchy sequence T (gn). d We define the translation operator τx; x 2 R acting on functions f d defined on R by (τxf)(y) = f(y − x). Obviously kτxfkp = kfkp. For fixed p d f 2 L (R ); 1 ≤ p < +1, the map x ! τxf is continuous. It is enough d proving continuity at zero. This is clear for g 2 Cc(R ) because τxg ! g uniformly as x ! 0; since the τxg have a common compact support, one has then kτxg − gkp ! 0 as well. Once we have established this for g in the d dense space Cc(R ) the general case follows through kτxf − fk ≤ kτxf − τxgk + kτxg − gk + kg − fk = 2kf − gk + kτxg − gk: 4 In the periodic situation, translation becomes rotation, of course. The reverse Hlder inequality can be used to prove that the dual space of Lp(X); 1 ≤ p < +1 is Lp0 (X), meaning that the general form of a continuous linear form ! : Lp(X) ! C is of the form Z f 7! g dµ; f for some g 2 Lp0 (X).