Stability inequalities for martingales and Riesz transforms ∗ nuelos † Rodrigo Ba˜ Purdue University Department of Mathematics West Lafayette, IN. 47906 May 19, 2017 ∗ With Adam Ose ¸kowski † Partially supported by NSF R. Ba˜ nuelos (Purdue) May 19, 2017
Stability (quantitative/deficit) sharp inequalities R. Ba˜ nuelos (Purdue) May 19, 2017
Stability (quantitative/deficit) sharp inequalities Optimal/sharp inequalities Suppose you have two functionals E and F on some normed (real) linear space M satisfying the functional inequality E � F in the sense that E ( x ) � F ( x ) , ∀ x ∈ M . The functional inequality E � F is sharp if for all λ < 1 there exist x ∈ M such that E ( x ) > λ F ( x ) The subset M 0 = { x ∈ M : E ( x ) = F ( x ) } is called the set of optimizers (extremals) of the inequality. When M 0 � = ∅ , the inequality is said to be optimal. (Note: An optimal functional inequality is sharp but not vice–versa.) R. Ba˜ nuelos (Purdue) May 19, 2017
Definition Let d be a metric on M (not necessarily the norm metric) and Φ a “rate function.” The optimal functional inequality E � F is ( d, Φ) – stable if F ( x ) − E ( x ) � Φ( d ( x, M 0 )) , ∀ x ∈ M In various examples, Φ( t ) = ct 2 and d ( x, y ) = � x − y � M and z ∈M 0 � x − z � 2 F ( x ) − E ( x ) � c inf M . R. Ba˜ nuelos (Purdue) May 19, 2017
Definition Let d be a metric on M (not necessarily the norm metric) and Φ a “rate function.” The optimal functional inequality E � F is ( d, Φ) – stable if F ( x ) − E ( x ) � Φ( d ( x, M 0 )) , ∀ x ∈ M In various examples, Φ( t ) = ct 2 and d ( x, y ) = � x − y � M and z ∈M 0 � x − z � 2 F ( x ) − E ( x ) � c inf M . Classical Sobolev in R n ( n � 3 ). n = n ( n − 2) k 2 | S n − 1 | 4 k 2 n � f � 2 n − 2 � �∇ f � 2 ∀ f ∈ H 1 0 ( R n ) = M , 2 , 2 n M 0 = { x → c ( a + b | x − x 0 | 2 ) − ( n − 2) / 2 , a, b > 0 , x 0 ∈ R n , c ∈ R } Optimality: Aubin (1976), Talenti (1976). Stability: Biachi-Egnell (1990) �∇ f � 2 2 − k 2 n � f � 2 g ∈M 0 �∇ ( f − g ) � 2 n − 2 � C inf 2 n 2 R. Ba˜ nuelos (Purdue) May 19, 2017
More general Sobolev ( 0 < α < n/ 2 ). n − 2 α � k n,α � ( − ∆) α/ 2 f � 2 � f � 2 n Optimality: Lieb (1983), Stability: Cheng-Frank-Weth (2013) Hardy-Littlewood-Sobolev Optimality: Lieb (1983). Stability: Carlen (2016), Log-Sobolev Gross (1975), Stability Fathi-Indrei-Ledoux (2015), Nash Optimality: Carlen-Loss (1993). Stability: Carlen-Lieb (2017) Housdorff-Young inequality: Sharpness Beckner 1975 (Lieb 1990) Stability: p Chris 2015. 1 � p � 2 , q = p − 1 � ˆ f � q � ( A p ) n � f � p A p = p 1 / 2 p q − 1 / 2 q A p is best contacts. Extremizers are general Gaussians: g ( x ) = ce Q ( x )+ x · v . R. Ba˜ nuelos (Purdue) May 19, 2017
More general Sobolev ( 0 < α < n/ 2 ). n − 2 α � k n,α � ( − ∆) α/ 2 f � 2 � f � 2 n Optimality: Lieb (1983), Stability: Cheng-Frank-Weth (2013) Hardy-Littlewood-Sobolev Optimality: Lieb (1983). Stability: Carlen (2016), Log-Sobolev Gross (1975), Stability Fathi-Indrei-Ledoux (2015), Nash Optimality: Carlen-Loss (1993). Stability: Carlen-Lieb (2017) Housdorff-Young inequality: Sharpness Beckner 1975 (Lieb 1990) Stability: p Chris 2015. 1 � p � 2 , q = p − 1 � ˆ f � q � ( A p ) n � f � p A p = p 1 / 2 p q − 1 / 2 q A p is best contacts. Extremizers are general Gaussians: g ( x ) = ce Q ( x )+ x · v . Brasco &De Philippis (2016). Torsional rigidity for Brownian motion. � � E z ( τ D ) dz � C n A ( D ) 2 E z ( τ D ∗ ) dz − D ∗ D ( Fraenkel Asymmetry) A ( D ) := inf {| D △ B | : B is a ball with | B | = | D |} . | D | Problem: Prove “it” for stable processes (any subordination of BM). R. Ba˜ nuelos (Purdue) May 19, 2017
Martingales: Sharp (but not optimal, i.e., M 0 = ∅ )) inequalities. (Reference: A. Ose ¸kowski, “Sharp martingale and semimartingale inequalities” , Monografie Matematyczne 72 , Birkh¨ auser, 2012.) Doob { f n } an L p , 1 < p � ∞ martingale. f ∗ = sup n | f n | maximal function. p � f ∗ � p � p − 1 � f � p Burkholder (1984), Wang (1991 for dyadic): Inequality is sharp. But M 0 = ∅ . Burkholder (1966) S ( f ) = ( � n ( f n − f n − 1 ) 2 ) 1 / 2 a p � f � p � � S ( f ) � p � b p � f � p 1 < p < ∞ Many sharp versions of these exists but none are optimal i.e., M 0 = ∅ , outside of the trivial case of p = 2 . (The first sharp case of these for Brownian martingales/stochastic integrals is due to B. Davis (1976).) R. Ba˜ nuelos (Purdue) May 19, 2017
X , Y c´ adl´ ag (right continuous/left limits) martingales: Y is differentially subordinate to X ( Y << X ), if the process { [ X, X ] t − [ Y, Y ] t } t � 0 is nonnegative and nondecreasing in t . They are orthogonal ( Y ⊥ X ) if [ X, Y ] = 0 . 1 < p < ∞ and . p ∗ = max { p, p/ ( p − 1) } . Set || X || p = sup t � 0 || X t || p Burkholder (1984) Y << X || Y || p � ( p ∗ − 1) || X || p . The constant ( p ∗ − 1) is best possible. Furthermore, the inequality is always strict unless p = 2 . That is, inequality is sharp and M 0 = ∅ unless p = 2 . R.B. G. Wang (1995) Y << X and Y ⊥ X � π � || Y || p � cot || X || p . 2 p ∗ � � π The constant cot is the best possible. Furthermore, the inequality is always 2 p ∗ strict unless p = 2 unless p = 2 . That is, inequality is sharp and M 0 = ∅ unless p = 2 . R. Ba˜ nuelos (Purdue) May 19, 2017
A careful analysis of proofs reveals that “almost” extremals used to proof sharpness are “almost” eigenfunctions. The dyadic maximal function in R n (dyadic martingales). � M d ( f )( x ) = sup { 1 | f ( y ) | dy : x ∈ Q, Q ∈ [0 , 1] n , dyadic cube } | Q | Q p Doob (for inequality) Wang (1995) for sharpness: � M d ( f ) � p � p − 1 � f � p , 1 < p � ∞ . (Here we may restrict to non-negative functions.) A. Melas (2015): If you take a sequence { f n } of almost externals then p lim n � M d ( f n ) − p − 1 f n � p = 0 Theorem (Melas 2015) Fix 2 < p < ∞ , ǫ > 0 (small enough). Suppose f � 0 (in L p ) is such that � � p � M d ( f ) � p � p − 1 − ε � f � p . Then p p − 1 f � p � c p ε 1 /p � f � p � M d ( f ) − for some constant c p depending only on p . R. Ba˜ nuelos (Purdue) May 19, 2017
Theorem (A.Ose ¸kowski & R.B. 2016: Y << X ) 1 (i) Let 1 < p < 2 and ε > 0 . || Y || p � ( p − 1 − ε ) || X || p . Then � �� �� � 1 � | Y ∞ | − � p � c p ε 1 / 2 || X || p . ( p − 1) | X ∞ | O ( ε 1 / 2 ) as ε → 0 is sharp. c p = O ((2 − p ) − 1 / 2 ) as p ↑ 2 and this is sharp. (ii) Let 2 < p < ∞ and ε > 0 . || Y || p � ( p − 1 − ε ) || X || p . � �� � �� � | Y ∞ | − ( p − 1) | X ∞ | � p � c p ε 1 /p || X || p , O ( ε 1 /p ) as ε → 0 is sharp. c p is O (( p − 2) − 1 /p ) as p ↓ 2 and O ( p ) as p → ∞ . These orders are sharp. (iii) For p = 2 , no c 2 and κ exist such that || Y || 2 � (1 − ε ) || X || 2 implies � �� � �� � | Y ∞ | − | X ∞ | � 2 � c 2 ε κ || X || 2 . In fact, there exist martingales Y and X , Y << X , such that �| Y | − | X |� 2 � Y � 2 = � X � 2 , and > 0 ( independent of ε ) � X � 2 R. Ba˜ nuelos (Purdue) May 19, 2017
¸kowski & R.B. 2016: Y << X and Y ⊥ X ) Theorem (A.Ose (i) � � � � � π � � � � � � � � � � c p ε 1 / 2 || X || p , 1 < p < 2 , � | Y ∞ | − tan | X ∞ | � � � 2 p p if X and Y are such that � � π � � || Y || p � − ε || X || p . tan 2 p Orders in ε as ε → 0 , and c p , as p ↑ 2 , are best possible. (ii) � � � � � π � � � � � � � � � � c p ε 1 /p || X || p , 2 < p < ∞ , � | Y ∞ | − cot | X ∞ | � � � 2 p p if � � cot π || Y || p � 2 p − ε || X || p (iii) As in previous theorem, no such estimate exists for p = 2 . R. Ba˜ nuelos (Purdue) May 19, 2017
Beurling-Ahlfors operator in complex plane C � Bf ( z ) = − 1 f ( w ) π p.v. ( z − w ) 2 d w. C It is a Calderón-Zygmund singular integral and � Bf � p � C p � f � p , 1 < p < ∞ . Conjecture T. Iwaniec 1984 The operator norm of B on L p is p ∗ − 1 : � B � p → p = ( p ∗ − 1) , 1 < p < ∞ Known: ( p ∗ − 1) � � B � p → p � 1 . 575( p ∗ − 1) Lower bound O. Lehto (1965) upper bound R.B and P. Janakiraman (2008). Lehto’s functions used to prove the lower bound have the property that | Bf ( z ) | ≈ ( p ∗ − 1) | f ( z ) | . That is, they are “near eigenfunctions.” R. Ba˜ nuelos (Purdue) May 19, 2017
Recommend
More recommend
