Boundedness of Fourier integral operators on Fourier Lebesgue spaces and related topics

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1 Boundedness of Fourier integral operators on Fourier Lebesgue spaces and related topics Fabio Nicola (joint work with Elena Cordero and Luigi Rodino) Dipartimento di Matematica Politecnico di Torino Applied Analysis and Fast Computation in Phase-Space November 24-28, 2008, Vienna Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

2 Outline 1 Classical theory: local L p boundedness 2 Boundedness on local FL p spaces 3 Global boundedness on L p, FL p and on the modulation spaces M p Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

3 Classical theory: local L p boundedness Fourier Integral Operators of Hörmander s type Definition A Fourier Integral Operator (FIO) is an operator of the form Af (x) = e 2πiΦ(x,η) σ(x, η)ˆf (η)dη, R d where Φ(x, η) is the phase and σ(x, η) the symbol of A.We assume that Φ(x, η) is C (R d (R d \ {0})), real-valued, with Φ(x, λη) = λφ(x, η), λ > 0 (positively homogeneous of degree 1 in η); σ(x, η) is a symbol of order m, i.e., σ C satisfies α η β x σ(x, η) C α,β (1 + η ) m α ; σ(x, η) is compactly supported in x; in a neighborhood of the support of σ ( 2 ) Φ det 0. x i η l Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

4 Classical theory: local L p boundedness Boundedness on local L 2 Theorem (Hörmander, 1971) If m 0, then A : L 2 comp L 2 comp continuously. Sketch of the proof. It suffices to consider the case m = 0. Then A A is a pseudodifferential operator with symbol of order m = 0, so that it is bounded L 2 L 2. So A is bounded too. Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

5 Classical theory: local L p boundedness Boundedness on local L p spaces Theorem (Seeger, Sogge and Stein, 1991) Let 1 < p <. If then A : L p comp L p comp continuously. m (d 1) p, Preliminary remark: if Φ(x, η) = φ(x)η is linear in η, and detφ 0, then Af (x) = e 2πiφ(x)η σ(x, η)ˆf (η) dη is the composition of a pseudodifferential operator of order m and the smooth change of variable x φ(x). Hence A : L p comp L p comp continuously, if m 0, i.e. without loss of derivatives. Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

6 Classical theory: local L p boundedness Sketch of the proof. In the general case, the proof goes as follows. Case p = 1. Let m = (d 1)/2. By a Littlewood-Paley decomposition on the frequency domain, one splits A into dyadic FIOs A j, j 0. Then each of them is further split into essentially 2 j(d 1)/2 FIOs with phases essentially linear in η, hence satisfying the desired estimates without loss of derivatives. By summing over j, one only obtains H 1 L 1 boundedness (H 1 being the Hardy space), which suffices for interpolation purposes. By analytic interpolation with the case p = 2 one obtains the result for 1 < p < 2, and by duality for 2 < p <. Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

7 Boundedness on local FL p spaces Boundedness on local FL p spaces Definition Let 1 p. The Fourier Lebesgue space FL p consists of the temperate distributions f such that ˆf L p. We set f FL p := ˆf L p. When is A : FL p FL p bounded? First attempt: A : FL p FL p, 1 < p <, is bounded F A F 1 : L p L p is bounded (F A F 1 ) : L p L p is bounded, where (F A F 1 ) f (x) = e 2πiΦ(η,x) σ(η, x)ˆf (η) dη. This problem is a genuinely global one: σ(η, x) is not compactly supported in x (moreover, Φ(η, x) is not homogeneous in η). This operator falls out of the classical L p -theory Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

8 Boundedness on local FL p spaces A model example in dimension d = 1 Af (x) = f (ϕ(x)) = R e 2πiϕ(x)η ˆf (η)dη, where ϕ : R R is a diffemorphism, with ϕ(x) = x for x 1, non-linear on ( 1, 1). Observe that A : C 0 (( 1, 1)) C 0 (( 1, 1)). However, For p 2 the estimate Af FL p C f FL p f C 0 (( 1, 1)) is false (cf. the Beurling-Helson theorem and also Okoudjou 2007, Ruzhansky, Sugimoto, Toft and Tomita, Preprint 2008). To see this, take f n (x) = χ(x)e 2πinx, with χ C 0 (( 1, 1)). Then, for 1 p < 2, f n FL p 1, Af n FL p n ( 1 p 1 2) +, as n, by an easy argument based on the van der Corput Lemma. For 2 < p one argues by duality. Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

9 Further examples Boundedness on local FL p spaces More generally, let Af (x) = with ϕ as before and G C 0 R e 2πiϕ(x)η G(x) η m ˆf (η)dη, (R), G(x) = 1 for x 1. Then A is bounded on (FL p ) comp, 1 < p < 2, (if and) only if ( 1 m p 1 ). 2 Similar examples hold for every 1 p, d 1, and give the threshold m d 1 p 1 2. Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

10 Boundedness on local FL p spaces Boundedness on local FL p spaces Theorem (Cordero, N., Rodino, 2008) If m d p, (1) then A is bounded on (FL p ) comp, whenever 1 p <. For p =, A is bounded on the closure of C 0 (Rd ) in FL (R d ) comp. The loss of derivatives in (1) is shown to be sharp for every 1 p and in any dimension d 1, even for phases linear in η. The proof makes use of techniques from time-frequency analysis: the modulation spaces M p. Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

11 Boundedness on local FL p spaces The use of modulation spaces Definition For 1 p the modulation spaces M p consists of the temperate distributions f such that f M p := f ( )g( x) FL p L p x <. Here g is a non-zero (so-called window) function in S(R d ). For heuristic purposes distributions in M p may be regarded as functions which are locally in FL p and decay at infinity like a function in L p. In particular: For distributions supported in a fixed compact subset K R d, there exists C K > 0 such that C 1 K u M p u FL p C K u M p. It suffices to prove the boundedness results on M p. We do this when p = 1 or p =, and m = d/2. The remaining cases follow by interpolation with the case p = 2, m = 0. Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

12 Boundedness on local FL p spaces Idea of the proof (first step) We know from [Concetti, Garello, Toft, 2007] and [Cordero, N., Rodino, 2007] that the the desired boundedness result on M p holds (without loss of derivatives) for phases Φ(x, η) satisfying the Shubin-type symbol estimates α x β η Φ(x, η) C α,β, α + β 2. For our phase (positively homogeneous of degree 1 in η), the derivatives 2 x i x l Φ fail to be bounded, in general. The phase is also singular at η = 0. Hence another argument is required to treat low frequencies (omitted for brevity). The high frequency part of A is then treated as follows. Let p = 1 or p =, and m = d/2. First step A is split into dyadic pieces A j localized where η 2 j, j 1. Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

13 Boundedness on local FL p spaces Idea of the proof (second step) Second step A j is then conjugated with the dilations operators U 2 j/2f (x) = f (2 j/2 x), so that where à j is a FIO with phase A j = U 2 j/2ãju 2 j/2, Φ j (x, η) = Φ(2 j/2 x, 2 j/2 η) = 2 j/2 Φ(2 j/2 x, η). Now Φ j (x, η) has derivatives of order 2 bounded on the support of the corresponding symbol, so that à j u M p 2 jd/2 u M p. Combining this estimate with the M p -bounds for the dilation operator [Sugimoto and Tomita, 2006], one deduces A j u M p u M p. Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

14 Boundedness on local FL p spaces Idea of the proof (third step) Third step Summing the estimates over j 1: one exploits the Proposition (Quasi-orthogonality property) If û is localized in the shell η 2 j, then Âu is localized in the neighbour shells. Heuristic proof. A moves the time-frequency concentration of a function u according to the canonical transformation generated by Φ: But (x, ξ) = χ(y, η). η ξ(y, η) in Lipschitz continuous, uniformly for y in bounded sets of R d. Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

15 Further refinements Boundedness on local FL p spaces Can the threshold m = d p be raised, under additional conditions? For phases Φ(x, η) linear in x, local boundedness on FL p holds without loss of derivatives: m = 0 [Ruzhansky, Sugimoto, Toft and Tomita 2008]. This agrees with the above counterexample, which deals with phases for which the graph of x Φ(x, η) has some curvature. Is there room for intermediate results, depending on the rank of Hessian dx 2 Φ(x, η)? Seeger, Sogge and Stein, 1991, showed that A is locally bounded on L p, 1 < p <, m r 1/2 1/p, if the rank dηφ(x, 2 ) is r, and a certain smooth factorization condition is satisfied (automatically satisfied in the case of constant rank). See also Ruzhansky s PhD thesis. Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

16 Further refinements Boundedness on local FL p spaces Definition (Spatial smooth factorization condition) Let 0 r d and suppose that for every (x 0, η 0 ) suppσ, η 0 = 1, there exists an open neighborhood Ω of x 0 and an open neighborhood Γ S d 1 of η 0, satisfying the following condition. For every η Γ there exists a smooth fibration of Ω, smoothly depending on η and with affine fibers of codimension r, such that x Φ(, η) is constant on every fiber. By a fibration of Ω, smoothly depending on η Γ and with fibers of codimension r we mean that a smooth function Π : Ω Γ R d is given, with d x Π having constant rank r. The fibers are the level sets of the mapping Π(, η). This condition implies the Hessian d 2 x Φ(x, η) to have rank r. Moreover it is always satisfied if r = d or if d 2 x Φ(x, η) has constant rank r (in particular, for phases linear in x, corresponding to r = 0). Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

17 Further refinements Boundedness on local FL p spaces Theorem (N., 2008) Assume that Φ satisfies the spatial smooth factorization condition for some r. If m r p, (2) then the corresponding FIO A is locally bounded on FL p, whenever 1 p <. For p =, A extends to a bounded operator on the closure of C0 (Rd ) in FL (R d ) comp. The threshold in (2) is sharp in any dimension d 1, even for phases Φ(x, η) which are linear in η (consider the phase Φ(x, η) = r k=1 ϕ(x k)η k + d k=r+1 x kη k, where ϕ : R R is a diffeomorphism, with ϕ(t) = t for t 1 and whose restriction to ( 1, 1) is non-linear). The proof relies on a suitable decomposition of the physical space, tailored to the above fibration. Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

18 Global boundedness on L p, FL p and on the modulation spaces M p Global boundedness: a striking example Consider the example, in dimension d = 1, given at the beginning: Af (x) = e 2πiϕ(x)η G(x) η m ˆf (η)dη, R where ϕ : R R is a diffeomorphism, with ϕ(x) = x for x 1, and G C0 (R), G(x) = 1 for ( x 1. ) We saw that A is not bounded on FLp, 1 1 < p < 2, unless m p 1 2. Hence the operator (F A F 1 ) f (x) = e 2πiϕ(η)x G(η) x mˆf (η) dη is not bounded on L p, 2 < p < unless m ( p ). New phenomenon: loss of decay Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

19 Global boundedness on L p, FL p and on the modulation spaces M p Global symbol classes It is natural to consider symbols no longer compactly supported in x, but satisfying decay estimates at infinity (SG or scattering classes of Parenti, Cordes, Melrose, Schrohe, Coriasco): α η β x σ(x, η) C α,β η m1 α x m2 β, (x, η) R 2d ; we write σ SG m1,m2. We also consider phases Φ SG 1,1, uniformly non-degenerate: ( ) 2 det Φ δ > 0, on R 2d. x i η l The above example Af (x) = R e 2πiϕ(η)x G(η) x mˆf (η) dη, G C 0 (R) has phase in SG 1,1 and symbol in SG,m. Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

20 Global boundedness on L p, FL p and on the modulation spaces M p Problems and results For such FIOs, natural problems are Global L p -boundedness [Coriasco and Ruzhansky, Preprint 2008]; Global FL p boundedness (follows from the previous investigation by conjugating with the Fourier transform and duality); Boundedness on the modulation spaces M p : Theorem (Cordero, N., Rodino, 2008) Let σ SG m1,m2 and Φ SG 1,1, uniformly non-degenerate. If m 1 d p, m 2 d p, (3) then the corresponding FIO A is bounded on M p, whenever 1 p <. For p =, A is bounded on the closure of the Schwartz space in M. Both the bounds in (3) are sharp. Namely, for any m 1 > d 1/2 1/p, or m 2 > d 1/2 1/p, there exists a FIO with σ SG m1,, σ SG,m2, respectively, (σ being compactly supported with respect to x and η respectively) which is not bounded on M p. Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21

21 Global boundedness on L p, FL p and on the modulation spaces M p F. Concetti, G. Garello, J. Toft. Trace Ideals for Fourier Integral Operators with Non-Smooth Symbols II. Preprint October Available at ArXiv: F. Concetti, G. Garello, J. Toft. Trace Ideals for Fourier Integral Operators with Non-Smooth Symbols III. Preprint February Available at ArXiv: E. Cordero, F. Nicola and L. Rodino. Boundedness of Fourier integral operators on FL p spaces. Trans. Amer. Math. Soc., to appear. Available at ArXiv: E. Cordero, F. Nicola and L. Rodino. On the global boundedness of Fourier integral operators. Preprint, April Available at ArXiv: L. Hörmander. Fourier integral operators I. Acta Math., 127:79 183, A. Seeger, C. D. Sogge and E. M. Stein. Regularity properties of Fourier integral operators. Ann. of Math. (2), 134(2): , Fabio Nicola (Politecnico di Torino) FIOs on Fourier Lebesgues spaces November 24-28, / 21