Cornell University ICM 04, Satellite Conference in Harmonic Analysis, Chosun University, Gwangju, Korea August 6, 04
Motivation the Coifman-Meyer theorem with classical paraproduct(979) B(f, f )(x) := IR m(ξ, ξ) f (ξ ) f (ξ )e πix(ξ +ξ ) dξ dξ where m(ξ, ξ ) satisfies α (m(ξ)) for sufficiently many ξ α multi-indices α. the bilinear Hilbert transform (by Lacey and Thiele, 997) BHT(f, f )(x) := f (ξ ) f (ξ )e πix(ξ +ξ ) dξ dξ ξ <ξ iterated trilinear Fourier integrals (called as Biest II) (by Muscalu, Tao and Thiele, 00) T(f, f, f 3)(x) := f (ξ ) f (ξ ) f 3(ξ 3)e πix(ξ +ξ +ξ 3) dξ dξ dξ 3 ξ <ξ <ξ 3 = IR χ 3 ξ <ξ χ ξ <ξ 3 f (ξ ) f (ξ ) f 3(ξ 3)e πix(ξ +ξ +ξ 3) dξ dξ dξ 3
Motivation the Coifman-Meyer theorem (classical paraproduct) B(f, f )(x) := IR m(ξ, ξ) f (ξ ) f (ξ )e πix(ξ +ξ ) dξ dξ where m(ξ, ξ ) satisfies α (m(ξ)) for sufficiently many ξ α multi-indices α. Flag paraproduct (by Muscalu, 007) T ab(f, f, f 3)(x) = IR a(ξ, ξ) b(ξ, ξ3) f (ξ ) f (ξ ) f 3(ξ 3)e πix(ξ +ξ +ξ 3) dξ dξ dξ 3 3 where a, b satisfies the classical Marcinkiewicz-Mikhlin-Hörmander condition.
Motivation the bilinear Hilbert transform (by Lacey and Thiele) BHT(f, f )(x) := IR χ ξ <ξ f (ξ ) f (ξ )e πix(ξ +ξ ) dξ dξ the Biest II(the Fourier case) (by Muscalu, Tao and Thiele) T(f, f, f 3)(x) = IR χ 3 ξ <ξ χ ξ <ξ 3 f (ξ ) f (ξ ) f 3(ξ 3)e πix(ξ +ξ +ξ 3) dξ dξ dξ 3
Motivation the bilinear Hilbert transform (by Lacey and Thiele) BHT(f, f )(x) := IR χ ξ <ξ f (ξ ) f (ξ )e πix(ξ +ξ ) dξ dξ Bilinear operator with more generic symbol having -dim. singularity B m(f, f )(x) := m(ξ, ξ ) f (ξ ) f (ξ )e πix(ξ +ξ ) dξ dξ where m(ξ, ξ ) is smooth away from the line Γ = {ξ = ξ } and satisfying α (m(ξ)) for every ξ IR \ Γ, sufficiently many dist(γ,ξ) α multi-indices α. Remark :B m has the same L p estimates as BHT by applying the same model operator (modulo nice decaying Fourier coefficient).
Motivation the Biest II(the Fourier case) T(f, f, f 3)(x) = IR χ 3 ξ <ξ χ ξ <ξ 3 f (ξ ) f (ξ ) f 3(ξ 3)e πix(ξ +ξ +ξ 3) dξ dξ dξ 3 Key idea : In the region { ξ 3 ξ ξ ξ },. χ ξ <ξ χ ξ <ξ 3 = χ ξ <ξ χ ξ +ξ <ξ 3 T m m (f, f, f 3)(x) = IR m(ξ, ξ)m(ξ, ξ3) f (ξ ) f (ξ ) f 3(ξ 3)e πix(ξ +ξ +ξ 3) dξ dξ dξ 3 3 where m i(ξ i, ξ i+) is smooth away from the line Γ i = {ξ i = ξ i+} for i =, and satisfying α (m i(ξ)) for every ξ IR \ Γ dist(γ i,ξ) i, α sufficiently many multi-indices α. Remark : We are no longer able to apply this key idea to T m m.
Theorem Let T m m (f, f, f 3)(x) = IR m(ξ, ξ)m(ξ, ξ3) f (ξ ) f (ξ ) f 3(ξ 3)e πix(ξ +ξ +ξ 3) dξ dξ dξ 3 3 where m i(ξ i, ξ i+) is smooth away from the line Γ i = {ξ i = ξ i+} for i =, and satisfying α (m i(ξ)) for every ξ IR \ Γ dist(γ i,ξ) i, sufficiently many α multi-indices α. Theorem Let < p, p, p 3 and 0 < p 4 < such that p + p + p 3 =. Then p 4 T m m maps T m m : L p L p L p 3 L p 4 as long as (/p, /p, /p 3, /p 4) D. Remark: By using duality, we show that the quadrilinear form Λ associated to T m m via the formula Λ(f, f, f 3, f 4) := IR Tm m (f, f, f 3)(x)f 4(x)dx is bounded on L p L p L p 3 L p 4 for < p 4 <.
the restricted weak-type interpolation theorems Let (/p, /p, /p 3, /p 4) := (α, α, α 3, α 4) := α. Definition A tuple α = (α, α, α 3, α 4) is called admissible, if 4 i= αi = with α i < for all i 4, and (possibly) only one α j < 0. A 4-linear form Λ is of restricted type α if for every sequence E, E, E 3, E 4 of subsets of IR with finite measure, there exists a major subset E j of E j for α j < 0 (if exists) such that Λ(f, f, f 3, f 4) E j α j E i α i for all functions f i supported on E i and such that f i. i j Theorem For every vertex of D there exist admissible tuples α arbitrarily close to the vertex such that the form Λ is of restricted type α. Lemma Let α be an admissible tuple such that α D. Then Λ is of restricted type α.
Reduction to the discretized model By a standard partition of unity, we obtain m (ξ, ξ )m (ξ, ξ 3) = Q m (ξ, ξ )φ Q(ξ, ξ ) Q m (ξ, ξ 3)φ Q (ξ, ξ 3) = + + Q Q Q Q Q Q Then m (ξ, ξ )φ Q(ξ, ξ )m (ξ, ξ 3)φ Q (ξ, ξ 3) Q Q = C n φ Q (ξ )φ Q (ξ )φ Q (ξ )φ Q (ξ 3) n Z 4 Q Q Question: φ Q (ξ ) φ Q (ξ + ξ )? By using a Taylor decomposition of φ Q (ξ ), we obtain that φ Q (ξ ) = φ Q ( ξ + ξ ) + M l= φ (l) Q ( ξ + ξ ) ( ) l ξ ξ + RM(ξ, ξ) l!
For l M, = Q Q # 000 φ Q (ξ )φ Q (ξ ) (#+)l [ Q,Q ; # Q Q φ (l) Q ( ξ + ξ ) ( ξ ξ ) ] l φ l! Q (ξ 3) φ Q (ξ ) φ Q (ξ ) φ Q,l (ξ + ξ )φ Q (ξ 3) by letting Q = k, Q = k for k, k Z, and # = k k. We denote m #,l := Q,Q ; φ Q (ξ ) φ Q (ξ ) φ Q,l (ξ + ξ )φ Q (ξ 3). # Q Q
Then the quadrilinear form Λ #,l associated to T m#,l via the formula Λ #,l (f, f, f 3, f 4) := IR Tm #,l(f, f, f3)(x)f4(x)dx = IR m #,l(ξ, ξ, ξ 3) f (ξ ) f (ξ ) f 3(ξ 3) f 4( ξ ξ ξ 3)dξ dξ dξ 3 3 = φ Q (ξ ) φ Q (ξ )φ Q3 (ξ + ξ ) φ Q,l (ξ + ξ ) = Q Q ; Q,Q ; # Q Q # Q Q ξ +ξ +ξ 3 +ξ 4 =0 φ Q (ξ 3)φ Q 3 (ξ + ξ + ξ 3) f (ξ ) f (ξ ) f 3(ξ 3) f 4(ξ 4)dξ dξ dξ 3dξ 4 = Q IR ξ +ξ +ξ 3 +ξ 4 =0 f ˇφ Q (ξ ) f ˇφ Q (ξ ) f 3 ˇφ Q (ξ 3) f 4 ˇφ Q 3 (ξ 4) dξ dξ dξ 3dξ 4 Q; # Q Q (f 3 ˇφ Q )(x)(f 4 ˇφ Q 3 )(x)dx ( (f ˇφ Q )(x)(f ˇφ Q )(x) ) ˇφ Q3 ˇφ Q3 ˇφ Q l (ξ + ξ ) ˇφ Q,l (x)
Define that a tri-tile P = (P, P, P 3) where each i-tile P i = I Pi ω Pi and ω Pi = Q i and the I Pi = I P are independent of i. Similarly, we define a tri-tile Q = (Q, Q, Q 3) with frequency cube Q = Q Q Q 3. Then for # 000 Λ # P, Q (f, f, f 3, f 4) := P P I P / B# P (f, f ), Φ P f 3, Φ P f 4, Φ P3 where and Let B # P (f, f ) := Q Q: ω Q3 ω P, # ω Q3 ω P Λ P, Q (f, f, f 3, f 4) := # 000 Λ # P, Q (f, f, f3, f4) = Q Q I Q / f, ΦQ f, ΦQ ΦQ 3 (#+)l Λ # P, Q (f, f, f 3, f 4). a() Q I Q / a () Q a (3),# Q 3 where a () Q := f, Φ Q a () Q := f, Φ Q a (3),# Q 3 := P P:ω Q3 ω P # ω Q3 ω P I P / f3, ΦP f4, ΦP 3 ΦP, ΦQ 3.
Definition (Tile norms) Let Q be a finite collection of tri-tiles, j =,, 3, and let (a Qj ) Q Q be a sequence of complex numbers. We define the size of this sequence by size j((a Qj ) Q Q ) := sup( T I Q T a Qj ) / where T ranges over all trees in Q which are i-trees for some i j. We also define the energy of the sequence by Q T energy j ((a Qj ) Q Q ) := sup sup n ( I T ) / n Z T T T where T ranges over all collections of strongly j-disjoint trees in Q such that ( Q T a Qj ) / n I T / for all T T, and ( Q T a Qj ) / n+ I T / for all sub-trees T T T.
Proposition Let Q be a finite collection of tri-tiles, and for each Q Q and j =,, 3 let a (j) Q j be a complex number. Then Q Q a() Q I Q / a () Q a (3) Q 3 3 j= size j((a (j) Q j ) Q Q )θ j energy j ((a (j) Q j ) Q Q ) θ j for any 0 θ, θ, θ 3 < with θ + θ + θ 3 =, with the implicit constant depending on the θ j. Let E j be sets of finite measure and f j be functions supported on E j and such that f j. size j((a (j) Q j ) Q Q ) sup χ M I Q Q Q for j =, E j I Q energy j ((a (j) Q j ) Q Q ) fj Ej / for j =, ( ) size 3((a (3),# Q 3 ) Q Q ) sup χ M θ ( I Q Q Q I Q energy 3 ((a (3),# Q 3 ) Q Q ) #/ ( E 4 / sup P P E 3 E 4 χ M I Q I Q ) θ E 3 χ M I P I P ) θ ( E 3 / sup P P E 4 χ M I P I P ) θ #/ E 4 ( θ)/ E 3 θ/
With these tile norms and by Proposition, we obtain that Λ # P, Q (f, f, f 3, f 4) #/ E α for given sequence E, E, E 3, E 4 with finite measure and f i supported on E i and such that f i. Hence, finally we have Λ P, Q (f, f, f 3, f 4) (#+)l Λ # P, Q (f, f, f 3, f 4) This complete the proof. # 000 # 000 E α. (#+)l #/ E α
Thank you!