Recent Results in Sparse Domination Michael Lacey Georgia Tech May - PowerPoint PPT Presentation

Recent Results in Sparse Domination Michael Lacey Georgia Tech May 31, 2018 Section 0.0 Slide 1

A Sparse Operator root A collection of cubes S is sparse if for each S ∈ S , there is a an 1 E S ⊂ S , so that | E S | > 100 | S | and { E S : S ∈ S} are disjoint. � Λ r , s ( f , g ) = | S |� f � S , r � g � S , s . S ∈S � � 1 � 1 / r � f � S , r = | f | dx . | S | S Section 0.0 Slide 2

Definition For sublinear operator T and 1 ≤ r , s < ∞ , � T : ( r , s ) � is the smallest constant C so that for all bounded compactly supported functions f , g |� Tf , g �| ≤ C sup Λ r , s ( f , g ) � �� � � �� � Λ messy, complicated positive, localized Definition only requires a bilinear form, not a linear operator. 1 The supremum over sparse forms is essentially obtained. 2 A (1 , r ) bound implies weak-type, for any r ≥ 1. 3 A ( r , s ) bound implies weighted inequalities: r < p < s ′ , 4 � T : L p ( w ) �→ L p ( w ) � � C ( � w � A ( p / r ) , � w � RH (( s ′ / p ) ′ ) ) Section 0.0 Slide 3

1 / s best better 1 / r Meh Section 0.0 Slide 4

The Sparse T 1 Theorem Theorem (L.-Mena) Let T be a Calder´ on-Zygmund operator with kernel satisfying |∇ α K ( x , y ) | ≤ | x − y | − 1 − α , α = 0 , 1 . � Q | T 1 Q | + | T ∗ 1 Q | dx � | Q | . Then Assume for all cubes Q we have � T : (1 , 1) � < ∞ Many people contributed to this: Lerner, Conde-Rey, Hyt¨ onen, Volberg, Petermichl, Frey, Bernicot, di Plinio, Ou, Culiuc,..... This implies virtually all the standard mapping properties of T , with sharp constants ( A 2 Theorem) Missing in this formulation: H 1 / BMO type estimates. Section 0.0 Slide 5

Why is this (1,1) sparse bound true? If f is supported on cube Q , then Tf is typically no more than � f � Q . � T : L 1 loc �→ L ∞ � “ < ∞ ” Section 0.0 Slide 6

Bilinear Hilbert Transform � � f 1 ( x − y ) f 2 ( x − 2 y ) f 3 ( x ) dy BHT ( f 1 , f 2 , f 3 ) = y dx Theorem (Culiuc, di Plinio, Ou) For admissible ( p 1 , p 2 , p 3 ) � BHT : ( p 1 , p 2 , p 3 ) � < ∞ . For instance (2 , 2 , 1) is at the boundary of admissible. Section 0.0 Slide 7

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Sparse bounds have been proved for a wide variety of operators. Virtually the entire A p literature has been completely rewritten in the last three years. Along the way, bounds have been extended, simplified, and quantified. Section 0.0 Slide 9

Littman/Strichartz Inequality � A t f ( x ) = | y | = t f ( x − y ) d σ ( t ) Theorem (Littman (1971), Strichartz (1971)) For (1 / p , 1 / q ) are in the L p improving triangle below, 1 / q n n ( n +1 , n +1 ) 1 � A 1 f , g � � � f � p � g � q 1 1 / p Section 0.0 Slide 10

Small Improvement, Inside the Triangle � ( A 1 − A 1 ◦ τ y ) f , g � � | y | δ p , q � f � p � g � q Combine this with the Calder´ on-Zygmund-Christ method to deduce sparse bounds. Section 0.0 Slide 11

Lacunary Spherical Maximal Function � S n − 1 f ( x − 2 j y ) σ ( dy ) M lac f ( x ) = sup j ∈ Z Theorem (L.) For (1 / p , 1 / q ) are in the L p improving triangle below, 1 / q n n ( n +1 , n +1 ) 1 � M lac : ( p , q ) � < ∞ 1 1 / p Section 0.0 Slide 12

Stein Maximal Function ˜ Mf = sup A t f 1 ≤ t ≤ 2 Theorem (Schlag and Sogge) For (1 / p , 1 / q ) are in the L p improving triangle below, 1 q P 4 (0 , 1) � ˜ M : ( p , q ) � < ∞ ( d − 1 , d − 1 ) d d ( d − 1 , 1 d ) d 1 p Section 0.0 Slide 13

Stein Maximal Function M full f = sup A t f t > 0 Theorem For (1 / p , 1 / q ) are in the L p improving triangle below, 1 q P 4 (0 , 1) � M full : ( p , q ) � < ∞ ( d − 1 , d − 1 ) d d ( d − 1 , 1 d ) d 1 p Section 0.0 Slide 14

Discrete Spherical Averages � 1 A λ f ( x ) = f ( x − n ) | Z d ∩ S λ | n : | n | 2 = λ 2 | Z d ∩ S λ | ≃ λ d − 2 , d ≥ 5 . Section 0.0 Slide 15

Started with Bourgain, and averages along square integers: 1 � N 1 n =1 f ( x − n 2 ) N Discrete implies Continuous, but the two cases are dramatically 2 different. Entails Hardy-Littlewood method, and sometimes some serious 3 number theory. Many new difficulties, and fine distinctions with the continuous case. 4 Deep recent developments, including work of Bourgain, Mirek, 5 Krause and Stein. Section 0.0 Slide 16

Magyar Stein Wainger Theorem (2002) Theorem For dimensions d ≥ 5 , d � sup A λ f � p � � f � p , d − 2 < p < ∞ λ d Compare to d − 1 in the continuous case. 1 The case of 2, 3, 4 dimensions are excluded here, due to 2 irregularities on the number of lattice points in these dimensions. Section 0.0 Slide 17

Theorem (Kesler, 2018) For (1 / p , 1 / q ) are in the triangle below, 1 q (0 , 1) ( d − 2 , d − 2 ) d d � sup A λ f : ( p , q ) � < ∞ λ ( d − 2 , 2 d ) d 1 p Section 0.0 Slide 18

The sparse bound implies the ℓ p -improving inequality, which is a 1 result w/o precedent in the subject. ℓ p -improving does NOT imply the sparse bound. The ’Holder 2 continuity’ gain fails in the discrete setting, and there is no replacement for it. Proof heavily expolits the representation of the multiplier from 3 Magyar, Stein, Wainger. The sparse bound implies a very rich set of weigthed and vector 4 valued consequences, which are entirely new in this subject. Section 0.0 Slide 19

ℓ p -improving in the fixed radius case Theorem (Kesler-L (2018)) � A λ f � ℓ p ′ � λ d (1 − 2 / p ) � f � p , d d − 2 < p < 2 1 � A λ f � ℓ p ′ � C ω ( λ 2 ) λ d (1 − 2 / p ) � f � p , d +1 d d − 1 < p ≤ d − 2 where 2 ω ( λ 2 ) = number of distinct prime factors of λ 2 . If for all ǫ > 0 , all λ , � A λ f � ℓ p ′ � λ ǫ + d (1 − 2 / p ) � f � p , then p ≥ d +1 d − 1 . 3 Section 0.0 Slide 20

The sufficient proof uses 1 Magyar’s very fine analysis of the ‘minor arcs.’ 1 Andre Weil’s estimates for Kloosterman sums. 2 A result of Bourgain on average values of Ramanjuan sums. 3 The necessary direction uses a subtle ‘self-improving’ aspect of the 2 sufficient direction. We do not know what the counterexample looks like! 3 We just know that it exists. These results hold in dimension d = 4, if λ 2 is odd. 4 Section 0.0 Slide 21

The theory of the discrete lacunary spherical maximal operator is 1 rather different than the continuous case. Due to an example of Zienkiewicz, there are lacunary radii λ k for 2 d which sup k A λ k f is unbounded for 1 < p < d − 1 . On the other hand, we should expect results for A lac f = sup k A p k / 2 f , 3 for prime p , since ω ( p k ) = 1, for all primes p and integers k . More evidence that the ℓ p -improving and sparse bounds decouple in 4 the discrete setting. Section 0.0 Slide 22

Sparse bounds of the discrete lacunary spherical maximal function Theorem (Kesler-L, 2018) For (1 / p , 1 / q ) are in the triangle below, 1 q (0 , 1) ( d − 1 d +1 , d − 1 d +1 ) � sup A lac f : ( p , q ) � < ∞ λ ( d − 2 1 d − 1 , d − 1 ) 1 p Section 0.0 Slide 23

Number Theory A λ f = C λ f + R λ f , q � � e ( − λ 2 a / q ) C a / q C λ f = f , λ 1 ≤ λ ≤ q a =1 � c a / q G ( a / q , ℓ )Φ q ( ξ − ℓ/ q ) � ( ξ ) := d σ λ ( ξ − ℓ/ q ) λ ℓ ∈ Z d G ( a / q , ℓ ) = q − d � e ( | n | 2 a / q + n · ℓ/ q ) . n ∈ Z d q � q e ( − λ 2 a / q ) G ( a / q , ℓ ) K ( λ, ℓ, q ) = a =1 Theorem (Magyar) � R λ � 2 → 2 � λ − d − 3 2 Section 0.0 Slide 24

Theorem (Weil) 2 � | K ( λ, ℓ, q ) | � q − d − 1 ( λ 2 , q odd ) q even � q e 2 π ina / q c q ( n ) = a =1 ( a , q )=1 Theorem (Bourgain) For n � = 0 � Q | c q ( n ) | � Q 1+ ǫ q =1 Section 0.0 Slide 25

Alan McIntosh Section 0.0 Slide 26