Variational and jump inequalities Pavel Zorin-Kranich University of - PowerPoint PPT Presentation

Variational and jump inequalities Pavel Zorin-Kranich University of Bonn 2019 May 10 1

Lépingle’s inequality (∑ continuity converges t f t . 1 / r Theorem (Lépingle, 1976) j t ( 0 )<⋯< t ( J ) sup V r ‖ V r Let f = ( f t ) be a martingale. For 1 < p < ∞ and 2 < r we have 2 t f t ‖ p ≤ C p , r ‖ f ‖ p , where V r is the r -variation norm t f t ∶= | f t ( j + 1 ) − f t ( j ) | r ) ▶ refjnes martingale maximal inequality: Mf ≤ f 0 + V r ▶ quantifjes martingale convergence: V r f t fjnite ⟹ f t ▶ V r is a parametrization-invariant version of 1 / r -Hölder

Some variational estimates in harmonic analysis Theorem (Jones+Seeger+Wright 2008) Riesz transforms. 3. V r 2. for any odd CZ kernel V r 1. 𝜈 is uniformly n-rectifjable Let 𝜈 be an n-dimensional AD regular Radon measure on ℝ d . TFAE: Theorem (Mas+Tolsa 2011, 2015) slide. They also prove an r = 2 “jump” endpoint to be explained in the next d Same for truncated Radon transforms along homogeneous curves. 1 < p < ∞, r > 2 . 3 If T t are truncations of a cancellative singular integral, then ‖ V r T t f ‖ p ≤ C p , r ‖ f ‖ p , Same for spherical averages on ℝ d for d − 1 < p < 2 d. t T t is L p bounded for 1 < p < ∞ , r > 2 , t R t is L 2 bounded for some r < ∞ , where R t are truncated

Lépingle’s inequality, endpoint version sup r -variational estimates in open ranges of p . This + real interpolation shows that jump inequalities imply 2 < r . ‖𝜇 N 1 / 2 𝜇> 0 Observation Theorem (Pisier, Xu 1988/Bourgain 1989) t ( 0 )<⋯< t ( J ) 𝜇> 0 ‖𝜇 N 1 / 2 2 ( f t ) ∶= sup J p For 1 < p < ∞ we have the jump inequality 4 𝜇 f t ‖ p ≤ C p ‖ f ‖ p , where N 𝜇 is the 𝜇 -jump counting function N 𝜇 f t ∶= #{ j | | f t ( j + 1 ) − f t ( j ) | > 𝜇}. ‖ V r f t ‖ p ,∞ ≤ C p , r sup 𝜇 f t ‖ p ,∞ ,

Proof of endpoint Lépingle inequality 𝜇 -jump counting function is morally extremized by can have power 1 / q instead of 1 / 2. For martingales with values in a Banach space with martingale cotype q Remark (vector valued) – square function of the stopped martingale f t ( j ) , bounded on L p . 1 / 2 j ≤ 2 (∑ 1 / 2 ) (𝜇/ 2 ) 2 j ≤ 𝜇(∑ 𝜇 𝜇 N 1 / 2 𝜇 𝜇/ 2 t ( 0 ) ∶= 0 , greedy selection of 𝜇/ 2-jumps: 5 t ( j + 1 ) ∶= min { s > t ( j ) | | f s − f t ( j ) | > 𝜇/ 2 }. | f t ( j + 1 ) − f t ( j ) | 2 | f t ( j + 1 ) − f t ( j ) | 2 )

Jumps as a real interpolation space Proof of Lépingle’s inequality gives for a given 𝜇 a decomposition j j Observation (Pisier+Xu 1988) This decomposition shows in fact that [ L ∞ ( V ∞ ), L 1 ( V 1 )] 1 / 2 ,∞ ( f t ) ≲ ‖ f ‖ 2 , where the LHS is a norm in a real interpolation space. More generally, it turns out that J p 2 ( f t ) ∼ [ L ∞ ( V ∞ ), L p 𝜄 ( V 2 𝜄 )] 𝜄,∞ ( f t ) ≲ ‖ f ‖ p for 1 < p < ∞ and 0 < 𝜄 < 1. 6 f t = ∑ 1 t ( j )≤ t < t ( j + 1 ) f t ( j ) + ∑ 1 t ( j )≤ t < t ( j + 1 ) ( f t − f t ( j ) ).

Application: difgusion semigroups Conditional expectation bounded on J p Bourgain 2013 ( q = ∞ ) to get larger range of p 3 / 2 < p < 4 . 2 ( A t f ) ≤ C p ‖ f ‖ p , J p Corollary Corollary (Mirek, Stein, ZK) Rota’s dilation theorem: T t f = 𝔽 ∘ martingale. Proof. 1 < p < ∞. 2 ( T t f ) ≤ C p ‖ f ‖ p , J p selft-adjoint, order positive, T t 1 = 1 ), then 7 If ( T t ) is a difgusion semigroup (i.e., contractive on L 1 and L ∞ , 2 by interpolation. Let G ⊂ ℝ d be a symmetric convex body and A t f ( x ) = | G | − 1 ∫ G f ( x + ty ) dy. Then ▶ maximal estimate by Bourgain ( L 2 ), Carbery ▶ variational estimate by Bourgain+Mirek+Stein+Wrobel ▶ can input results for ℓ q balls by Müller 1990 ( q < ∞ ),

Periodic multipliers ‖ m per ‖ mult 2 L p → J p ℓ p → J p ‖ m per ‖ mult Corollary L p →[ L ∞ ( X 0 ), L p 𝜄 ( X 1 )] 𝜄;∞ ‖ m per ‖ mult Theorem (Mirek+Stein+ZK) L p → L p ( X ) Let ( m t ) be a sequence of multipliers supported on [− 8 For any Banach space X of functions in t and 1 ≤ p ≤ ∞ we have Theorem (Magyar+Stein+Wainger 2002) m t (𝜊 − l / d ). l ∈ℤ d m per positive integer. Defjne periodic multipliers 2 q ] d , q 1 2 q , 1 t (𝜊) ∶= ∑ ℓ p →ℓ p ( X ) ≤ C p , d ‖ m ‖ mult For any Banach spaces X 0 , X 1 of functions in t and 1 ≤ p 𝜄 we have ℓ p →[ℓ ∞ ( X 0 ),ℓ p 𝜄 ( X 1 )] 𝜄;∞ ≤ C p , d ‖ m ‖ mult 2 ≤ C p , d ‖ m ‖ mult

Application: discrete Radon transforms Let A N f ( x ) ∶= 1 < p < ∞. 2 ( A N f ) ≲ ‖ f ‖ ℓ p (ℤ) , J p Theorem (Mirek+Stein+ZK) denominators r > 2 . 1 < p < ∞, ‖ V r Theorem (Mirek+Stein+Trojan 2015) 1 9 N ∑ N n = 1 f ( x − n 2 ) . N A N f ‖ ℓ p (ℤ) ≲ ‖ f ‖ ℓ p (ℤ) , ▶ Circle method approach by Bourgain ▶ Ionescu–Wainger multipliers select rationals with small ▶ Use periodic multipliers on major arcs

What are correct endpoint variational inequalities? Theorem (S.J. Taylor 1972) Jump inequalities: 𝜔( t ) = t r , r > 2 . Variational inequalities: What is the best 𝜔 -variational estimate for general martingales? Question are reparametrizations of Brownian motion. Same is true for all martingales with continuous paths, since they is a.s. fjnite with the Young function t 0 <⋯< t J < T sup 𝜔( V t < T )( B t ) = If ( B t ) is the standard Brownian motion, then 10 ‖ B t j + 1 − B t j ‖ 𝜔( L ) j , 𝜔( t ) = t 2 / log ∗ log ∗ t . 𝜔( t ) = t 2 /( log ∗ t ) 1 +𝜗 .

Variational estimates in time-frequency analysis t j <𝜊 1 <𝜊 2 < t j + 1 sup t 0 <⋯< t J (∑ j | |∫ e 2 𝜌 ix (𝜊 1 +𝜊 2 ) ˆ Theorem (Oberlin+Seeger+Tao+Thiele+Wright 2009) f 1 (𝜊 1 )ˆ f 2 (𝜊 2 ) d 𝜊 1 d 𝜊 2 | | r / 2 ) 2 / r The variationally truncated bilinear Hilbert transform Theorem (Do+Muscalu+Thiele 2016) 11 | The variationally truncated partial Fourier integral sup t 0 <⋯< t J (∑ j |∫ t j <𝜊< t j + 1 f (𝜊) d 𝜊| | r ) 1 / r e 2 𝜌 ix 𝜊 ̂ is bounded L 2 → L 2 for r > 2 . ▶ Quantitative form of Carleson’s theorem is bounded L 2 × L 2 → L 1 for r > 2 . ▶ Uses a variational estimate for paraproducts

Martingale paraproduct dg j s ( g , f ) t s t s df j t s t s df j dg k . s ≤ j < k ≤ t ∑ s ( f , g ) ∶= Π t the truncated paraproduct (or area process ) is defjned by 12 For martingales ( f j ) j , ( g j ) j and martingale difgerences df j = ( f j − f j − 1 ) ( f t − f s )( g t − g s ) = Π t s ( f , g ) + df s + 1 dg s + 1 + ⋯ + df t dg t + Π t

Variational estimate for martingale paraproduct t ( j ) ( f , g )| > 𝜇}. t ( j ) #{ j | |Π t ( j + 1 ) t ( 0 )<⋯< t ( J ) sup Proof idea: for 𝜇 > 0 estimate the jump counting function ≤ C p 1 , p 2 ‖ f ‖ p 1 ‖ g ‖ p 2 3 ‖ p ′ 2 / r ‖ r / 2 ) | ( f , g )| |Π t ( j + 1 ) Theorem (Do+Muscalu+Thiele 2012 (doubling), Kovač+ZK | j (∑ t 0 <⋯< t J ‖ 1 1 1 2018 (non-doubling)) 13 For 1 < p 1 , p 2 < ∞ with p 1 + p 2 + p 3 = 1 and 2 < r we have ‖ sup

Application: stochastic integrals j ≤ C p 1 , p 2 , r ‖ X ‖ p 1 ‖ Y ‖ p 2 . 3 ‖ p ′ 2 / r ‖ r / 2 ) | ( X s − − X t ( j ) ) dY s | ( t ( j ), t ( j + 1 )] |∫ Corollary | (∑ 1 Let ( X t ) , ( Y t ) be càdlàg continuous time martingales. Then for 1 t 0 <⋯< t J 14 1 ‖ 1 < p 1 , p 2 < ∞ with p 1 + p 2 + p 3 = 1 and 2 < r we have ‖ sup ▶ Chevyrev+Friz 2018: diagonal case p 1 = p 2 . ▶ Friz+Victoir 2006: martingales with continuous paths. ▶ Classically X , Y are Brownian motions. ▶ Useful in Lyons’s theory of rough paths.