HARMONIC ANALYSIS Contents. Introduction 2. Hardy-Littlewood maximal function 3. Approximation by convolution 4. Muckenhaupt weights 4.. Calderón-Zygmund decomposition 5. Fourier transform 6. BMO (bounded mean oscillation) 7. Singular integrals 7.. Hilbert and Riesz transforms 7.2. Singular integrals of convolution type 7.3. Singular integrals of nonconvolution type: Calderón- Zygmund operators Date: 9.9.2.
2 HARMONIC ANALYSIS. Introduction This lecture note contains a sketch of the lectures. More illustrations and examples are presented during the lectures. The tools of the harmonic analysis have a wide spectrum of applications in mathematical theory. The theory has strong real world applications at the background as well: Signal processing: Fourier transform, Fourier multipliers, Singular integrals. Solving PDEs: Poisson integral, Hilbert transform, Singular integrals. Regularity of PDEs: Hardy-Littlewood maximal function, approximation by convolution, Calderón-Zygmund decomposition, BMO. Example.. We consider a problem u = f in R n where f L p (R n ). The solution u is of the form f(y) u(x) = C n 2 dy. x y R n One of the questions in the regularity theory of PDEs is, does u have the second derivatives in L p i.e. 2 u x i x j L p (R n )? If we formally differentiate u, we get 2 u 2 = C f(y) x i x j R x n i x j x y n 2 dy. }{{} C/ x y n It follows that R n f(y) 2 x i x j x y n 2 dy defines a singular integral T f(x). A typical theorem in the theory of singular integrals says and thus we can deduce that T f p C f p 2 u x i x j L p (R n ). Example.2. Suppose that we have three different signals f, f 2, f 3 with different frequencies but only one channel, and that we receive f = f + f 2 + f 3 from the channel. The Fourier transform F(f) gives us a spectrum of the signal f with three spikes in F(f). We would like to recover the
HARMONIC ANALYSIS 3 signal f. Thus we take a multiplier (filter) {, y (a, b), a (y) := χ (a,b) (y) =, otherwise, where the interval (a, b) contains the frequency of f. Thus formally by taking the inverse Fourier transform, we get f = F (a F(f)) =: T f(x). This, again formally, defines an operator T which turns out to be of the form sin(cy) c f(x y) dy R y with some constants c, C. This operator is of a convolution type. However, sin(cy)/y is not integrable over the whole R, so this requires some care! 2. Hardy-Littlewood maximal function Definition 2.. Let f L loc (Rn ) and m a Lebesgue measure. Hardy-Littlewood maximal function Mf : R n [, ] is Mf(x) = sup f(y) dy =: sup f(y) dy, x m() x where the supremum is taken over all the cubes with sides parallel to the coordinate axis and that contain the point x. Above we used the shorthand notation f(x) dx = f(x) dx m() for the integral average. Notation 2.2. We denote an open cube by = (x, l) = {y R n : max i n y i x i < l/2}, l() is a side length of the cube, m() = l() n, diam() = l() n. Example 2.3. f : R R, f(x) = χ (,) (x), x >, x Mf(x) =, x,, x <. x Observe that f L (R) but Mf / L (R). A
4 HARMONIC ANALYSIS Remark 2.4. (i) Mf is defined at every point x R n and if f = g almost everywhere (a.e.), then Mf(x) = Mg(x) at every x R n. (ii) It may well be that Mf = for every x R n. Let for example n = and f(x) = x 2. (iii) There are several definitions in the literature which are often equivalent. Let Mf(x) = sup f(y) dy, l> (x,l) where the supremum is taken over all cubes (x, l) centered at x. Then clearly Mf(x) Mf(x) for all x R n. On the other hand, if is a cube such that x, then = (x, l ) (x, 2l ) and f(x) dy m((x, 2l )) f(y) dy m((x, l )) m((x, 2l )) 2 n Mf(x) because m((x, 2l )) m((x, l )) = (2l ) n = 2 n. l n It follows that Mf(x) 2 n Mf(x) and Mf(x) Mf(x) 2 n Mf(x) (x,2l ) for every x R n. We obtain a similar result, if cubes are replaced for example with balls. Next we state some immediate properties of the maximal function. The proofs are left for the reader. Lemma 2.5. Let f, g L loc (Rn ). Then (i) Mf(x) for all x R n (positivity). (ii) M(f + g)(x) Mf(x) + Mg(x) (sublinearity) (iii) M(αf)(x) = α Mf(x), α R (homogeneity). (iv) M(τ y f) = (τ y Mf)(x) = Mf(x + y) (translation invariance). Lemma 2.6. If f C(R n ), then for all x R n. f(x) Mf(x)
Proof. Let f C(R n ), x R n. Then HARMONIC ANALYSIS 5 ε > δ > s.t. f(x) f(y) < ε whenever x y < δ. From this and the triangle inequality, it follows that f(x) dy f(x) dy = ( ) = f(y) f(x) dy f(y) f(x) dy f(y) f(x) dy < ε whenever diam() = n l() < δ. Thus f(x) = lim x,l() f(x) dy sup x f(x) dy = Mf(x). Remember that f : R n [, ] is lower semicontinuous if {x R n : f(x) > λ} = f ((λ, ]) is open for all λ R. Thus for example, χ U is lower semicontinuous whenever U R n is open. It also follows that if f is lower semicontinuous then it is measurable. Lemma 2.7. M f is lower semicontinuous and thus measurable. Proof. We denote E λ = {x R n : Mf(x) > λ}, λ >. Whenever x E λ it follows that there exists x such that f(y) dy > λ. Further for every z, and thus Mf(z) f(y) dy > λ E λ. Lemma 2.8. If f L (R n ), then Mf L (R n ) and Mf f. Proof. f(y) dy f dx = f, (x) for every x R n. From this it follows that Mf f.
6 HARMONIC ANALYSIS Lemma 2.9. Let E be a measurable set. Then for each < p <, we have E f(x) p dx = p Proof. Sketch: f(x) p dx = χ E (x)p E R n Fubini = p = p λ p m({x E : f(x) > λ}) dλ f(x) λ p dλ dx λ p R n χ {x E : f(x) >λ} (x) dx dλ λ p m({x E : f(x) > λ}) dλ. Definition 2.. Let f : R n [, ] be measurable. The function f belongs to weak L (R n ) if there exists a constant C such that C < such that m({x R n : f(x) > λ}) C λ 7.9.2 for all λ >. Remark 2.. (i) L (R n ) weak L (R n ) because m({x R n : f(x) > λ}) = dx {x R n : f(x) >λ} {x R n : f(x) >λ} f(x) }{{ λ } dx f λ, for every λ >. (ii) weak L (R n ) is not included into L (R n ). This can be seen by considering Indeed, f(x) dx = B(,) f : R n [, ], f(x) = x n. B(,) x n dx = = B(,r) = ω n r n ds(x) dr r n ds(x) dr B(,r) }{{} ω n r n dr =, r