SINGULAR OSCILLATORY INTEGRAL OPERATORS

Transcription

1 SINGULAR OSCILLATORY INTEGRAL OPERATORS D.H. Phong Columbia University Conference in honor of Elias M. Stein Princeton University, May 16-21, 2011 May 16, 2011

2 Outline of the Talk

3 Outline of the Talk 1. Reminiscences about the 70 s

4 Outline of the Talk 1. Reminiscences about the 70 s 2. Generalized Radon transforms

5 Outline of the Talk 1. Reminiscences about the 70 s 2. Generalized Radon transforms The framework of Fourier integral operators

6 Outline of the Talk 1. Reminiscences about the 70 s 2. Generalized Radon transforms The framework of Fourier integral operators Dirac measures on subvarieties

7 Outline of the Talk 1. Reminiscences about the 70 s 2. Generalized Radon transforms The framework of Fourier integral operators Dirac measures on subvarieties Some analytic and geometric questions

8 Outline of the Talk 1. Reminiscences about the 70 s 2. Generalized Radon transforms The framework of Fourier integral operators Dirac measures on subvarieties Some analytic and geometric questions 3. Some partial answers

9 Outline of the Talk 1. Reminiscences about the 70 s 2. Generalized Radon transforms The framework of Fourier integral operators Dirac measures on subvarieties Some analytic and geometric questions 3. Some partial answers Estimates for degenerate oscillatory integrals

10 Outline of the Talk 1. Reminiscences about the 70 s 2. Generalized Radon transforms The framework of Fourier integral operators Dirac measures on subvarieties Some analytic and geometric questions 3. Some partial answers Estimates for degenerate oscillatory integrals Estimates for sublevel sets

11 Outline of the Talk 1. Reminiscences about the 70 s 2. Generalized Radon transforms The framework of Fourier integral operators Dirac measures on subvarieties Some analytic and geometric questions 3. Some partial answers Estimates for degenerate oscillatory integrals Estimates for sublevel sets Degenerate oscillatory integral operators

12 Outline of the Talk 1. Reminiscences about the 70 s 2. Generalized Radon transforms The framework of Fourier integral operators Dirac measures on subvarieties Some analytic and geometric questions 3. Some partial answers Estimates for degenerate oscillatory integrals Estimates for sublevel sets Degenerate oscillatory integral operators 4. Jugendtraum

13 Reminiscences about the 70 s Pseudo-differential operators Calculus of Kohn-Nirenberg, inspired by Kohn s L 2 solution of the and b problems; subelliptic estimates, weakly pseudoconvex domains Exotic classes Sρ,δ m of Hörmander and SM,m Φ,φ of Beals-Fefferman The almost-orthogonality lemma of Cotlar-Stein, L 2 boundedness theorem of Calderón-Vaillancourt

14 Reminiscences about the 70 s Pseudo-differential operators Calculus of Kohn-Nirenberg, inspired by Kohn s L 2 solution of the and b problems; subelliptic estimates, weakly pseudoconvex domains Exotic classes Sρ,δ m of Hörmander and SM,m Φ,φ of Beals-Fefferman The almost-orthogonality lemma of Cotlar-Stein, L 2 boundedness theorem of Calderón-Vaillancourt Fourier integral operators Early ideas of Maslov and Egorov Theory of Hörmander and Duistermaat-Hörmander for real phases Complex phases of Melin-Sjöstrand

15 Reminiscences about the 70 s Pseudo-differential operators Calculus of Kohn-Nirenberg, inspired by Kohn s L 2 solution of the and b problems; subelliptic estimates, weakly pseudoconvex domains Exotic classes Sρ,δ m of Hörmander and SM,m Φ,φ of Beals-Fefferman The almost-orthogonality lemma of Cotlar-Stein, L 2 boundedness theorem of Calderón-Vaillancourt Fourier integral operators Early ideas of Maslov and Egorov Theory of Hörmander and Duistermaat-Hörmander for real phases Complex phases of Melin-Sjöstrand New singular integral operators Folland-Stein s fundamental solution for b Greiner-Stein s L p estimates for the Neumann problem Rothschild-Stein s fundamental solution for N j=1 X2 j +ix 0 Fefferman s expansion for the Bergman kernel, subsequently simplified by Kerzman-Stein, and refined by Boutet de Monvel-Sjöstrand.

16 Green s function for the -Neumann problem The model case is the Siegel upper half-space U = {(z,z n+1 C n+1 ; Imz n+1 > z 2 }, which can be identified with H n R + via (z,z n+1 ) (ζ,ρ), ζ = (z,t), t = Rez n+1, ρ = Imz n+1 z 2. Here H n is the Heisenberg group H n = {C n R;(z,t) (z,t ) = (z +z,t +t +2Imz z )}. The -Neumann problem is the following boundary value problem u = f on H n R +, ( ρ +i t)u = 0 when ρ = 0.

17 Green s function for the -Neumann problem The model case is the Siegel upper half-space U = {(z,z n+1 C n+1 ; Imz n+1 > z 2 }, which can be identified with H n R + via (z,z n+1 ) (ζ,ρ), ζ = (z,t), t = Rez n+1, ρ = Imz n+1 z 2. Here H n is the Heisenberg group H n = {C n R;(z,t) (z,t ) = (z +z,t +t +2Imz z )}. The -Neumann problem is the following boundary value problem u = f on H n R +, ( ρ +i t)u = 0 when ρ = 0. An explicit formula for the Green s function u(ζ,ρ) = N(ζ 1 η, ρ µ )f(η,µ) K(ζ 1 η,ρ+µ)f(η,µ) H n R + H n R + where N(ζ,ρ) 1 (2 z 2 +t 2 +ρ 2 ) n, K(ζ,ρ) 1 (2 z 2 +t 2 +ρ 2 ) k (2 z 2 +ρ it) l

18 Green s function for the -Neumann problem The model case is the Siegel upper half-space U = {(z,z n+1 C n+1 ; Imz n+1 > z 2 }, which can be identified with H n R + via (z,z n+1 ) (ζ,ρ), ζ = (z,t), t = Rez n+1, ρ = Imz n+1 z 2. Here H n is the Heisenberg group H n = {C n R;(z,t) (z,t ) = (z +z,t +t +2Imz z )}. The -Neumann problem is the following boundary value problem u = f on H n R +, ( ρ +i t)u = 0 when ρ = 0. An explicit formula for the Green s function u(ζ,ρ) = N(ζ 1 η, ρ µ )f(η,µ) K(ζ 1 η,ρ+µ)f(η,µ) H n R + H n R + where N(ζ,ρ) 1 (2 z 2 +t 2 +ρ 2 ) n, K(ζ,ρ) 1 (2 z 2 +t 2 +ρ 2 ) k (2 z 2 +ρ it) l Key features of K K(ζ, ρ) is a mixture of elliptic and parabolic homogeneities K C (U \0), but K has hidden singularities along t = ρ = 0.

19 A distribution of hypersurfaces Ω 0 = {(z,0);z C} H n Ω ζ = ζ Ω 0

20 A distribution of hypersurfaces Ω 0 = {(z,0);z C} H n Ω ζ = ζ Ω 0 Propagation of hidden singularities along Ω ζ { } D 2 u +(ζ,ρ) = K ζ,η (z w,ρ+µ)t sf(η,µ)dσ Ωζ (η) dsdµ, 0 Ω ζ where ζ = (z,t), η = (w,s), and K ζ,η (w,ρ) is a Calderón-Zygmund kernel on Ω ζ, with norm O((s 2 +µ 2 ) 1 ), T sv is a translation of v by s. Singular Radon transforms Rv(ζ) = Ω ζ K(ζ,η)v(η)dσ Ωζ (η)

21 A distribution of hypersurfaces Ω 0 = {(z,0);z C} H n Ω ζ = ζ Ω 0 Propagation of hidden singularities along Ω ζ { } D 2 u +(ζ,ρ) = K ζ,η (z w,ρ+µ)t sf(η,µ)dσ Ωζ (η) dsdµ, 0 Ω ζ where ζ = (z,t), η = (w,s), and K ζ,η (w,ρ) is a Calderón-Zygmund kernel on Ω ζ, with norm O((s 2 +µ 2 ) 1 ), T sv is a translation of v by s. Singular Radon transforms Rv(ζ) = K(ζ,η)v(η)dσ Ωζ (η) Ω ζ Group Fourier transform proof by Geller-Stein Analogue of the Hilbert transforms along curves introduced by Nagel-Riviere-Wainger WF(R) = N (C) : works of Guillemin, and especially Greenleaf-Uhlmann on Gelfand s problem, namely to identify family of curves that suffice to invert the X-ray transform along curves. Most general version of L p boundedness by Christ-Nagel-Stein-Wainger

22 Generalized Radon transforms Let X,Y be smooth manifolds, and C X Y a smooth submanifold. Then a Dirac measure δ C (x,y) supported on C defines a generalized Radon transform, Rf(x) = δ C (x,y)f(y) C x with C x = {y Y;(x,y) C}.

23 Generalized Radon transforms Let X,Y be smooth manifolds, and C X Y a smooth submanifold. Then a Dirac measure δ C (x,y) supported on C defines a generalized Radon transform, Rf(x) = δ C (x,y)f(y) C x with C x = {y Y;(x,y) C}. The framework of Fourier integral operators If C = {ϕ 1 (x,y) = = ϕ l (x,y) = 0} locally, then δ C (x,y) = e i l k=1 θ k ϕ k (x,y) a(x,y,θ)dθ, so R is a Fourier integral operator with Lagrangian Λ = N (C) T (X) T (Y).

24 Generalized Radon transforms Let X,Y be smooth manifolds, and C X Y a smooth submanifold. Then a Dirac measure δ C (x,y) supported on C defines a generalized Radon transform, Rf(x) = δ C (x,y)f(y) C x with C x = {y Y;(x,y) C}. The framework of Fourier integral operators If C = {ϕ 1 (x,y) = = ϕ l (x,y) = 0} locally, then δ C (x,y) = e i l k=1 θ k ϕ k (x,y) a(x,y,θ)dθ, so R is a Fourier integral operator with Lagrangian Λ = N (C) T (X) T (Y). General theory of Hörmander: if Λ is a local graph over T (X) (equivalently, over T (Y)), then R is smoothing of order (n l)/2 = dimc x/2. The local graph condition can be written down explicitly as, θ R l \0, ( ) 0 dyϕ j det d xϕ k dxy 2 l m=1 θmϕm(x,y) 0. In general, R is smoothing of order 1 2 (dimcx dimkerdπ X), with π X the projection π X : T (X Y) T (X).

25 Dirac measure of subvarieties Consider the case X = Y = R n, and C x is the translate to x of a submanifold V passing through the origin. Then the order of smoothing of R is the rate of decay of the Fourier transform of the Dirac measure on V, ˆδ V (ξ) C ξ δ

26 Dirac measure of subvarieties Consider the case X = Y = R n, and C x is the translate to x of a submanifold V passing through the origin. Then the order of smoothing of R is the rate of decay of the Fourier transform of the Dirac measure on V, ˆδ V (ξ) C ξ δ When V is a hypersurface, the graph condition holds when the Gaussian curvature of V is not 0. The Radon transform R is then smoothing of order δ = (n 1)/2. When V is a curve, the graph condition cannot hold if dimx 3. Hörmander s theorem shows only that R is smoothing of order δ = 0. When V is a curve with torsion, the van der Corput lemma shows that R is smoothing of order δ = 1/n.

27 Dirac measure of subvarieties Consider the case X = Y = R n, and C x is the translate to x of a submanifold V passing through the origin. Then the order of smoothing of R is the rate of decay of the Fourier transform of the Dirac measure on V, ˆδ V (ξ) C ξ δ When V is a hypersurface, the graph condition holds when the Gaussian curvature of V is not 0. The Radon transform R is then smoothing of order δ = (n 1)/2. When V is a curve, the graph condition cannot hold if dimx 3. Hörmander s theorem shows only that R is smoothing of order δ = 0. When V is a curve with torsion, the van der Corput lemma shows that R is smoothing of order δ = 1/n. Higher codimension lead to higher order degeneracies, which are beyond the scope of the standard method of stationary phase, and the corresponding conditions on second order derivatives.

28 A closer look Let R d t x(t) R n be a local parametrization of V. Then ˆδ V (ξ) = e i n j=1 ξ j x j (t) χ(t)dt Setting ξ = λω, λ = ξ, ω S n 1, ˆδ V (ξ) = e iλφω(t) χ(t)dt, where Φ ω(t) = n j=1 ω jx j (t), we can formulate the following

29 A closer look Let R d t x(t) R n be a local parametrization of V. Then ˆδ V (ξ) = e i n j=1 ξ j x j (t) χ(t)dt Setting ξ = λω, λ = ξ, ω S n 1, ˆδ V (ξ) = e iλφω(t) χ(t)dt, where Φ ω(t) = n j=1 ω jx j (t), we can formulate the following Analytic questions Given Φ ω(t), what is the decay rate C ω λ δω of the above oscillatory integral with phase Φ ω(t)? When are δ ω and C ω semicontinuous ( stable ) as ω varies?

30 A closer look Let R d t x(t) R n be a local parametrization of V. Then ˆδ V (ξ) = e i n j=1 ξ j x j (t) χ(t)dt Setting ξ = λω, λ = ξ, ω S n 1, ˆδ V (ξ) = e iλφω(t) χ(t)dt, where Φ ω(t) = n j=1 ω jx j (t), we can formulate the following Analytic questions Given Φ ω(t), what is the decay rate C ω λ δω of the above oscillatory integral with phase Φ ω(t)? When are δ ω and C ω semicontinuous ( stable ) as ω varies? Geometric questions Given n = dimx, d = dimv, what are the best possible orders δ(n,d) of smoothing? (e.g., δ(n,n 1) = (n 1)/2, δ(n,1) = 1/n) What are the geometric conditions on V which guarantee this best possible order of smoothing?

31 Multiplicities or Milnor numbers Intuitively, smoothing should require the map V V x 1 + +x N R n to be locally surjective for N large enough. This implies that no direction ø R n is orthogonal to V at N points. This suggests a measure µ of (higher-order) curvature of V is the maximum number of points admitting a given direction among its normals. Set then, for f C ω, and a an isolated critical point of f, µ = dima(a)/i[ 1 f,, d f] with A(a) the space of germs of analytic functions at a, and I[ 1 f,, d f] the ideal generated by the partial derivatives of f at a. We say that V has non-vanishing µ-curvature if ω R n \0, the phase Φ ω(t) has multiplicity at most µ at any critical point. (Note that µ = 1 for a hypersurface and µ = n for a curve correspond respectively to non-vanishing Gaussian curvature and non-vanishing torsion).

32 Multiplicities or Milnor numbers Intuitively, smoothing should require the map V V x 1 + +x N R n to be locally surjective for N large enough. This implies that no direction ø R n is orthogonal to V at N points. This suggests a measure µ of (higher-order) curvature of V is the maximum number of points admitting a given direction among its normals. Set then, for f C ω, and a an isolated critical point of f, µ = dima(a)/i[ 1 f,, d f] with A(a) the space of germs of analytic functions at a, and I[ 1 f,, d f] the ideal generated by the partial derivatives of f at a. We say that V has non-vanishing µ-curvature if ω R n \0, the phase Φ ω(t) has multiplicity at most µ at any critical point. (Note that µ = 1 for a hypersurface and µ = n for a curve correspond respectively to non-vanishing Gaussian curvature and non-vanishing torsion). A naive conjecture for the optimal order of smoothing for Radon transforms defined by d-dimensional submanifolds with non-vanishing curvature is d δ = µ d 1 +1 (0.1)

33 Multiplicities or Milnor numbers Intuitively, smoothing should require the map V V x 1 + +x N R n to be locally surjective for N large enough. This implies that no direction ø R n is orthogonal to V at N points. This suggests a measure µ of (higher-order) curvature of V is the maximum number of points admitting a given direction among its normals. Set then, for f C ω, and a an isolated critical point of f, µ = dima(a)/i[ 1 f,, d f] with A(a) the space of germs of analytic functions at a, and I[ 1 f,, d f] the ideal generated by the partial derivatives of f at a. We say that V has non-vanishing µ-curvature if ω R n \0, the phase Φ ω(t) has multiplicity at most µ at any critical point. (Note that µ = 1 for a hypersurface and µ = n for a curve correspond respectively to non-vanishing Gaussian curvature and non-vanishing torsion). A naive conjecture for the optimal order of smoothing for Radon transforms defined by d-dimensional submanifolds with non-vanishing curvature is d δ = µ d 1 +1 (0.1) This reduces to (n 1)/2 for hypersurfaces with non-vanishing curvature, and 1/n for curves with torsion. For general d, µ provides a non-linear interpolation between these extreme cases.

34 Multiplicities or Milnor numbers Intuitively, smoothing should require the map V V x 1 + +x N R n to be locally surjective for N large enough. This implies that no direction ø R n is orthogonal to V at N points. This suggests a measure µ of (higher-order) curvature of V is the maximum number of points admitting a given direction among its normals. Set then, for f C ω, and a an isolated critical point of f, µ = dima(a)/i[ 1 f,, d f] with A(a) the space of germs of analytic functions at a, and I[ 1 f,, d f] the ideal generated by the partial derivatives of f at a. We say that V has non-vanishing µ-curvature if ω R n \0, the phase Φ ω(t) has multiplicity at most µ at any critical point. (Note that µ = 1 for a hypersurface and µ = n for a curve correspond respectively to non-vanishing Gaussian curvature and non-vanishing torsion). A naive conjecture for the optimal order of smoothing for Radon transforms defined by d-dimensional submanifolds with non-vanishing curvature is d δ = µ d 1 +1 (0.1) This reduces to (n 1)/2 for hypersurfaces with non-vanishing curvature, and 1/n for curves with torsion. For general d, µ provides a non-linear interpolation between these extreme cases. The case d = 2 can be proved (P.-Stein, with a loss of ǫ derivatives, ǫ arbitrarily small), using results of Varchenko, Karpushkin, and Kushnirenko.

35 Estimates for Degenerate Oscillatory Integrals

36 Estimates for Degenerate Oscillatory Integrals The van der Corput lemma Let Φ(t) be a smooth real-valued function on [a,b]. If Φ (k) (t) 1 then b e iλφ(t) dt C k λ k 1 a if k 2, or k = 1 and Φ (t) is monotone. Here C k is a constant depending only on k.

37 Estimates for Degenerate Oscillatory Integrals The van der Corput lemma Let Φ(t) be a smooth real-valued function on [a,b]. If Φ (k) (t) 1 then b e iλφ(t) dt C k λ k 1 a if k 2, or k = 1 and Φ (t) is monotone. Here C k is a constant depending only on k. Varchenko s theorem Let Φ(t) be a real-valued function on R d with 0 as a critical point. The Newton diagram of Φ is the convex hull of the upper quadrants in R d with vertices at those k = (k 1,,k d ) with the monomial t k appearing in the Taylor expansion of Φ. The Newton distance α is defined by the condition that (α 1,,α 1 ) be the intersection of the line k 1 = = k d with a face of the Newton diagram. For each face γ of the Newton diagram, let P γ be the polynomial in the Taylor expansion of Φ with monomials in the face γ. Assume dp γ 0 in R d \0. Then e iλφ(t) χ(t)dt C λ α (log λ ) β.

38 Estimates for Degenerate Oscillatory Integrals The van der Corput lemma Let Φ(t) be a smooth real-valued function on [a,b]. If Φ (k) (t) 1 then b e iλφ(t) dt C k λ k 1 a if k 2, or k = 1 and Φ (t) is monotone. Here C k is a constant depending only on k. Varchenko s theorem Let Φ(t) be a real-valued function on R d with 0 as a critical point. The Newton diagram of Φ is the convex hull of the upper quadrants in R d with vertices at those k = (k 1,,k d ) with the monomial t k appearing in the Taylor expansion of Φ. The Newton distance α is defined by the condition that (α 1,,α 1 ) be the intersection of the line k 1 = = k d with a face of the Newton diagram. For each face γ of the Newton diagram, let P γ be the polynomial in the Taylor expansion of Φ with monomials in the face γ. Assume dp γ 0 in R d \0. Then e iλφ(t) χ(t)dt C λ α (log λ ) β. Stability theorem of Karpushkin Let Φ be a C ω function with t <r Φ(t) δ < for some δ > 0, d = 2. Then there exists 0 < s < r and ǫ > 0 so that for all C ω Ψ with Φ Ψ C 0 ( t <r,t C 2 ) < ǫ, Ψ δ < t <s

39 Criteria for adapted coordinate systems The decay rate δ is reparametrization invariant, while the Newton distance α is coordinate dependent. The equality δ = α can only hold generically. It is very useful to have criteria for when it holds.

40 Criteria for adapted coordinate systems The decay rate δ is reparametrization invariant, while the Newton distance α is coordinate dependent. The equality δ = α can only hold generically. It is very useful to have criteria for when it holds. In dimension d = 2, roots r j (x) of polynomials Φ(x,y) are given by Puiseux series. Define the clustering Ξ to be the number of elements in the largest cluster of roots, where a cluster of roots is an equivalence class of roots, with r j r k if r j r k r j 1 0. The criterion for adapted coordinate systems is Ξ α 1 and such coordinate systems always exist (P.-Stein-Sturm; also simpler and self-contained proof of the stability theorem of Karpushkin).

41 Criteria for adapted coordinate systems The decay rate δ is reparametrization invariant, while the Newton distance α is coordinate dependent. The equality δ = α can only hold generically. It is very useful to have criteria for when it holds. In dimension d = 2, roots r j (x) of polynomials Φ(x,y) are given by Puiseux series. Define the clustering Ξ to be the number of elements in the largest cluster of roots, where a cluster of roots is an equivalence class of roots, with r j r k if r j r k r j 1 0. The criterion for adapted coordinate systems is Ξ α 1 and such coordinate systems always exist (P.-Stein-Sturm; also simpler and self-contained proof of the stability theorem of Karpushkin). Extensions to C phases and other constructions have been given by Greenblatt, Ikromov-Müller, Ikromov-Kempe-Müller, and others.

42 Criteria for adapted coordinate systems The decay rate δ is reparametrization invariant, while the Newton distance α is coordinate dependent. The equality δ = α can only hold generically. It is very useful to have criteria for when it holds. In dimension d = 2, roots r j (x) of polynomials Φ(x,y) are given by Puiseux series. Define the clustering Ξ to be the number of elements in the largest cluster of roots, where a cluster of roots is an equivalence class of roots, with r j r k if r j r k r j 1 0. The criterion for adapted coordinate systems is Ξ α 1 and such coordinate systems always exist (P.-Stein-Sturm; also simpler and self-contained proof of the stability theorem of Karpushkin). Extensions to C phases and other constructions have been given by Greenblatt, Ikromov-Müller, Ikromov-Kempe-Müller, and others. The recent work of Collins-Greenleaf-Pramanik In general dimension d, for a given non-constant C ω function Φ, there exists a finite collection C of coordinate transformations, so that if α(f) denotes the Newton distance in the coordinate system F F, we have δ = inf F C α(f). The construction of the class C is actually algorithmic. Criteria for whether a specific coordinate system in C is adapted can be formulated in terms of projections onto diagrams in 2 variables, and using the 2-dimensional criteria formulated above.

43 Estimates for Sublevel Sets It is not difficult to see that the decay rate of oscillatory integrals with phase Φ is essentially the same as the growth rate of the volume of its level sets {t B; Φ(t) M} C M δ Stable estimates for oscillatory integrals correspond to volume estimates with bounds C uniform in Φ. In fact, certain even stronger bounds are known, which depend only on lower bounds for certain derivatives of Φ.

44 Estimates for Sublevel Sets It is not difficult to see that the decay rate of oscillatory integrals with phase Φ is essentially the same as the growth rate of the volume of its level sets {t B; Φ(t) M} C M δ Stable estimates for oscillatory integrals correspond to volume estimates with bounds C uniform in Φ. In fact, certain even stronger bounds are known, which depend only on lower bounds for certain derivatives of Φ. Sublevel set estimates of Carbery-Christ-Wright For any multi-index k, there exists δ > 0 and C, depending only on k and d, so that the above estimate holds, for any function Φ satisfying the lower bound k Φ 1 on B.

45 Estimates for Sublevel Sets It is not difficult to see that the decay rate of oscillatory integrals with phase Φ is essentially the same as the growth rate of the volume of its level sets {t B; Φ(t) M} C M δ Stable estimates for oscillatory integrals correspond to volume estimates with bounds C uniform in Φ. In fact, certain even stronger bounds are known, which depend only on lower bounds for certain derivatives of Φ. Sublevel set estimates of Carbery-Christ-Wright For any multi-index k, there exists δ > 0 and C, depending only on k and d, so that the above estimate holds, for any function Φ satisfying the lower bound k Φ 1 on B. Sublevel set operators of P.-Stein-Sturm For any given set β (1),,β (K) N d \0, define the multilinear operator W M (f 1,,f d ) = f 1 (x 1 ) f d (x d )dx 1 dx d β(j) Φ >1, 1 j K Assume that Φ is a polynomial of degree m. Then there exists a constant C depending only β (1),,β (K) N d \0, so that W M (f 1,,f d ) C M d 1 α log d 2 (2 + 1 d M ) f i L i=1 d d 1 where α is the Newton distance for the diagram with vertices at β (j), 1 j K.

46 The new estimates of Gressman Define inductively the classes L κ,ρ of operators by L 1,(0,,0) consists of the identity; L κ,ρ consists of operators of the form LΦ = det ti1 L 1 Φ ti1 L nφ tin L 1 Φ tin L nφ Here L p L κp,ρp, κ = κ 1 + +κ n, ρ = ρ 1 + +ρ p +(1,,1,0,,0), the 1 occurring at i p. If Φ C ω (B), and for any closed set D B, there exists a constant C so that {t D; Φ(t) M} C M β +1 α (inf t D LΦ ) 1 α β +1 α For Pfaffian functions Φ, C depends only on d, L, and the Pfaffian type of Φ. The proof makes use of works of Khovanskii and Gabrielov.

47 Degenerate Oscillatory Integral Operators Low order of degeneracies Lagrangians with two-sided Whitney folds: smoothing with loss of 1 6 derivatives (Melrose-Taylor) Lagrangians with one-sided Whitney fold: smoothing with loss of 1 4 derivatives (Greenleaf-Uhlmann) Lagrangians with two-sided cusps: loss of 1 4 (Comech-Cuccagna, Greenleaf-Seeger) Radon transforms and finite-type conditions in the plane: Seeger

48 Degenerate Oscillatory Integral Operators Low order of degeneracies Lagrangians with two-sided Whitney folds: smoothing with loss of 1 6 derivatives (Melrose-Taylor) Lagrangians with one-sided Whitney fold: smoothing with loss of 1 4 derivatives (Greenleaf-Uhlmann) Lagrangians with two-sided cusps: loss of 1 4 (Comech-Cuccagna, Greenleaf-Seeger) Radon transforms and finite-type conditions in the plane: Seeger Arbitrary degeneracies in dimensions (P.-Stein) Let Φ(x, y) be a real-analytic phase function in 2 dimensions. Then the oscillatory integral operator T λ defined by Tf(x) = e iλφ(x,y) χ(x,y)f(y)dy R for χ C 0 (R2 ) with sufficiently small support near 0, is bounded on L 2 (R) with norm T C λ 1 2 δ where δ is the reduced Newton distance of Φ at 0

49 Degenerate Oscillatory Integral Operators Low order of degeneracies Lagrangians with two-sided Whitney folds: smoothing with loss of 1 6 derivatives (Melrose-Taylor) Lagrangians with one-sided Whitney fold: smoothing with loss of 1 4 derivatives (Greenleaf-Uhlmann) Lagrangians with two-sided cusps: loss of 1 4 (Comech-Cuccagna, Greenleaf-Seeger) Radon transforms and finite-type conditions in the plane: Seeger Arbitrary degeneracies in dimensions (P.-Stein) Let Φ(x, y) be a real-analytic phase function in 2 dimensions. Then the oscillatory integral operator T λ defined by Tf(x) = e iλφ(x,y) χ(x,y)f(y)dy R for χ C 0 (R2 ) with sufficiently small support near 0, is bounded on L 2 (R) with norm T C λ 1 2 δ where δ is the reduced Newton distance of Φ at 0 Extensions to C phases were obtained by Rychkov, Greenblatt. A simpler proof for polynomial phases was given later by P.-Stein-Sturm, using sublevel set multilinear functionals, and the Hardy-Littlewood maximal function.

50 Damped oscillatory integral operators (P.-Stein) Let Φ(x,y) and χ(x,y) be as previously. Then the damped oscillatory integral operator Df(x) = e iλφ(x,y) Φ xy(x,y) 2χ(x,y)f(y)dy 1 R is bounded on L 2 (R) with norm D C λ 1 2 Earlier works on damped operators are in Sogge-Stein, Cowling-Disney-Mauceri -Müller, and P.-Stein, where they are used for the study of L p L q smoothing.

51 Damped oscillatory integral operators (P.-Stein) Let Φ(x,y) and χ(x,y) be as previously. Then the damped oscillatory integral operator Df(x) = e iλφ(x,y) Φ xy(x,y) 2χ(x,y)f(y)dy 1 R is bounded on L 2 (R) with norm D C λ 1 2 Earlier works on damped operators are in Sogge-Stein, Cowling-Disney-Mauceri -Müller, and P.-Stein, where they are used for the study of L p L q smoothing. Related non-oscillating operator (P.-Stein) Let E be the following operator, where I is a small interval around 0, Ef(x) = Φ(x,y) µ f(y)dy I Then E is a bounded operator on L 2 (R) for µ < 1 2 δ 0 where δ 0 is the Newton distance for Φ at 0. It is still bounded on L 2 (R) when µ = 1 2 δ 0, except possibly when the main face reduces to a single vertex, or is parallel to one of the axes, or to the line p +q = 0.

52 The general strategy: operator van der Corput Decompose the set {Φ xy 0} into {Φ xy 0} = k{ Φ xy 2 k } with corresponding partition χ k (x,y) and decomposition T = k T k.

53 The general strategy: operator van der Corput Decompose the set {Φ xy 0} into {Φ xy 0} = k{ Φ xy 2 k } with corresponding partition χ k (x,y) and decomposition T = k T k. Establish an oscillatory estimate and a size estimate T k C (2 k λ ) 1 2 T k (I k J k ) 1 2 where I k and J k are the widths along the x and y axes of the set { Φ xy 2 k }.

54 The general strategy: operator van der Corput Decompose the set {Φ xy 0} into {Φ xy 0} = k{ Φ xy 2 k } with corresponding partition χ k (x,y) and decomposition T = k T k. Establish an oscillatory estimate and a size estimate T k C (2 k λ ) 1 2 T k (I k J k ) 1 2 where I k and J k are the widths along the x and y axes of the set { Φ xy 2 k }. Resum in k, exploiting the better of the oscillatory or size estimate.

55 The general strategy: operator van der Corput Decompose the set {Φ xy 0} into {Φ xy 0} = k{ Φ xy 2 k } with corresponding partition χ k (x,y) and decomposition T = k T k. Establish an oscillatory estimate and a size estimate T k C (2 k λ ) 1 2 T k (I k J k ) 1 2 where I k and J k are the widths along the x and y axes of the set { Φ xy 2 k }. Resum in k, exploiting the better of the oscillatory or size estimate. The key difficulty The sets { Φ xy 2 k } are usually very complicated geometrically, and the partition χ k (x,y) necessarily complicated also. It is essential that the oscillatory estimate be uniform in χ k. and this requires very precise versions of the oscillatory integral estimates.

56 The general strategy: operator van der Corput Decompose the set {Φ xy 0} into {Φ xy 0} = k{ Φ xy 2 k } with corresponding partition χ k (x,y) and decomposition T = k T k. Establish an oscillatory estimate and a size estimate T k C (2 k λ ) 1 2 T k (I k J k ) 1 2 where I k and J k are the widths along the x and y axes of the set { Φ xy 2 k }. Resum in k, exploiting the better of the oscillatory or size estimate. The key difficulty The sets { Φ xy 2 k } are usually very complicated geometrically, and the partition χ k (x,y) necessarily complicated also. It is essential that the oscillatory estimate be uniform in χ k. and this requires very precise versions of the oscillatory integral estimates. Curved Box Lemma: Let a curved box B be a set of the form B = {(x,y);φ(x) < y < φ(x)+δ, a < x < b} for some monotone function φ(x). Assume that the cut-off function satisfies y nχ(x,y) δ n, and that Φ xy is a polynomial satisfying µ Φ xy Aµ on B. Then the corresponding operator T satisfies T C(λµ) 1 2 with C depending only on A and the degree of Φ(x,y).

57 The general strategy: operator van der Corput Decompose the set {Φ xy 0} into {Φ xy 0} = k{ Φ xy 2 k } with corresponding partition χ k (x,y) and decomposition T = k T k. Establish an oscillatory estimate and a size estimate T k C (2 k λ ) 1 2 T k (I k J k ) 1 2 where I k and J k are the widths along the x and y axes of the set { Φ xy 2 k }. Resum in k, exploiting the better of the oscillatory or size estimate. The key difficulty The sets { Φ xy 2 k } are usually very complicated geometrically, and the partition χ k (x,y) necessarily complicated also. It is essential that the oscillatory estimate be uniform in χ k. and this requires very precise versions of the oscillatory integral estimates. Curved Box Lemma: Let a curved box B be a set of the form B = {(x,y);φ(x) < y < φ(x)+δ, a < x < b} for some monotone function φ(x). Assume that the cut-off function satisfies y nχ(x,y) δ n, and that Φ xy is a polynomial satisfying µ Φ xy Aµ on B. Then the corresponding operator T satisfies T C(λµ) 1 2 with C depending only on A and the degree of Φ(x,y). Curved Trapezoid Lemma: requires Hardy-Littlewood maximal function (P.-Stein-Sturm)

58 Further Developments Works of Kempe-Ikromov-Müller L p boundedness of maximal Radon transforms for smooth hypersurfaces in R 3, p > h(s), where h(s) is the supremum over Newton distances. Applications to conjectures of Stein, Iosevich-Sawyer, and to restriction theorems.

59 Further Developments Works of Kempe-Ikromov-Müller L p boundedness of maximal Radon transforms for smooth hypersurfaces in R 3, p > h(s), where h(s) is the supremum over Newton distances. Applications to conjectures of Stein, Iosevich-Sawyer, and to restriction theorems. Works of Tang, Greenleaf-Pramanik-Tang, Gressman: Higher Dimensions Tang s result: Let Φ(x,z) = m 1 j=1 P j(x)z m j be a homogeneous polynomial of degree m in R 2 R. Assume that the first and the last non-vanishing polynomials P jmin and P jmax are non-degenerate (dp(x) 0 for x 0), and that j min 2m jmax. Then for m 4, 3 T C λ 3 2m.

60 Further Developments Works of Kempe-Ikromov-Müller L p boundedness of maximal Radon transforms for smooth hypersurfaces in R 3, p > h(s), where h(s) is the supremum over Newton distances. Applications to conjectures of Stein, Iosevich-Sawyer, and to restriction theorems. Works of Tang, Greenleaf-Pramanik-Tang, Gressman: Higher Dimensions Tang s result: Let Φ(x,z) = m 1 j=1 P j(x)z m j be a homogeneous polynomial of degree m in R 2 R. Assume that the first and the last non-vanishing polynomials P jmin and P jmax are non-degenerate (dp(x) 0 for x 0), and that j min 2m jmax. Then for m 4, 3 T C λ 3 2m. Greenleaf-Pramanik-Tang s result: Let Φ(x, z) be a homogeneous polynomial of degree m in R n X R n Y. Assume that S (x,z) has at least one non-zero entry at every point of R n X+n Y \0. Then T C λ n X +n Y 2m if m > n X +n Y, and T C λ 1/2 log λ if m = n X +n Y, and T C λ 1/2 if 2 m < n X +n Y.

61 Further Developments Works of Kempe-Ikromov-Müller L p boundedness of maximal Radon transforms for smooth hypersurfaces in R 3, p > h(s), where h(s) is the supremum over Newton distances. Applications to conjectures of Stein, Iosevich-Sawyer, and to restriction theorems. Works of Tang, Greenleaf-Pramanik-Tang, Gressman: Higher Dimensions Tang s result: Let Φ(x,z) = m 1 j=1 P j(x)z m j be a homogeneous polynomial of degree m in R 2 R. Assume that the first and the last non-vanishing polynomials P jmin and P jmax are non-degenerate (dp(x) 0 for x 0), and that j min 2m jmax. Then for m 4, 3 T C λ 3 2m. Greenleaf-Pramanik-Tang s result: Let Φ(x, z) be a homogeneous polynomial of degree m in R n X R n Y. Assume that S (x,z) has at least one non-zero entry at every point of R n X+n Y \0. Then T C λ n X +n Y 2m if m > n X +n Y, and T C λ 1/2 log λ if m = n X +n Y, and T C λ 1/2 if 2 m < n X +n Y. Cubic phases: Greenleaf-Pramanik-Tang (n X = n Y = 2); also Gressman.

62 Jugendtraum Problem: formulate uniform estimates with interplay between the decay rate and the configuration of critical points.

63 Jugendtraum Problem: formulate uniform estimates with interplay between the decay rate and the configuration of critical points. The one-dimensional model: let Φ(x) be a monic polynomial of degree N in R, and let r j C, 1 j N be its roots. Then there exists constant C N, depending only on N, so that {x R; Φ(x) < M} C N max 1 j N min S j ( M k/ S r k r j ) 1 S where S ranges over all subsets of {1,2,,N} which contain j, and S denotes the number of elements in S.

64 Jugendtraum Problem: formulate uniform estimates with interplay between the decay rate and the configuration of critical points. The one-dimensional model: let Φ(x) be a monic polynomial of degree N in R, and let r j C, 1 j N be its roots. Then there exists constant C N, depending only on N, so that {x R; Φ(x) < M} C N max 1 j N min S j ( M k/ S r k r j ) 1 S where S ranges over all subsets of {1,2,,N} which contain j, and S denotes the number of elements in S. Can this lead to a geometry on the space of phase functions, which can help identify compact sets within the subspace of phase functions with s specific volume growth rate?