SINGULAR OSCILLATORY INTEGRAL...

SINGULAR OSCILLATORY INTEGRAL

OPERATORS

D.H. PhongColumbia University

Conference in honor of Elias M. SteinPrinceton University, May 16-21, 2011

May 16, 2011

Outline of the Talk

1. Reminiscences about the 70’s

Outline of the Talk

2. Generalized Radon transforms

Outline of the Talk

The framework of Fourier integral operators

Outline of the Talk

Dirac measures on subvarieties

Outline of the Talk

Some analytic and geometric questions

Outline of the Talk

3. Some partial answers

Outline of the Talk

Estimates for degenerate oscillatory integrals

Outline of the Talk

Estimates for sublevel sets

Outline of the Talk

Degenerate oscillatory integral operators

Outline of the Talk

Degenerate oscillatory integral operators

4. Jugendtraum

Reminiscences about the 70’s

Pseudo-differential operators

Calculus of Kohn-Nirenberg, inspired by Kohn’s L2 solution of the ∂ and ∂bproblems; subelliptic estimates, weakly pseudoconvex domains

Exotic classes Smρ,δ of Hormander and S

M,mΦ,φ of Beals-Fefferman

The almost-orthogonality lemma of Cotlar-Stein, L2 boundedness theorem ofCalderon-Vaillancourt

Fourier integral operators

Early ideas of Maslov and Egorov

Theory of Hormander and Duistermaat-Hormander for real phases

Complex phases of Melin-Sjostrand

Fourier integral operators

Early ideas of Maslov and Egorov

Theory of Hormander and Duistermaat-Hormander for real phases

Complex phases of Melin-Sjostrand

New singular integral operators

Folland-Stein’s fundamental solution for ∂b Greiner-Stein’s Lp estimates for the ∂ Neumann problem

Rothschild-Stein’s fundamental solution for∑N

j=1 X2j+ iX0

Fefferman’s expansion for the Bergman kernel, subsequently simplified byKerzman-Stein, and refined by Boutet de Monvel-Sjostrand.

Green’s function for the ∂-Neumann problem

The model case is the Siegel upper half-space U = (z , zn+1 ∈ Cn+1; Im zn+1 > |z |2,

which can be identified with Hn × R+ via (z , zn+1) ↔ (ζ, ρ), ζ = (z , t), t = Re zn+1,ρ = Im zn+1 − |z |2. Here Hn is the Heisenberg group

Hn = Cn × R; (z , t) · (z ′, t′) = (z + z ′, t + t′ + 2Im zz ′).

The ∂-Neumann problem is the following boundary value problem

2u = f on Hn × R+, (∂ρ + i∂t)u = 0 when ρ = 0.

The model case is the Siegel upper half-space U = (z , zn+1 ∈ Cn+1; Im zn+1 > |z |2,which can be identified with Hn × R+ via (z , zn+1) ↔ (ζ, ρ), ζ = (z , t), t = Re zn+1,ρ = Im zn+1 − |z |2. Here Hn is the Heisenberg group

An explicit formula for the Green’s function

u(ζ, ρ) =

Hn×R+

N(ζ−1 · η, |ρ− µ|)f (η, µ) −

Hn×R+

K(ζ−1 · η, ρ+ µ)f (η, µ)

N(ζ, ρ) ∼1

(2|z |2 + t2 + ρ2)n, K(ζ, ρ) ∼

(2|z |2 + t2 + ρ2)k (2|z |2 + ρ− it)ℓ

The model case is the Siegel upper half-space U = (z , zn+1 ∈ Cn+1; Im zn+1 > |z |2,which can be identified with Hn × R+ via (z , zn+1) ↔ (ζ, ρ), ζ = (z , t), t = Re zn+1,ρ = Im zn+1 − |z |2. Here Hn is the Heisenberg group

An explicit formula for the Green’s function

u(ζ, ρ) =

Hn×R+

N(ζ−1 · η, |ρ− µ|)f (η, µ) −

Hn×R+

K(ζ−1 · η, ρ+ µ)f (η, µ)

N(ζ, ρ) ∼1

(2|z |2 + t2 + ρ2)n, K(ζ, ρ) ∼

(2|z |2 + t2 + ρ2)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.

A distribution of hypersurfaces

Ω0 = (z , 0); z ∈ C ⊂ Hn

Ωζ = ζ · Ω0

Ω0 = (z , 0); z ∈ C ⊂ Hn

Ωζ = ζ · Ω0

Propagation of hidden singularities along Ωζ

D2u+(ζ, ρ) =

∫ ∞

−∞

Kζ,η(z − w , ρ+ µ)Ts f (η, µ) dσΩζ(η)

dsdµ,

where ζ = (z , t), η = (w , s), and Kζ,η(w , ρ) is a Calderon-Zygmund kernel on Ωζ ,with norm O((s2 + µ2)−1), Tsv is a translation of v by s.

Singular Radon transforms

Rv(ζ) =

K(ζ, η)v(η)dσΩζ(η)

Ω0 = (z , 0); z ∈ C ⊂ Hn

Ωζ = ζ · Ω0

Propagation of hidden singularities along Ωζ

D2u+(ζ, ρ) =

∫ ∞

−∞

Kζ,η(z − w , ρ+ µ)Ts f (η, µ) dσΩζ(η)

dsdµ,

where ζ = (z , t), η = (w , s), and Kζ,η(w , ρ) is a Calderon-Zygmund kernel on Ωζ ,with norm O((s2 + µ2)−1), Tsv is a translation of v by s.

Singular Radon transforms

Rv(ζ) =

K(ζ, η)v(η)dσΩζ(η)

Group Fourier transform proof by Geller-Stein

Analogue of the Hilbert transforms along curves introduced byNagel-Riviere-Wainger

WF (R) = N∗(C) ∪∆: works of Guillemin, and especially Greenleaf-Uhlmann onGelfand’s problem, namely to identify family of curves that suffice to invert theX -ray transform along curves.

Most general version of Lp boundedness by Christ-Nagel-Stein-Wainger

Generalized Radon transforms

Let X ,Y be smooth manifolds, and C ⊂ X × Y a smooth submanifold. Then a Diracmeasure δC(x , y) supported on C defines a generalized Radon transform,

Rf (x) =

δC (x , y)f (y)

with Cx = y ∈ Y ; (x , y) ∈ C.

Rf (x) =

δC (x , y)f (y)

with Cx = y ∈ Y ; (x , y) ∈ C.

If C = ϕ1(x , y) = · · · = ϕℓ(x , y) = 0 locally, then

δC(x , y) =

e i∑ℓ

k=1 θkϕk (x,y)a(x , y , θ)dθ,

so R is a Fourier integral operator with LagrangianΛ = N∗(C) ⊂ T∗(X ) × T∗(Y ).

Rf (x) =

δC (x , y)f (y)

with Cx = y ∈ Y ; (x , y) ∈ C.

If C = ϕ1(x , y) = · · · = ϕℓ(x , y) = 0 locally, then

δC(x , y) =

e i∑ℓ

k=1 θkϕk (x,y)a(x , y , θ)dθ,

so R is a Fourier integral operator with LagrangianΛ = N∗(C) ⊂ T∗(X ) × T∗(Y ).

General theory of Hormander: if Λ is a local graph over T∗(X ) (equivalently,over T∗(Y )), then R is smoothing of order (n − ℓ)/2 = dimCx/2.

The local graph condition can be written down explicitly as, ∀θ ∈ Rℓ \ 0,

0 dyϕj

dxϕk d2xy

∑ℓm=1 θmϕm(x , y)

In general, R is smoothing of order 12(dimCx − dimKer dπX ), with πX the

projection πX : T∗(X × Y ) → T∗(X ).

Dirac measure of subvarieties

Consider the case X = Y = Rn, and Cx 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 ofthe Fourier transform of the Dirac measure on V ,

|δV (ξ)| ≤ C |ξ|−δ

When V is a hypersurface, the graph condition holds when the Gaussiancurvature of V is not 0. The Radon transform R is then smoothing of orderδ = (n − 1)/2.

When V is a curve, the graph condition cannot hold if dimX ≥ 3. Hormander’stheorem 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 issmoothing of order δ = 1/n.

|δV (ξ)| ≤ C |ξ|−δ

When V is a hypersurface, the graph condition holds when the Gaussiancurvature of V is not 0. The Radon transform R is then smoothing of orderδ = (n − 1)/2.

When V is a curve, the graph condition cannot hold if dimX ≥ 3. Hormander’stheorem 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 issmoothing of order δ = 1/n.

Higher codimension lead to higher order degeneracies, which are beyond thescope of the standard method of stationary phase, and the correspondingconditions on second order derivatives.

A closer look

Let Rd ∋ t → x(t) ∈ R

n be a local parametrization of V . Then

δV (ξ) =

ei∑n

j=1 ξj xj (t)χ(t)dt

Setting ξ = λω, λ = |ξ|, ω ∈ Sn−1,

δV (ξ) =

e iλΦω(t)χ(t)dt,

where Φω(t) =∑n

j=1 ωjxj(t), we can formulate the following

A closer look

δV (ξ) =

ei∑n

δV (ξ) =

where Φω(t) =∑n

Analytic questions

Given Φω(t), what is the decay rate Cω |λ|−δω of the above oscillatory integralwith phase Φω(t) ?

When are δω and Cω semicontinuous (“stable”) as ω varies ?

A closer look

δV (ξ) =

ei∑n

δV (ξ) =

where Φω(t) =∑n

Analytic questions

Given Φω(t), what is the decay rate Cω |λ|−δω of the above oscillatory integralwith phase Φω(t) ?

When are δω and Cω semicontinuous (“stable”) as ω varies ?

Geometric questions

Given n = dimX , d = dimV , what are the best possible orders δ(n, d) ofsmoothing ? (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 possibleorder of smoothing ?

Multiplicities or Milnor numbersIntuitively, smoothing should require the map V × · · · ×V → x1 + · · ·+ xN ∈ R

n to belocally surjective for N large enough. This implies that no direction ø ∈ R

n isorthogonal to V at N points. This suggests a measure µ of (higher-order) curvatureof V is the maximum number of points admitting a given direction among its normals.Set then, for f ∈ Cω, and a an isolated critical point of f ,

µ = dimA(a)/I[∂1f , · · · , ∂d f ]

with A(a) the space of germs of analytic functions at a, and I[∂1f , · · · , ∂d f ] theideal generated by the partial derivatives of f at a. We say that V has non-vanishingµ-curvature if ∀ω ∈ Rn \ 0, the phase Φω(t) has multiplicity at most µ at any criticalpoint. (Note that µ = 1 for a hypersurface and µ = n for a curve correspondrespectively to non-vanishing Gaussian curvature and non-vanishing torsion).

µ = dimA(a)/I[∂1f , · · · , ∂d f ]

A naive conjecture for the optimal order of smoothing for Radon transformsdefined by d-dimensional submanifolds with non-vanishing curvature is

µ1d + 1

µ = dimA(a)/I[∂1f , · · · , ∂d f ]

µ1d + 1

This reduces to (n − 1)/2 for hypersurfaces with non-vanishing curvature, and1/n for curves with torsion.

For general d, µ provides a non-linear interpolation between these extreme cases.

µ = dimA(a)/I[∂1f , · · · , ∂d f ]

µ1d + 1

This reduces to (n − 1)/2 for hypersurfaces with non-vanishing curvature, and1/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 arbitrarilysmall), using results of Varchenko, Karpushkin, and Kushnirenko.

Estimates for Degenerate Oscillatory Integrals

The van der Corput lemmaLet Φ(t) be a smooth real-valued function on [a,b]. If |Φ(k)(t)| ≥ 1 then

e iλΦ(t)dt| ≤ Ck |λ|− 1

if k ≥ 2, or k = 1 and Φ′(t) is monotone. Here Ck is a constant depending only on k.

Varchenko’s theorem

Let Φ(t) be a real-valued function on Rd with 0 as a critical point. The Newtondiagram of Φ is the convex hull of the upper quadrants in Rd with vertices atthose k = (k1, · · · , kd ) with the monomial tk appearing in the Taylor expansionof Φ. The Newton distance α is defined by the condition that (α−1, · · · , α−1)be the intersection of the line k1 = · · · = kd with a face of the Newton diagram.

For each face γ of the Newton diagram, let Pγ be the polynomial in the Taylorexpansion of Φ with monomials in the face γ. Assume dPγ 6= 0 in R

d \ 0. Then

e iλΦ(t)χ(t)dt| ≤ C |λ|−α(log |λ|)β .

Varchenko’s theorem

Let Φ(t) be a real-valued function on Rd with 0 as a critical point. The Newtondiagram of Φ is the convex hull of the upper quadrants in Rd with vertices atthose k = (k1, · · · , kd ) with the monomial tk appearing in the Taylor expansionof Φ. The Newton distance α is defined by the condition that (α−1, · · · , α−1)be the intersection of the line k1 = · · · = kd with a face of the Newton diagram.

For each face γ of the Newton diagram, let Pγ be the polynomial in the Taylorexpansion of Φ with monomials in the face γ. Assume dPγ 6= 0 in R

d \ 0. Then

e iλΦ(t)χ(t)dt| ≤ C |λ|−α(log |λ|)β .

Stability theorem of KarpushkinLet Φ be a Cω 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 Cω Ψ with ‖Φ−Ψ‖C0(|t|<r,t∈C2) < ǫ,∫

|Ψ|−δ < ∞

Criteria for adapted coordinate systems

The decay rate δ is reparametrization invariant, while the Newton distance α iscoordinate dependent. The equality

δ = α

can only hold generically. It is very useful to have criteria for when it holds.

δ = α

can only hold generically. It is very useful to have criteria for when it holds. In dimension d = 2, roots rj (x) of polynomials Φ(x , y) are given by Puiseux

series. Define the “clustering Ξ” to be the number of elements in the largestcluster of roots, where a cluster of roots is an equivalence class of roots, withrj ∼ rk if |rj − rk | · |rj |

−1 → 0. The criterion for adapted coordinate systems is

Ξ ≤ α−1

and such coordinate systems always exist (P.-Stein-Sturm; also simpler andself-contained proof of the stability theorem of Karpushkin).

δ = α

Ξ ≤ α−1

Extensions to C∞ phases and other constructions have been given byGreenblatt, Ikromov-Muller, Ikromov-Kempe-Muller, and others.

δ = α

Ξ ≤ α−1

Extensions to C∞ phases and other constructions have been given byGreenblatt, Ikromov-Muller, Ikromov-Kempe-Muller, and others.

The recent work of Collins-Greenleaf-Pramanik

In general dimension d, for a given non-constant Cω function Φ, there exists afinite collection C of coordinate transformations, so that if α(F ) denotes theNewton distance in the coordinate system F ∈ F , we have

δ = infF∈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 the2-dimensional criteria formulated above.

Estimates for Sublevel Sets

It is not difficult to see that the decay rate of oscillatory integrals with phase Φ isessentially the same as the growth rate of the volume of its level sets

|t ∈ B; |Φ(t)| ≤ M| ≤ C Mδ

Stable estimates for oscillatory integrals correspond to volume estimates with boundsC uniform in Φ. In fact, certain even stronger bounds are known, which depend onlyon lower bounds for certain derivatives of Φ.

|t ∈ B; |Φ(t)| ≤ M| ≤ C Mδ

Sublevel set estimates of Carbery-Christ-WrightFor any multi-index k, there exists δ > 0 and C , depending only on k and d, so thatthe above estimate holds, for any function Φ satisfying the lower bound

|∂kΦ| ≥ 1 on B.

|t ∈ B; |Φ(t)| ≤ M| ≤ C Mδ

Sublevel set estimates of Carbery-Christ-WrightFor any multi-index k, there exists δ > 0 and C , depending only on k and d, so thatthe above estimate holds, for any function Φ satisfying the lower bound

|∂kΦ| ≥ 1 on B.

Sublevel set operators of P.-Stein-SturmFor any given set β(1), · · · , β(K ) ∈ N

d \ 0, define the multilinear operator

WM(f1, · · · , fd ) =

|∂β(j)Φ|>1, 1≤j≤K

f1(x1) · · · fd(xd ) dx1 · · · dxd

Assume that Φ is a polynomial of degree m. Then there exists a constant C

depending only β(1), · · · , β(K ) ∈ Nd \ 0, so that

|WM(f1, · · · , fd )| ≤ C M1dα logd−2(2 +

‖fi‖L

dd−1

where α is the Newton distance for the diagram with vertices at β(j), 1 ≤ j ≤ K .

The new estimates of Gressman

Define inductively the classes Lκ,ρ of operators byL1,(0,··· ,0) consists of the identity;Lκ,ρ consists of operators of the form

LΦ = det

∂ti1 L1Φ · · · ∂ti1 LnΦ

· · ·∂tin L1Φ · · · ∂tin LnΦ

Here Lp ∈ Lκp ,ρp , κ = κ1 + · · ·+ κn, ρ = ρ1 + · · ·+ ρp + (1, · · · , 1, 0, · · · , 0),the 1 occurring at ip .

If Φ ∈ Cω(B), and for any closed set D ⊂ B, there exists a constant C so that

|t ∈ D; |Φ(t)| ≤ M| ≤ C Mα

|β|+1−α (inft∈D |LΦ|)− 1

|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.

Degenerate Oscillatory Integral Operators

Low order of degeneracies

Lagrangians with two-sided Whitney folds: smoothing with loss of 16derivatives

(Melrose-Taylor) Lagrangians with one-sided Whitney fold: smoothing with loss of 1

4derivatives

(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

4derivatives

4(Comech-Cuccagna,

Arbitrary degeneracies in 1 + 1 dimensions

(P.-Stein) Let Φ(x , y) be a real-analytic phase function in 2 dimensions. Thenthe oscillatory integral operator Tλ defined by

Tf (x) =

e iλΦ(x,y)χ(x , y)f (y)dy

for χ ∈ C∞0 (R2) with sufficiently small support near 0, is bounded on L2(R)

with norm

‖T‖ ≤ C |λ|−12δ

where δ is the reduced Newton distance of Φ at 0

4derivatives

4(Comech-Cuccagna,

Arbitrary degeneracies in 1 + 1 dimensions

(P.-Stein) Let Φ(x , y) be a real-analytic phase function in 2 dimensions. Thenthe oscillatory integral operator Tλ defined by

Tf (x) =

e iλΦ(x,y)χ(x , y)f (y)dy

for χ ∈ C∞0 (R2) with sufficiently small support near 0, is bounded on L2(R)

with norm

‖T‖ ≤ C |λ|−12δ

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 setmultilinear functionals, and the Hardy-Littlewood maximal function.

Damped oscillatory integral operators

(P.-Stein) Let Φ(x , y) and χ(x , y) be as previously. Then the damped oscillatoryintegral operator

Df (x) =

e iλΦ(x,y)|Φ′′xy (x , y)|

12 χ(x , y)f (y)dy

is bounded on L2(R) with norm

‖D‖ ≤ C |λ|−12

Earlier works on damped operators are in Sogge-Stein, Cowling-Disney-Mauceri-Muller, and P.-Stein, where they are used for the study of Lp − Lq smoothing.

Damped oscillatory integral operators

(P.-Stein) Let Φ(x , y) and χ(x , y) be as previously. Then the damped oscillatoryintegral operator

Df (x) =

e iλΦ(x,y)|Φ′′xy (x , y)|

12 χ(x , y)f (y)dy

is bounded on L2(R) with norm

‖D‖ ≤ C |λ|−12

Earlier works on damped operators are in Sogge-Stein, Cowling-Disney-Mauceri-Muller, and P.-Stein, where they are used for the study of Lp − Lq 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)|−µf (y) dy

Then E is a bounded operator on L2(R) for

where δ0 is the Newton distance for Φ at 0. It is still bounded on L2(R) whenµ = 1

2δ0, 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.

The general strategy: “operator van der Corput”

Decompose the set Φ′′xy 6= 0 into Φ′′

xy 6= 0 = ∪k|Φ′′xy | ∼ 2−k with

corresponding partition χk(x , y) and decomposition T =∑

k Tk .

xy 6= 0 = ∪k|Φ′′xy | ∼ 2−k with

k Tk . Establish an “oscillatory estimate” and a “size estimate”

‖Tk‖ ≤ C (2−k |λ|)−12

‖Tk‖ ≤ (IkJk)12

where Ik and Jk are the widths along the x and y axes of the set |Φ′′xy | ∼ 2−k.

xy 6= 0 = ∪k|Φ′′xy | ∼ 2−k with

‖Tk‖ ≤ C (2−k |λ|)−12

Resum in k, exploiting the better of the oscillatory or size estimate.

xy 6= 0 = ∪k|Φ′′xy | ∼ 2−k with

‖Tk‖ ≤ C (2−k |λ|)−12

The key difficulty

The sets |Φ′′xy | ∼ 2−k are usually very complicated geometrically, and the

partition χk(x , y) necessarily complicated also. It is essential that the oscillatoryestimate be uniform in χk . and this requires very precise versions of theoscillatory integral estimates.

xy 6= 0 = ∪k|Φ′′xy | ∼ 2−k with

‖Tk‖ ≤ C (2−k |λ|)−12

The key difficulty

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|∂n

yχ(x , y)| ≤ δ−n, and that Φ′′xy is a polynomial satisfying µ ≤ |Φ′′

xy | ≤ Aµ onB. Then the corresponding operator T satisfies

‖T‖ ≤ C(λµ)−12

with C depending only on A and the degree of Φ(x , y).

xy 6= 0 = ∪k|Φ′′xy | ∼ 2−k with

‖Tk‖ ≤ C (2−k |λ|)−12

The key difficulty

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|∂n

yχ(x , y)| ≤ δ−n, and that Φ′′xy is a polynomial satisfying µ ≤ |Φ′′

xy | ≤ Aµ onB. Then the corresponding operator T satisfies

‖T‖ ≤ C(λµ)−12

with C depending only on A and the degree of Φ(x , y). Curved Trapezoid Lemma: requires Hardy-Littlewood maximal function

(P.-Stein-Sturm)

Further Developments

Works of Kempe-Ikromov-MullerLp boundedness of maximal Radon transforms for smooth hypersurfaces in R

3,p > h(S), where h(S) is the supremum over Newton distances. Applications toconjectures 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 Pj (x)zm−j be a homogeneous polynomial of

degree m in R2 × R. Assume that the first and the last non-vanishing

polynomials Pjminand Pjmax are non-degenerate (dP(x) 6= 0 for x 6= 0), and that

jmin ≤ 2m3

≤ jmax . Then for m ≥ 4,

‖T‖ ≤ C |λ|−32m .

jmin ≤ 2m3

‖T‖ ≤ C |λ|−32m .

Greenleaf-Pramanik-Tang’s result: Let Φ(x , z) be a homogeneous polynomial ofdegree m in R

nX × RnY . Assume that S ′′(x , z) has at least one non-zero entry

at every point of RnX+nY \ 0. Then

‖T‖ ≤ C |λ|−nX+nY

2m if m > nX + nY ,

and ‖T‖ ≤ C |λ|−1/2 log |λ| if m = nX + nY , and ‖T‖ ≤ C |λ|−1/2 if2 ≤ m < nX + nY .

jmin ≤ 2m3

‖T‖ ≤ C |λ|−32m .

Greenleaf-Pramanik-Tang’s result: Let Φ(x , z) be a homogeneous polynomial ofdegree m in R

nX × RnY . Assume that S ′′(x , z) has at least one non-zero entry

at every point of RnX+nY \ 0. Then

‖T‖ ≤ C |λ|−nX+nY

2m if m > nX + nY ,

and ‖T‖ ≤ C |λ|−1/2 log |λ| if m = nX + nY , and ‖T‖ ≤ C |λ|−1/2 if2 ≤ m < nX + nY .

Cubic phases: Greenleaf-Pramanik-Tang (nX = nY = 2); also Gressman.

Jugendtraum

Problem: formulate uniform estimates with interplay between the decay rateand the configuration of critical points.

Jugendtraum

The one-dimensional model: let Φ(x) be a monic polynomial of degree N in R,and let rj ∈ C, 1 ≤ j ≤ N be its roots. Then there exists constant CN ,depending only on N, so that

|x ∈ R; |Φ(x)| < M| ≤ CN max1≤j≤NminS∋j

( M∏

k /∈S |rk − rj |

where S ranges over all subsets of 1, 2, · · · ,N which contain j , and |S|denotes the number of elements in S.

Jugendtraum

The one-dimensional model: let Φ(x) be a monic polynomial of degree N in R,and let rj ∈ C, 1 ≤ j ≤ N be its roots. Then there exists constant CN ,depending only on N, so that

|x ∈ R; |Φ(x)| < M| ≤ CN max1≤j≤NminS∋j

( M∏

k /∈S |rk − rj |

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 helpidentify compact sets within the subspace of phase functions with s specificvolume growth rate ?

SINGULAR OSCILLATORY INTEGRAL...

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