On maximal and variational Fourier restriction
- 1. |||| |||| |||| On maximal and variational Fourier restriction Vjekoslav Kovaˇc (U of Zagreb) HRZZ UIP-2017-05-4129 (MUNHANAP) joint work with Diogo Oliveira e Silva (U of Birmingham) Hausdorff Institute, Universit¨at Bonn May 9, 2019
- 2. Restriction of the Fourier transform |||| |||| |||| S = a (hyper)surface in Rd (e.g. a paraboloid, a cone, a sphere) Is it possible to give a meaning to f |S when f ∈ Lp (R)? (Stein, late 1960s) p = 1 YES, because f is continuous p = 2 NO, since f is an arbitrary L2 function What can be said for 1 < p < 2? A question depending on S and p
- 3. A priori estimate |||| |||| |||| σ = a measure on S, e.g. (appropriately weighted) surface measure We want an estimate for the restriction operator Rf := f |S of the form f Lq (S,σ) ≤ C f Lp (Rd ) for some 1 ≤ q ≤ ∞ and all functions f ∈ S(Rd ) How to “compute” f |S ? χ ∈ S(Rd ), Rd χ = 1, χε(x) := ε−d χ(ε−1x) lim ε→0+ f ∗ χε S exists in the norm of Lq (S, σ)
- 4. Adjoint formulation |||| |||| |||| Equivalently, we want an estimate for the extension operator (Eg)(x) = S e2πix·ξ g(ξ) dσ(ξ) Rf , g L2 (S,σ) = f , Eg L2 (Rd ) R: Lp (Rd ) → Lq (S, σ) ⇐⇒ E : Lq (S, σ) → Lp (Rd )
- 5. Restriction conjecture in d = 2 |||| |||| |||| Essentially solved: p < 4 3, q ≤ p 3 For S = S1 Zygmund, 1974 For compact C2 curves S with curvature κ ≥ 0 dσ = arclength measure weighted by κ1/3 Carleson and Sj¨olin, 1972; Sj¨olin, 1974
- 6. Restriction conjecture in d ≥ 3 |||| |||| |||| Largely open, even for the three classical hypersurfaces paraboloid {ξ = (η, −2πk|η|2) : η ∈ Rd−1} with dσ(ξ) = dη q = 2 Strichartz estimates for the Schr¨odinger equation i∂tu + k∆u = 0 in Rd−1 u(·, 0) = u0 Conjecture: p < 2d d+1 , q = (d−1)p d+1 (note p < q)
- 7. Restriction conjecture in d ≥ 3 |||| |||| |||| cone {ξ = (η, ±k|η|) : η ∈ Rd−1} with dσ(ξ) = dη/|η| q = 2 Strichartz estimates for the wave equation ∂2 t u − k2∆u = 0 in Rd−1 u(·, 0) = u0, ∂tu(·, 0) = u1 Conjecture: p < 2(d−1) d , q = (d−2)p d (note p < q) sphere {ξ ∈ Rd : |ξ| = k/2π} with surface measure σ q = 2 The Helmholtz equation ∆u + k2 u = 0 in Rd Conjecture: p < 2d d+1 , q ≤ (d−1)p d+1
- 8. Maximal Fourier restriction |||| |||| |||| Theorem (M¨uller, Ricci, and Wright, 2016) d = 2, S = C2 curve with κ ≥ 0, dσ = κ1/3 dl, χ ∈ S(Rd ), p < 4 3, q ≤ p 3 sup t>0 |f ∗ χt| Lq (S,σ) ≤ C f Lp (Rd ) For f ∈ Lp (Rd ) the restriction f |S makes sense pointwise, i.e. lim ε→0+ f ∗ χε (ξ) exists for σ-a.e. ξ ∈ S (pointwise convergence on S(Rd ) + maximal estimate)
- 9. Higher dimensions |||| |||| |||| Theorem (Vitturi, 2017) d ≥ 3, S = Sd−1, σ = surface measure, χ ∈ S(Rd ), p ≤ 4 3, q ≤ (d−1)p d+1 (strict subset of the Tomas–Stein range (i.e. q = 2) when d ≥ 4) sup t>0 |f ∗ χt| Lq (Sd−1,σ) ≤ C f Lp (Rd ) Idea: inserting the maximal function inside the non-oscillatory restriction estimate
- 10. Variational Fourier restriction |||| |||| |||| Can we do it “quantitatively”? Variational (semi)norms Theorem (K. and Oliveira e Silva, 2018) σ = surface measure on S2 ⊂ R3, χ ∈ S(R3) or χ = 1B(0,1), > 2 sup 0<t0<t1<···<tm m j=1 f ∗χtj −f ∗χtj−1 1/ L2 (S2,σ) ≤ C f L4/3 (R3) If f ∈ L4/3 (R3), then for σ-a.e. ξ ∈ S2: sup 0<t0<t1<···<tm m j=1 (f ∗ χtj )(ξ) − (f ∗ χtj−1 )(ξ) 1/ < ∞ =⇒ f ∗ χε (ξ) ε>0 makes O(δ− ) jumps of size ≥ δ
- 11. Abstract principle |||| |||| |||| Theorem (K., 2018) S ⊆ Rd a measurable set, σ = a measure on S, µ = a complex measure on Rd , µt(E) := µ(t−1E), µ ∈ C∞ , η > 0 | µ(x)| ≤ D(1 + |x|)−1−η Suppose that for some 1 ≤ p ≤ 2, p < q < ∞ the a priori F. r. estimate holds. Then we have the maximal F. r. estimate: sup t∈(0,∞) f ∗ µt Lq (S,σ) ≤ C f Lp (Rd ) and for p < < ∞ we have the variational F. r. estimate: sup 0<t0<t1<···<tm m j=1 f ∗ µtj − f ∗ µtj−1 1/ Lq (S,σ) ≤ C f Lp (Rd )
- 12. Consequences |||| |||| |||| Covers the full Tomas–Stein range for the sphere Sd−1 Covers any known range for the paraboloid or the cone One can take dµ(x) = χ(x) dx, χ ∈ S(Rd ) or χ = 1B(0,1) One can take µ to be the surface measure on Sd−1 in dimensions d ≥ 4 spherical averages of f : 1 µ(Sd−1) Sd−1 f (ξ + εζ) dµ(ζ)
- 13. Subsequent research |||| |||| |||| Theorem (Ramos, 2019) µ = a measure on R2, p < 4 3, q ≤ p 3 Maximal F. r. for S1 holds as soon as Mµg := supt>0 |g ∗ µt| is bounded on Lr (R2) for r > 2. µ = a measure on R3, p ≤ 4 3, q ≤ 2 Maximal F. r. for S2 holds as soon as Mµg := supt>0 |g ∗ µt| is bounded on L2 (R3). ⇐= spherical averages in d = 2, 3 (Bourgain, 1986; Stein, 1976)
- 14. The Christ–Kiselev trick |||| |||| |||| Idea of the proof of the maximal F. r.: Reduce to the supremum over t ∈ {t1 > t2 > · · · > tN} Linearize the supremum: f ∗ µt(ξ) (ξ) Lq ξ(S,σ) ≤ C f Lp (Rd ) Dualize: Λ(f , g) := S f ∗ µt(y) (y)g(y) dσ(y) = S Rd f (x)g(y)e−2πix·y ˇµ(t(y)x) dx dσ(y) Denote Sn := {y ∈ S : t(y) = tn} “Imagine” that ˇµ(tnx) = 1Fn (x) with F1 ⊆ F2 ⊆ · · · ⊆ FN En := Fn Fn−1 ∈ Rd =⇒ Fn = E1 ˙∪E2 ˙∪ · · · ˙∪En This is where we cheat a little
- 15. The Christ–Kiselev trick |||| |||| |||| We obtain: Λ(f , g) = S Rd f (x)g(y)e−2πix·y m,n 1≤m≤n≤N 1Em (x)1Sn (y) dx dσ(y) This is a block-triangular truncation of K(x, y) = e−2πix·y We need: |Λ(f , g)| ≤ C f Lp (Rd ) g Lq (S,σ) with a constant independent of N It can be shown by induction(!) on N using only p < q (Tao, 2006)
- 16. The actual proof |||| |||| |||| Rather begin by proving the variational estimate Represent µ as a superposition of “nice” Schwartz cutoffs Split into long and short variations (Jones, Seeger, and Wright, 2008) For long variations use a smooth version of the lemma by Christ and Kiselev, 2001 + variational upgrade by Oberlin, Seeger, Tao, Thiele, and Wright, 2010 Short variations are trivial(!) (an off-diagonal square function estimate)
- 17. Thank you for your attention!