Variational estimates for the bilinear iterated Fourier integral

Transcription

1 Variational estimates for the bilinear iterated Fourier integral Yen Do, University of Virginia joint with Camil Muscalu and Christoph Thiele

2 Two classical operators in time-frequency analysis: (i) The Carleson operator Cf (x) = sup e ixξ f (ξ)dξ N ξ<n L p (p > 1) bounds by Carleson Hunt (66/67), implies a.e. convergence of Fourier series for L p. (ii) The bilinear Hilbert transform B(f, g)(x) = p.v. f (x + y)g(x y) dy y R L p esimates (p > 2/3) by Lacey Thiele (99/00), settling a conjecture of Calderón.

3 B is a bilinear Fourier multiplier operator: up to normalization, B(f, g)(x) = e ix(ξ 1+ξ 2 ) f (ξ1 )ĝ(ξ 2 )dξ 1 dξ 2 ξ 1 <ξ 2 Muscalu-Tao-Thiele (06): same estimates hold for the maximal truncation of B (the bi-carleson operator): B (f, g)(x) = sup e ix(ξ 1+ξ 2 ) f (ξ1 )ĝ(ξ 2 )dξ 1 dξ 2 N ξ 1 <ξ 2 <N C, B, and B appear in the multilinear expansion of a nonlinear analogue of the Fourier transform.

4 Consider the matrix valued ODE, parameter λ. d dx φ = (iλj+w )φ, φ(x) = eiλxj [I n +o(1)], x, J diagonal distinct entries, W zero diagonal entries. Open question: given W L 2, prove boundedness of the solution curve {φ(x), x R} for a.e. λ R. Christ-Kiselev (98): proof for L p, p < 2. In L 2 we may write W = F for F L 2. The multilinear expansion of φ essentially consists of ξ 1 < <ξ k e iλ(α 1ξ 1 + +α k ξ k ) f1 (ξ 1 )... f k (ξ k )dξ 1... dξ k.

5 Muscalu-Tao-Thiele: For L 2, higher multi-linear terms unbounded if consecutive α j sums to 0. This difficulty could be turned off by assuming extra structures for W. T. Lyons rough path theory for ODEs G (x) = V (x)g(x) driven by rough signal V : If for some r < 3 the r-variation sum of the linear parts M<x<N V (x)dx and the r/2-variation sum of the bilinear parts M<x 1 <x 2 <N V (x 1)V (x 2 ) are bounded then sup x G(x) <.

6 For w defined on increasing pairs {(M, N), M N} its r-variation sum is the sup over increasing sequences w V r = sup (N j )( j w(n j 1, N j ) r ) 1/r For AKNS systems, the r-variation sum of linear parts is the variation norm Carleson operator (Oberlin-Seeger-Tao-Thiele-Wright, 08). Need r > 2. D.-Muscalu-Thiele consider the r/2-variational estimates for the bilinear parts.

7 Theorem: Let r > 2 and T (f, g)(x) be e ix(ξ 1+ξ 2 ) f (ξ1 )ĝ(ξ 2 )dξ 1 dξ 2 V r/2 M<ξ 1 <ξ 2 <N then T is bounded from L p 1 Lp 2 Lp 3 for 2r max(1, 3r 4 ) < p 1, p 2, max( 2 3, r 2 ) < p 3 <. By T. Lyons theory, this also implies diagonal estimates L p L p L p/k for r/k-variation sum of the k-linear parts for p > max(r, 3/2). This is the special case α 1 = α 2 = 1 of the general setup with e ix(α 1ξ 1 +α 2 ξ 2 ) (future work).

8 Our proof extends the proof of the bi-carleson case (r = ), however the proof for bi-carleson breaks down at r = 4. (And we need r < 3.) The reason is the bi-carleson proof requires boundedness of the r/2-variation Carleson operator. To get around this problem one uses L p spaces defined over outer measure spaces (D.-Thiele, 15). Given an outer measure space (X, E, µ) and a notion of size S capturing the size of a function f on E E, one could define suitably the distribution function µ(s(f ) > λ) and build L p spaces from there.

9 Let (φ P ) be wave packets, say L 1 -normalized. The discrete r/2-variation norm operator satisfies P P I P a(p) φ P L p a Lp (P,µ,S r ) and if r < 4 then the wavelet projection f a(p) = f, φ P does not map into these outer L p (P, µ, S r ). One needs bilinear outer measure estimates for those (a(p)) arising in our consideration. for example a(p) is a local wavelet projection of (discretized) bilinear Hilbert operators. proofs of these estimates involve singular integral operators on outer measure spaces.