Graduate Texts in Mathematics

The great response to the publication of the book Classical and Modern Fourier Analysis was very gratifying. All of these individuals have provided me with invaluable assistance that has led to the improved presentation of the present second edition.

L p and Weak L p

The Distribution Function

Knowledge of the distribution functiondf provides sufficient information to evaluate the Lpnorm of a function f accurately. We state and prove the following important description of the Lpnorm with respect to the distribution function.

Fig. 1.1 The graph of a simple function f =∑ 3 k=1 a k χ E k and its distribution function d f (α)

Convergence in Measure

It does not immediately follow from their definition that the weak spaces Lp are complete with respect to the quasi-norm. One way to achieve this is with the following theorem, which is a useful version of Theorem 1.1.11.

A First Glimpse at Interpolation

Show that df(α0+tn)↑df(α0) using a convergence theorem. a) Show that the product is in Lpand that. Hint: Part (a): Given 0<ε

Convolution and Approximate Identities

• Examples of Topological Groups
• Convolution
• Basic Convolution Inequalities
• Approximate Identities

Throughout the remainder of this section, fix a locally compact group and a left invariant Haar measureλ onG. Consider Gas the group of all 2×2 matrices with bottom row(0,1) and non-zero top left entry.] Show that a left Haar measures onGis.

Fig. 1.2 The Fej´er kernel F 5 plotted on the interval [− 1 2 , 12 ].

Interpolation

• Real Method: The Marcinkiewicz Interpolation Theorem
• Complex Method: The Riesz–Thorin Interpolation Theorem
• Interpolation of Analytic Families of Operators
• Proofs of Lemmas 1.3.5 and 1.3.8

Returning to the proof of Theorem 1.3.4, we see that F is analytic in the open strip S and continuous at its closure. As in the proof of Theorem 1.3.4, we work with simple functions f onX engonY. Fix 0<θ<1 and also fix simple functions f,gso that. whereak>0,bj>0,αk,β are real, Ak are pairwise disjoint subsets of X with finite measure, and Bja are pairwise disjoint subsets of Y with finite measure.

Lorentz Spaces

Decreasing Rearrangements

From the previous examples it is clear that f∗ is continuous from the right and decreasing. For an arbitrary measurable function f, find a set of non-negative simple functions fn such that fn↑ |f|and apply (9).

Fig. 1.3 The graph of a simple function f(x) and its decreasing rearrangement f ∗ (t).

Lorentz Spaces

A simple calculation shows that the triangle inequality for these functions is relative to the norm. Then for all 0

Duals of Lorentz Spaces

Raise (1.4.10) to powerq, multiply bytq/p, integrate with respect todt/tover (0,∞), and apply Fatou's lemma to obtain. It is a general fact that if a Cauchy sequence has a convergent subsequence in a quasinorm space, then the sequence is convergent to the same. For example, R with Lebesgue measure is nonatomic, but any measure space with count measure is atomic.

The fact that the integral in (1.4.16) converges is a consequence of the observation in which the function f∗ is defined.

The Off-Diagonal Marcinkiewicz Interpolation Theorem

Let T be either a quasilinear operator defined on Lp0(X) +Lp1(X) and taking values ​​in the set of measurable functions on Y or a linear operator defined on the set of simple functions on X and taking values ​​as before. It follows from part (c) that RX f1g dµ=R0∞f∗(s)g∗(s)ds. The case of a general function f follows from that in which f is simple using Exercise 1.4.1 and approximation. be a nonnegative functional on a vector space X that satisfies An extension of the Marcinkiewicz interpolation theorem to Lorentz spaces was obtained by Hunt [133].

We have already seen that the convolution of a function with a fixed density is a smoothing operation that produces a certain mean of the function.

Maximal Functions

The Hardy–Littlewood Maximal Operator

Equation (2.1.1) implies that M(f) is neverinL1(Rn)if f 6=0 a.e., a strong property that reflects a certain behavior of the maximal function. This shows that the union of the unselected balls is contained in the union of the triples of the selected balls. Therefore, the union of all balls is contained in the union of the triples of the chosen balls.

We are now ready to prove the main theorem about the boundedness of the centered and uncentered maximal functionsMandM, respectively.

Control of Other Maximal Operators

Let us now fix a radial, continuous and compactly supported function K with support in the ball B(0,R), which satisfies (2.1.8). Here we have used partial integration and the fact that the area measure of the unit sphere Sn−1 is tonvn. If K is an L1 function on Rn whose absolute value above is bounded by a continuously integrable radial function K0 satisfying (2.1.8), then (2.1.9) holds.

Solving the Dirichlet problem (2.1.12) motivates the study of almost everywhere convergence of the expressions f∗Pt.

Applications to Differentiation Theory

In view of Theorem 2.1.10, the supremum of the family of linear operatorsTε(f) = f∗Pε is governed by the Hardy–Littlewood maximal function, and thus it maps LptoLp,∞for 1≤p<∞. Divide each cube in the mesh into 2 congruent cubes by halving each of the sides. Then use the argument from the proof of Theorem 2.1.6 and the inner regularity of µ. 2.1.2. Reconsider the maximum function Mµ of Exercise 2.1.1. a) (W. H. Young) Prove the following covering lemma.

Using a covering lemma, show that M00 is a weak type(1,1) with a limit proportional to the square of the eccentricity.

The Schwartz Class and the Fourier Transform

• The Class of Schwartz Functions
• The Fourier Transform of a Schwartz Function
• The Inverse Fourier Transform and Fourier Inversion
• The Fourier Transform on L 1 + L 2

The Fourier transform is a homeomorphism of the Schwartz class and the Fourier inversion holds in it. It is clear that the inverse Fourier transform shares the same properties as the Fourier transform. In view of the result in Exercise 2.2.8, the Fourier transform is an isometry L2 to L1∩L2, which is a dense subspace of L2.

This calculation gives the Fourier transform of the Poisson kernel. a) Prove that Schwartz functions f on a line satisfy the estimate.

The Class of Tempered Distributions

• Spaces of Test Functions
• Spaces of Functionals on Test Functions
• The Space of Tempered Distributions
• The Space of Tempered Distributions Modulo Polynomials

The dual spaces (that is, the spaces of continuous linear functionals on the sets of test functions) that we introduced are denoted by . Example 2.3.10The function e|x|2 is not in S0(Rn) and therefore its Fourier transform is not defined as a distribution. Example 2.3.19The distribution|x|2+δa1+δa2, wherea1,a2inRn, coincides with the function|x|2on any open set that does not contain the pointsa1anda2.

Next, we give a proposal that extends the properties of the Fourier transform to tempered distributions.

More About Distributions and the Fourier Transform

Distributions Supported at a Point

Hint: First prove the corresponding theorems for functions ϕ inS or S∞ with convergence in the topology of these spaces. Asbu is supported in the singleton{ξ0}, then u is a finite linear combination of functions(−2πiξ)αe2πiξ·ξ0, whereα is a multi-index.

The Laplacian

Homogeneous Distributions

This example illustrates the interplay between the smoothness of a function and the decay of its Fourier transform. Hint: Let ϕ be a smooth nonzero function whose Fourier transform is supported in the interval [−1/2,1/2], and let ϕ be a smooth nonconstant real-valued periodic function with period 1. If the function on Rn(n≥2 ) constant on all (n−2)-dimensional spheres rectangular toe, its Fourier transform has the same property.

Use the result in part (a) to show that the Fourier transform of the function eiπ|x|2 in R is equal to eiπn4e−iπ|ξ|2.

Convolution Operators on L p Spaces and Multipliers

• Operators That Commute with Translations
• The Transpose and the Adjoint of a Linear Operator
• The Spaces M p,q (R n )
• Characterizations of M 1,1 (R n ) and M 2,2 (R n )
• The Space of Fourier Multipliers M p (R n )

In this case, the norm of the operator is equal to the total variation of the measure. Operators given by convolution with finite complex-valued Borel measures naturally map L∞(Rn)toL∞(Rn); so M1,1(Rn) is a subspace of M∞,∞(Rn). It is a consequence of Theorem 1.3.4 that the normed spacesMpairs are nested, that is, for 1≤p≤q≤2 we have.

If ψ is a function on Rn whose inverse Fourier transform is an integrable function, then prove this. a) Prove that the mapT(f) =f∗∂γδ0 continuously maps into S. Use Theorem 1.4.24 to show that the operator.

Oscillatory Integrals

Phases with No Critical Points

Sinceϕ0 has no zeros, it must be strictly positive or strictly negative everywhere on the support ofψ. It follows that ϕ is monotonic based on ψ and that we are allowed to change variables. Theorem 2.6.4.Suppose thatψ is a compactly supported smooth function on Rn and that ϕ is a real-valued C1 function on Rn that has no critical points on the support of ψ.

For this purpose we fix ak and we choose such that the support of ψ ζkis is contained in a sphere B(yj,ryj).

Sublevel Set Estimates and the Van der Corput Lemma

The class LlogL was introduced by Zygmund to provide a sufficient condition for the local integrability of the Hardy-Littlewood maximal operator. The exact value of the operator norm of the uncentered Hardy-Littlewood maximal function onLp(R) was shown by Grafakos and Montgomery-Smith [109] to be the unique positive solution of the equation (p−1)xp−pxp−1− 1=0 . This constant raised to powernis the operator norm for the strong maximal function MsonLp(Rn) for 1

The discussion of these results in the text is based on the article by Carbery, Christ and Wright [44], which also explores higher-dimensional analogies of the theory.

Fourier Coefficients

• The n-Torus T n
• Fourier Coefficients
• The Dirichlet and Fej´er Kernels
• Reproduction of Functions from Their Fourier Coefficients
• The Poisson Summation Formula

Translation invariance of the Lebesgue measure and the periodicity of functions on Clearly we have for all f onTn. This implies that the partial sums∑|m|≤Nbf(m)e2πim·x of the Fourier series of f given in (3.1.5) can be obtained by convolving f with the functions. This is analogous to the fact that the Fej´er kernelFN is the average of the Dirichlet kernelsD0,D1.

A fundamental problem is in what sense the partial sums of the Fourier series converge back to the function asN → ∞.

Fig. 3.1 The graph of the two-dimensional torus T 2 .

Decay of Fourier Coefficients

Decay of Fourier Coefficients of Arbitrary Integrable FunctionsFunctions

Continue inductively in this way and construct a subsequencek0kj+kj−k2j−1 for j≥1 . We are given a series of positive numbers{am}m∈Zwhich converges to zero if|m| →∞and we would like to find an integrable function onT1with|bf(m)| ≥am. Apply Lemma 3.2.3 to the sequence {am+a−m}m≥0 to find a convex sequence {cm}m≥0 that dominates {am+a−m}m≥0 and decreases to zero asm →∞.

The convexity of the sequence and the positivity of the Fej´er kernel imply that f≥0.

Decay of Fourier Coefficients of Smooth Functions

2πimj)sdx, (3.2.8) where the boundary terms all vanish due to the periodicity of the integrand. Take absolute values ​​and|m| use ≤√. We now move on to the second part of the statement. 0)be the element of the torusTnwhose jth coordinate is one and all others are zero.

The conclusion follows from the fact that the norm for what is measured is the total variation of e.g.

Functions with Absolutely Summable Fourier Coefficients

If f is a function of bounded variation, then the Lebesgue–Stieltjes integral with respect to f is well defined. Assume that f is a function defined on Tnall, whose partial derivatives of order s lie in the space Λ˙α. For 1≤j≤n, let the element Rn be zero with entries, except for the jth coordinate, which is 1. Let l be a positive integer and let lhj=2−l−2ej. n} chosen so that|mj|=supk|mk|we have.

3.2.7.(a) Prove that the product of two functions inA(Tn) is also inA(Tn) and that. b) Prove that the convolution of two square integrable functions on Tn always gives a function inA(Tn).

Pointwise Convergence of Fourier Series

Pointwise Convergence of the Fej´er Means

If x0 is a continuity point of f, and the quadratic partial sums of the Fourier series of f converge at x0, then they must converge to f(x0). b) In dimension1, if f(x) has left and right limits as x→x0 and the partial sum of the Fourier series of f converges, then they must converge to 12 f(x0+) +f(x0−). If(D(n,N)∗f)(x0)→A(x0)asN→∞, then the arithmetic means of this sequence must converge to the same number as the sequence.

Almost Everywhere Convergence of the Fej´er Means