Endpoint behavior of modulation invariant singular integrals - PowerPoint PPT Presentation

Endpoint behavior of modulation invariant singular integrals Francesco Di Plinio INdAM-Cofund Marie Curie Fellow at Universit` a degli Studi Roma “Tor Vergata” Institute of Scientific Computing and Applied Mathematics at Indiana University, Fellow XXXIII Convegno Nazionale di Analisi Armonica Alba, 17-20 Giugno 2013

Partly joint work with Ciprian Demeter and Christoph Thiele. C. Demeter, F. Di Plinio, Endpoint bounds for the quartile operator , arXiv preprint, to appear on Journal of Fourier Analysis and Applications F. Di Plinio, Lacunary Fourier and Walsh-Fourier series near L 1 , arXiv preprint

The Bilinear Hilbert Transform ˆ f ( x − t ) g ( x + t )d t ˆ g ( η )e ix ( ξ + η ) d ξ d η. ˆ BHT( f, g )( x ) = p . v . t ∼ f ( ξ )ˆ R ξ>η Multiplier singular on a line { ξ = η } ❀ modulation invariant paraproduct : BHT( M ζ f, M ζ g ) = M 2 ζ BHT( f, g ) Same scaling as pointwise product ❀ H¨ older-type bounds: 1 + 1 = 1 BHT : L p 1 ,q 1 ( R ) × L p 2 ,q 2 ( R ) → L p 3 ,q 3 ( R ) = ⇒ p 1 p 2 p 3 Lacey and Thiele showed BHT : L p 1 ( R ) × L p 2 ( R ) → L p 3 ( R ) older tuple with p 1 , p 2 > 1 and p 3 > 2 whenever ( p 1 , p 2 , p 3 ) H¨ 3 . F. Di Plinio (Rome Tor Vergata) Endpoint bounds 3 / 15

The Bilinear Hilbert Transform ˆ f ( x − t ) g ( x + t )d t ˆ g ( η )e ix ( ξ + η ) d ξ d η. ˆ BHT( f, g )( x ) = p . v . t ∼ f ( ξ )ˆ R ξ>η Multiplier singular on a line { ξ = η } ❀ modulation invariant paraproduct : BHT( M ζ f, M ζ g ) = M 2 ζ BHT( f, g ) Same scaling as pointwise product ❀ H¨ older-type bounds: 1 + 1 = 1 BHT : L p 1 ,q 1 ( R ) × L p 2 ,q 2 ( R ) → L p 3 ,q 3 ( R ) = ⇒ p 1 p 2 p 3 Lacey and Thiele showed BHT : L p 1 ( R ) × L p 2 ( R ) → L p 3 ( R ) older tuple with p 1 , p 2 > 1 and p 3 > 2 whenever ( p 1 , p 2 , p 3 ) H¨ 3 . F. Di Plinio (Rome Tor Vergata) Endpoint bounds 3 / 15

The Bilinear Hilbert Transform ˆ f ( x − t ) g ( x + t )d t ˆ g ( η )e ix ( ξ + η ) d ξ d η. ˆ BHT( f, g )( x ) = p . v . t ∼ f ( ξ )ˆ R ξ>η Multiplier singular on a line { ξ = η } ❀ modulation invariant paraproduct : BHT( M ζ f, M ζ g ) = M 2 ζ BHT( f, g ) Same scaling as pointwise product ❀ H¨ older-type bounds: 1 + 1 = 1 BHT : L p 1 ,q 1 ( R ) × L p 2 ,q 2 ( R ) → L p 3 ,q 3 ( R ) = ⇒ p 1 p 2 p 3 Lacey and Thiele showed BHT : L p 1 ( R ) × L p 2 ( R ) → L p 3 ( R ) older tuple with p 1 , p 2 > 1 and p 3 > 2 whenever ( p 1 , p 2 , p 3 ) H¨ 3 . F. Di Plinio (Rome Tor Vergata) Endpoint bounds 3 / 15

The Bilinear Hilbert Transform ˆ f ( x − t ) g ( x + t )d t ˆ g ( η )e ix ( ξ + η ) d ξ d η. ˆ BHT( f, g )( x ) = p . v . t ∼ f ( ξ )ˆ R ξ>η Multiplier singular on a line { ξ = η } ❀ modulation invariant paraproduct : BHT( M ζ f, M ζ g ) = M 2 ζ BHT( f, g ) Same scaling as pointwise product ❀ H¨ older-type bounds: 1 + 1 = 1 BHT : L p 1 ,q 1 ( R ) × L p 2 ,q 2 ( R ) → L p 3 ,q 3 ( R ) = ⇒ p 1 p 2 p 3 Lacey and Thiele showed BHT : L p 1 ( R ) × L p 2 ( R ) → L p 3 ( R ) older tuple with p 1 , p 2 > 1 and p 3 > 2 whenever ( p 1 , p 2 , p 3 ) H¨ 3 . F. Di Plinio (Rome Tor Vergata) Endpoint bounds 3 / 15

Fourier and Walsh models for BHT • Discretize into squares Q = ω × | ω | + ω ❀ dist( Q, { ξ = η } ) ∼ | ω | � • supp ˆ BHT( f, g ) = supp � f ⊂ ω, supp ˆ g ⊂ | ω | + ω ❀ supp fg = ⊂ 2 | ω | + ω Model sum Setting s = I s × ω s , s 1 = I s × ( ω s + | ω s | ) , s 2 = I s × ( ω s + 2 | ω s | ) , � 1 BHT( f, g )( x ) ∼ BHT ( f, g )( x ) = � f, ϕ s �� g, ϕ s 1 � ϕ s 2 ( x ) � | I s | s ∈ S u Rmk. The models BHT fail to map into L p 3 for p 3 < 2 3 . Not known for BHT . 3 candidate range for BHT on L 1 × L 2 . 2 What about p 3 = 2 3 ? L F. Di Plinio (Rome Tor Vergata) Endpoint bounds 4 / 15

Fourier and Walsh models for BHT • Discretize into squares Q = ω × | ω | + ω ❀ dist( Q, { ξ = η } ) ∼ | ω | � • supp ˆ BHT( f, g ) = supp � f ⊂ ω, supp ˆ g ⊂ | ω | + ω ❀ supp fg = ⊂ 2 | ω | + ω Model sum Setting s = I s × ω s , s 1 = I s × ( ω s + | ω s | ) , s 2 = I s × ( ω s + 2 | ω s | ) , � 1 BHT( f, g )( x ) ∼ BHT ( f, g )( x ) = � f, ϕ s �� g, ϕ s 1 � ϕ s 2 ( x ) � | I s | s ∈ S u Rmk. The models BHT fail to map into L p 3 for p 3 < 2 3 . Not known for BHT . 3 candidate range for BHT on L 1 × L 2 . 2 What about p 3 = 2 3 ? L F. Di Plinio (Rome Tor Vergata) Endpoint bounds 4 / 15

Fourier and Walsh models for BHT • Discretize into squares Q = ω × | ω | + ω ❀ dist( Q, { ξ = η } ) ∼ | ω | � • supp ˆ BHT( f, g ) = supp � f ⊂ ω, supp ˆ g ⊂ | ω | + ω ❀ supp fg = ⊂ 2 | ω | + ω Model sum Setting s = I s × ω s , s 1 = I s × ( ω s + | ω s | ) , s 2 = I s × ( ω s + 2 | ω s | ) , � 1 BHT( f, g )( x ) ∼ BHT ( f, g )( x ) = � f, ϕ s �� g, ϕ s 1 � ϕ s 2 ( x ) � | I s | s ∈ S u Rmk. The models BHT fail to map into L p 3 for p 3 < 2 3 . Not known for BHT . 3 candidate range for BHT on L 1 × L 2 . 2 What about p 3 = 2 3 ? L F. Di Plinio (Rome Tor Vergata) Endpoint bounds 4 / 15

Fourier and Walsh models for BHT • Discretize into squares Q = ω × | ω | + ω ❀ dist( Q, { ξ = η } ) ∼ | ω | � • supp ˆ BHT( f, g ) = supp � f ⊂ ω, supp ˆ g ⊂ | ω | + ω ❀ supp fg = ⊂ 2 | ω | + ω Model sum Setting s = I s × ω s , s 1 = I s × ( ω s + | ω s | ) , s 2 = I s × ( ω s + 2 | ω s | ) , � 1 BHT( f, g )( x ) ∼ BHT ( f, g )( x ) = � f, ϕ s �� g, ϕ s 1 � ϕ s 2 ( x ) � | I s | s ∈ S u Rmk. The models BHT fail to map into L p 3 for p 3 < 2 3 . Not known for BHT . 3 candidate range for BHT on L 1 × L 2 . 2 What about p 3 = 2 3 ? L F. Di Plinio (Rome Tor Vergata) Endpoint bounds 4 / 15

RWT bounds for BHT and near-endpoint results Conj. BHT : L p 1 × L p 2 → L 2 3 , ∞ p 1 + 1 1 p 2 = 3 ∀ 1 < p 1 , p 2 < 2 , 2 . Appropriate substitutes near L 1 × L 2 . LT result follows via (generalized) RWT interpolation of 1 p 2 | G 3 | 1 − 1 1 p 1 | F 2 | (RWE) |� BHT ( f 1 , f 2 ) , f 3 �| � | F 1 | p 3 , | f j | ≤ 1 F j where F 3 ⊂ G 3 major set , in the open range 1 < p 1 , p 2 ≤ ∞ , p 3 > 2 3 Estimate (RWE) for p 3 = 2 3 (weaker than Conj.) is OPEN Theorem (Bilyk-Grafakos06) Log-bumped version near (1 , 2 , 2 3 ) : for | F 1 | ≤| F 2 | � � 2 . | F 3 | 2 2 | F 3 | − 1 1 2 log (BG) |� BHT ( f 1 , f 2 ) , f 3 �| � | F 1 || F 2 | e + | F 1 || F 2 | F. Di Plinio (Rome Tor Vergata) Endpoint bounds 5 / 15

RWT bounds for BHT and near-endpoint results Conj. BHT : L p 1 × L p 2 → L 2 3 , ∞ p 1 + 1 1 p 2 = 3 ∀ 1 < p 1 , p 2 < 2 , 2 . Appropriate substitutes near L 1 × L 2 . LT result follows via (generalized) RWT interpolation of 1 p 2 | G 3 | 1 − 1 1 p 1 | F 2 | (RWE) |� BHT ( f 1 , f 2 ) , f 3 �| � | F 1 | p 3 , | f j | ≤ 1 F j where F 3 ⊂ G 3 major set , in the open range 1 < p 1 , p 2 ≤ ∞ , p 3 > 2 3 Estimate (RWE) for p 3 = 2 3 (weaker than Conj.) is OPEN Theorem (Bilyk-Grafakos06) Log-bumped version near (1 , 2 , 2 3 ) : for | F 1 | ≤| F 2 | � � 2 . | F 3 | 2 2 | F 3 | − 1 1 2 log (BG) |� BHT ( f 1 , f 2 ) , f 3 �| � | F 1 || F 2 | e + | F 1 || F 2 | F. Di Plinio (Rome Tor Vergata) Endpoint bounds 5 / 15

RWT bounds for BHT and near-endpoint results Conj. BHT : L p 1 × L p 2 → L 2 3 , ∞ p 1 + 1 1 p 2 = 3 ∀ 1 < p 1 , p 2 < 2 , 2 . Appropriate substitutes near L 1 × L 2 . LT result follows via (generalized) RWT interpolation of 1 p 2 | G 3 | 1 − 1 1 p 1 | F 2 | (RWE) |� BHT ( f 1 , f 2 ) , f 3 �| � | F 1 | p 3 , | f j | ≤ 1 F j where F 3 ⊂ G 3 major set , in the open range 1 < p 1 , p 2 ≤ ∞ , p 3 > 2 3 Estimate (RWE) for p 3 = 2 3 (weaker than Conj.) is OPEN Theorem (Bilyk-Grafakos06) Log-bumped version near (1 , 2 , 2 3 ) : for | F 1 | ≤| F 2 | � � 2 . | F 3 | 2 2 | F 3 | − 1 1 2 log (BG) |� BHT ( f 1 , f 2 ) , f 3 �| � | F 1 || F 2 | e + | F 1 || F 2 | F. Di Plinio (Rome Tor Vergata) Endpoint bounds 5 / 15

RWT bounds for BHT and near-endpoint results Conj. BHT : L p 1 × L p 2 → L 2 3 , ∞ p 1 + 1 1 p 2 = 3 ∀ 1 < p 1 , p 2 < 2 , 2 . Appropriate substitutes near L 1 × L 2 . LT result follows via (generalized) RWT interpolation of 1 p 2 | G 3 | 1 − 1 1 p 1 | F 2 | (RWE) |� BHT ( f 1 , f 2 ) , f 3 �| � | F 1 | p 3 , | f j | ≤ 1 F j where F 3 ⊂ G 3 major set , in the open range 1 < p 1 , p 2 ≤ ∞ , p 3 > 2 3 Estimate (RWE) for p 3 = 2 3 (weaker than Conj.) is OPEN Theorem (Bilyk-Grafakos06) Log-bumped version near (1 , 2 , 2 3 ) : for | F 1 | ≤| F 2 | � � 2 . | F 3 | 2 2 | F 3 | − 1 1 2 log (BG) |� BHT ( f 1 , f 2 ) , f 3 �| � | F 1 || F 2 | e + | F 1 || F 2 | F. Di Plinio (Rome Tor Vergata) Endpoint bounds 5 / 15

RWT bounds for BHT and near-endpoint results Theorems (Carro-Grafakos-Martell-Soria09) 3 , ∞ : weighted Lorentz quasi-Banach space � 2 2 3 , ∞ Proxy for L L 3 log(e+ t ) f ∗ ( t ) . t 2 � f � � 3 , ∞ = sup 2 t> 0 L Using (RWE) and bilinear extrapolation 3 → � BHT : L 1 , 2 3 × L 2 , 2 4 4 2 3 , ∞ 3 (log L ) 3 (log L ) (CGMS1) L Using (RWE) and ( ε, δ ) -atomic approximability of BHT 3 → � 2 + η × L 2 , 2 1 1 4 2 (CGMS2) BHT : L (log L ) 2 (log 2 L ) 3 , ∞ 2 (log 3 L ) 3 (log L ) L F. Di Plinio (Rome Tor Vergata) Endpoint bounds 6 / 15