The great response to the publication of the book Classical and Modern Fourier Analysis was very gratifying. Finally, I would like to thank Stephen Montgomery-Smith for creating the beautiful figure that appears on the cover of this and the previous volume.
Smoothness and Function Spaces
Riesz Potentials, Bessel Potentials, and Fractional Integrals
- Riesz Potentials
- Bessel Potentials
The significance of the Riesz potentials lies in the fact that they are actually smoothing operators. Nevertheless, for reasons of explanation, we choose to give another independent proof of Theorem 6.1.3.
Exercises
Sobolev Spaces
- Definition and Basic Properties of General Sobolev Spaces
- Littlewood–Paley Characterization of Inhomogeneous Sobolev Spaces
- Littlewood–Paley Characterization of Homogeneous Sobolev SpacesSpaces
Recalling the definition of Δj (see the discussion before the statement of Theorem 6.2.6), we note that the function. This proves that the homogeneous Sobolev norm for f is governed by a multiple of the expression in (6.2.23).
Lipschitz Spaces
- Introduction to Lipschitz Spaces
- Littlewood–Paley Characterization of Homogeneous Lipschitz SpacesLipschitz Spaces
- Littlewood–Paley Characterization of Inhomogeneous Lipschitz SpacesLipschitz Spaces
We have seen that quantities involving Littlewood–PaleyDj operators characterize homogeneous Lipschitz spaces. As in the Littlewood-Paley characterization of inhomogeneous Sobolev spaces, we must treat separately the contribution of near-zero frequencies.
Hardy Spaces
- Definition of Hardy Spaces
- Quasinorm Equivalence of Several Maximal Functions
- Consequences of the Characterizations of Hardy Spaces
- Vector-Valued H p and Its Characterizations
- Singular Integrals on Hardy Spaces
- The Littlewood–Paley Characterization of Hardy Spaces
The reverse inequality is due to the fact that sup. where M is the Hardy–Littlewood maximal operator. To obtain a similar estimate for the first term on the right in (6.4.17), we argue as follows: a) We choose a continuous and integrable function ψ(s) on the interval [1,∞), which decays faster than the reciprocal value of any polynomial (i.e. |ψ(s)| ≤CNs−N for all N>0), so that .
Besov–Lipschitz and Triebel–Lizorkin Spaces
- Introduction of Function Spaces
- Equivalence of Definitions
We show that if Ω is another function satisfying (6.5.1) and Θ is defined in terms of eΩ in the same way that Φ is defined in terms of eΨ, [i.e., via (6.5.2)], then the rates of defined in definition 6.5 .1 with respect to the pairs (Φ,Ψ) and (Θ,Ω) are comparable. Then there are constants C1 and C2. which depend only on n, c0 and r) such that for all t>0 and for all C1 functions u in Rn whose Fourier transform rests on the ball|ξ| ≤c0t and that satisfy. Choose a Schwartz function ψ whose Fourier transform rests on top|ξ| ≤2c0and is equal to 1 in the smallest ball|ξ| ≤c0.
Suppose that Ω is another collision whose Fourier transform rests on the ring 1−17≤ |ξ| ≤2 and satisfying (6.5.1). LeΨandΩbe the Schwartz functions whose Fourier transforms rest on the annulus 12≤ |ξ| ≤2 and satisfies (6.5.1). We start with a Schwartz function Θ whose Fourier transform is nonnegative, resting on the annulus 1−27≤ |ξ| ≤2, and satisfies.
In the case of the inhomogeneous spaces, we let SΨ0 and SΩ0 be the operators given by convolution with the bulgesΦ andΘ, respectively (remember that these are defined in terms ofΨ andΩ).
Atomic Decomposition
- The Space of Sequences ˙ f p α ,q
- The Smooth Atomic Decomposition of ˙ F p α,q
- The Nonsmooth Atomic Decomposition of ˙ F p α ,q
- Atomic Decomposition of Hardy Spaces
We now discuss the main theorem of this section, the non-smooth atomic decomposition of the homogeneous Triebel-Lizorkin spaces ˙Fpα,q, which includes in particular that of the Hardy spaces Hp. We now turn to one of the most important theorems of this chapter, the atomic decomposition of Hp(Rn) for 0 It remains to estimate the Lpquasinorm of the square root of the second series in (6.6.16), increased by (Q∗)c. According to the definition of the ∞-atom for ˙fp0,2, there exists a dyadic cube Q0j such that rj,Q=0 for all dyadic cubes Q that do not appear in Q0j. This observation can be very useful in certain applications. a) Prove that there exists a Schwartz functionΘ supported in the unit ball. Hint: In the case r=1 use the L1→L∞ boundedness of the Fourier transform and in the case r=∞ use Plancherel's theorem. The proof of this theorem provides another classical application of the atomic decomposition of Hp. Hence T(f) =G and we conclude that T(fM) converges to T(f) in Hp, i.e., the series∑jλjT(aj) converges to T(f) in Hp. Bessel potentials were introduced by Aronszajn and Smith [7] and also by Calder'on [41], who was the first to notice that the potential space Lsp (i.e., the Sobolev space Lsp) coincides with the space Lkp given in classical definition 6.2 .1 when s=k is an integer. The extension of the definition of Bes spaces to the case of p<1 is also due to Peetre [256]. Uchiyama obtained two alternative characterizations of Hardy spaces in terms of the generalized Littlewood–Paley g-function [319] and in terms of Fourier multipliers [320]. Another simple proof of the L2-atomic decomposition for Hp (starting from a non-tangential Poisson maximum function) was obtained by Wilson [332]. For a careful study of the operation of singular integrals on function spaces we refer to the book by Torres [315]. But the role of space BMO is deeper and more far-reaching than that. This space arises crucially in many situations in analysis, such as in the characterization of the L2boundedness of nonconvolution singular integral operators with standard kernels. The power of the Carleson measurement techniques becomes apparent in Chapter 8, where they play a crucial role in the proof of several important results. Also, the maximum and minimum of two functions can be expressed in terms of the absolute value of their difference. It turns out that this is a general property of BMO functions, and this is the content of the following statement. Set Q(0)=Q and divide Q(0) into 2 equal closed subcubes with a side length equal to half the side length of Q. Now (C-k) follows from the upper inequality in (B-k). E-k) is a consequence of Lebesgue's differentiation theorem, since for each point in Q(k−1)j \(jQ(k)j there is a sequence of cubes shrinking to it, and the average of. Suppose there are positive constants m and b such that for all cubes Q in Rna and for all 0 By the Riesz representation theorem for the Hilbert space L20(Q), there is an element FQin(L20(Q))∗=L2(Q)/{constant} such that. Thus for any cube Q in Rn, there is square integrable function FQ supported in Q such that (7.2.8) is satisfied. Indeed, given a cube Q choose the smallest m such that Q is contained in Qm and let CQ=−AvgQ1(FQm) +D(Q,Qm) where D(Q,Qm) is the constant value of the function FQm− FQ on Q. We have now found a locally integrable function b such that for all cubes Q and all g∈L20(Q) we have. This function is obtained by taking the maximum of the values of F within conΓ(x). Likewise, let P be an affine plane in Rn+1 and define a measureν by letting ν(A) be the n-dimensional Lebesgue measure of the set A∩P for every A⊆Rn+1+. We now turn to the study of some interesting properties of the limits of functions on Rn+1+ with respect to Carleson measures. It is a decomposition of a general open set Ω into R into a union of disjoint cubes whose lengths are proportional to their distance from the boundary of the open set. There exists a dimensional constant Such that for allα >0, all measureμ≥0 on Rn+1+, and allμ-measurable functions F on Rn+1+, we have. The measure μ is defined such that for every μ-integrable function F on Rn+1+ we have. 7.3.15). Let F be a function on Rn+1+, let F∗ be the nontangential maximal function derived from F, and let μ≥0 be a measure on Rn+1+. By proving (7.4.4) we can assume that for someξj∈Qj we have M#(f)(ξj)≤γλ; otherwise there is nothing to prove. Then for every p with p0≤p<∞ there is a constant Cn(p) such that for all functions f with Md(f)∈Lp0(Rn) we have. So for any p with p0≤p<∞ and for all locally integrable functions f with Md(f)∈Lp0(Rn) we have. Then for all p with p0 Prove that for every 1 We introduce some material needed in the study of the boundedness of the commutator. In what follows we use a certain maximum operator, defined in terms of the corresponding Orlicz norm. The main estimate follows from the following local version of the inverse weak type estimate (1,1) from exercise 2.1.4(b). Another interesting property of BMO is that it is preserved under the action of the Hardy-Littlewood maximum operator. Weighted Lpestimates for the commutator in terms of the double iteration of the Hardy–Littlewood maximum operator can be derived as a result of Lemma 7.5.5; see the article by P´erez [261]. This space is the closure in the BMO norm of the subspace of BMO(T1) consisting of all uniformly continuous functions on T1. One of the important features of V MO(Rn) is that it is the predual of H1(Rn), as was shown by Coifman and Weiss [86]. In the following we denote by CZO(δ,A,B) the class of all Calder´on-Zygmund operators associated with standard kernels in SK(δ,A) that admit L2-bounded extensions with norm at most B. the point that there can be multiple Calder'on-Zygmund operators associated with a given standard kernel K. In light of Proposition 8.1.11 and Remark 8.1.12, a Calder'on-Zygmund operator is equivalent to a Calder'on – Zygmund simple integral operator plus a bounded function times the identity operator. For this reason, the study of Calder'on-Zygmund operators is equivalent to the study of Calder'on-Zygmund singular integral operators, and in almost all situations it is sufficient to limit attention to the latter. We are now interested in defining the action of a Calder'on-Zygmund operator T on bounded and smooth functions. Indeed, ifζ is another function satisfying 0≤ζ ≤1 that is also equal to 1 in a neighborhood of the support ofϕ, then f(η−ζ)andϕ have disjoint supports, and from (8.1.7) we have integral absolutely convergent realization. To summarize, if T is a Calder'on-Zygmund operator and f lies in L∞(Rn), then T(f) has a well-defined action. Extend the result of Theorem 8.1.11 to the case that the space L2 is replaced by Lq for approximately 1 The result for 2 To obtain the Lpboundedness of T(∗) for 1Singular Integrals on Function Spaces
Functions of Bounded Mean Oscillation
Duality between H 1 and BMO
Nontangential Maximal Functions and Carleson Measures
The Sharp Maximal Function
Commutators of Singular Integrals with BMO Functions
Singular Integrals of Nonconvolution Type
General Background and the Role of BMO
Consequences of L 2 Boundedness
