Paraproducts in One and Several Variables M. Lacey and J. Metcalfe

Transcription

1 Paraproducts in One and Several Variables M. Lacey and J. Metcalfe Kelly Bickel Washington University St. Louis, Missouri IAS Workshop on Multiparameter Harmonic Analysis June 19, 2012

2 What is a Paraproduct?! A Paraproduct is a bilinear operator that is similar to, but nicer than, a product of two functions. Consider the following operator: Π(f, g)(s) := s f (t)g(t) dt, f, g C 1 0 (R). Then Π satisfies: Product Reconstruction: By Leibniz s rule, fg = Π(f, g) + Π(g, f ). Linearization Formula: For G C (R), we have G(f ) = G(0) + Π(f, G (f )). A Leibniz-type Rule: It follows immediately that Π(f, g) = f g. Then Π is almost a paraproduct. (generally, paraproducts also satisfy a Hölder s inequality.)

3 A Paraproduct is... A Paraproduct Π is a bilinear operator satisfying: product reconstruction, a linearization formula, a Hölder-type inequality, and a Leibniz-type rule: α Π(f, g) = Π ( α f, g). We study One-Parameter Model Paraproducts: Let I be an interval. Then φ I is a bump function adapted to I iff φ I 2 = 1 and D n φ I (x) I n 1/2 ( 1 + ) N x c(i ), n = 0, 1, I where c(i ) is the center of I and N is sufficiently large. Let D be the set of dyadic intervals. Define: B(f 1, f 2 ) := 2 I 1/2 φ 3,I f j, φ j,i, I D j=1 where, for each I, each φ j,i is adapted to I and two of the φ j,i have integral zero.

4 An Example For each dyadic interval I, the Haar function adapted to I is h I := I 1/2 (1 Il 1 Ir ), where I l is the left half of I, and I r is the right half of I. Moreover, define h 0 I := h I and h 1 I := h I. Then, the Haar Paraproducts are given by: B k1,k2,k3 (f 1, f 2 ) = I D I 1/2 f 1, h k1 I f 2, h k2 I h k3 I, where k j {0, 1} and two of the k j are zero. It can be shown that f 1 f 2 = B 1,0,0 (f 1, f 2 ) + B 0,1,0 (f 1, f 2 ) + B 0,0,1 (f 1, f 2 ).

5 Increase the Parameters! We also study Bi-parameter Model Paraproducts: Let R be the set of dyadic rectangles in R 2. A function φ R is adapted to the rectangle R, where R = R 1 R 2, if φ R (x) = φ R1 (x 1 )φ R2 (x 2 ), where each φ Rk is adapted to R k. The bi-parameter model paraproducts are of the form: B(f 1, f 2 ) = 2 R 1/2 φ 3,R f j, φ j,r, R R j=1 where each φ j,r is adapted to R and for each coordinate x k, k = 1, 2, there are two positions in j = 1, 2, 3 such that φ j,r (x 1, x 2 )dx k = 0 x i x k and R R. R Then we say B has x k zeros in the j th position (or {φ j,r } has x k zeros).

6 Main Results The main results proved in the paper are that both the one and bi-parameter model paraproducts satisfy a Hölder-type inequality: Theorem 1 (Coifman, Meyer 78), (Muscalu, Pipher, Tao, Thiele 04) Whenever 1 < p 1, p 2, 1/r = 1/p 1 + 1/p 2, and 0 < r <, B(f 1, f 2 ) r f 1 p1 f 2 p2.

7 Classical Coifman-Meyer Theorem Theorem 1 is a discrete version of the following one and bi-parameter results: Let m be a bounded function on R 2, smooth away from the origin and satisfying α m(ζ) 1 ζ α, for sufficiently many multi-indices α and define the bilinear operator T m (1) T m (1) (f, g) = m(ζ)ˆf (ζ 1 )ĝ(ζ 2 )e 2πix(ζ1+ζ2) dζ, R 2 for Schwartz functions f, g S(R). We can generalize this by allowing m to be defined on R 2n and f, g S(R n ). Theorem 2 (Coifman, Meyer, 78) If m is a symbol satisfying the above estimates, then the bilinear operator T (1) m maps L p L q L r whenever 1 < p, q, 1/r = 1/p + 1/q, and 0 < r <. by

8 Bi-parameter Coifman-Meyer Theorem Let m(ζ, η) be a bounded function on R 4, smooth away from {(ζ 1, η 1 ) = 0} {(ζ 2, η 2 ) = 0} and satisfying the estimate α ζ β η m(ζ, η) 1 (ζ 1, η 1 ) α1+β1 1 (ζ 2, η 2 ) α2+β2, for sufficiently many multi-indices α and β. Then we can define the bilinear operator T m (2) as follows: T m (2) (f, g) = m(ζ, η)ˆf (ζ)ĝ(η)e 2πix(ζ+η) dζdη, R 4 where f, g S(R 2 ). Theorem 3 (Muscalu, Pipher, Tao, Thiele 04) If m is a symbol satisfying the above estimates, then the bilinear operator T (2) m maps L p L q L r whenever 1 < p, q, 1/r = 1/p + 1/q, and 0 < r <.

9 Fractional Derivative Estimates Let f, g S(R 2 ) and for α > 0, define the fractional derivative D α by D α f (ζ) = ζ αˆf (ζ). There are paraproducts Π j for j = 0, 1, 2, 3 such that the Coifman-Meyer theorem applies to each Π j and 3 fg = Π j (f, g). j=0 Using the structure of the Π j, one can find paraproducts Π 1 and Π 2 with D α( Π 1 (f, g) ) = Π 1(f, D α g) and D α( Π 2 (f, g) ) = Π 2(D α f, g), and similar Π 0 and Π 3 paraproducts. Using the Coifman-Meyer theorem, calculate D α (fg) r 3 D α (Π j (f, g)) r j=0 D α f p g q + f p D α g q, for 1 < p, q, 1/r = 1/p + 1/q, and 0 < r <.

10 Fractional Partial Derivative Estimates For f S(R 2 ) and for α, β > 0, define the partial differential operator D α 1 Dβ 2 by D α 1 Dβ 2 f (ζ) = ζ 1 α ζ 2 βˆf (ζ). Then, for f, g S(R 2 ), using the bi-parameter Coifman-Meyer theorem and analogous manipulations of paraproducts, we have D α 1 D β 2 (fg) r D α 1 D β 2 f p g q + f p D α 1 D β 2 g q + D α 1 f p D β 2 g q + D β 2 f p D α 1 g q, for 1 < p, q, 1/r = 1/p + 1/q, and 0 < r <.

11 Square and Maximal Functions We are considering the following paraproducts: B(f 1, f 2 ) := I D 2 I 1/2 φ 3,I f j, φ j,i, where, for each I, each φ j,i is adapted to I and two of the φ j,i have integral zero. Without loss of generality,we can assume φ 2,I and φ 3,I always have integral zero. We will need the following variations of the maximal function and square function: g, φ 1,I M 1 g := sup 1 I I D I j=1 S j g := [ ] 1/2 g, φ j,i 2 1 I, for j = 2, 3. I I D

12 Maximal Function Bounds f, φ 1,I M 1 (f )(x) = sup 1 I Mf (x), I D I where M denotes the typical Hardy-Littlewood maximal function. Fix I D, say I = 2 K+1. Translate I so that it is centered at 0 and let y be in I. Then: f, φ I I 1 f (x) (1 + x / I ) 2 dx I I 1 R I f + I 1 R I f (x) (1 + x 2 / I 2 ) 1 dx Mf (y) + 2 K 1 f (x) dx( (j 1) /2 2(K+1) ) 1 j>k I j Mf (y) + 2 k j 1 2 j+1 f (x) dx j>k I j Mf (y), where I j is the interval centered around zero with length 2 j+1. As y I, it is clear that y I j.

13 Square Function Bounds Sf is more or less large only where f is large, which is reflected by Sf p f p 1 < p <. Partially follows because φ I, φ I is usually small. In particular, if all {φ I } have integral zero, then φ I, φ I C. I D For p = 2, restrict to finite sums, let f 2 = 1 and calculate: f, φ I 2 f, φ I φ I 2, I using Cauchy-Schwarz and then calculate ( ) 2 f, φ I 2 I I I 2 I I f, φ I φ I, f φ I, φ I f, φ I 2 I φ I, φ I.

14 Proof of One-Parameter Result Recall: We are trying to show that B maps L p1 L p2 L r whenever 1 < p 1, p 2, 1/r = 1/p 1 + 1/p 2, and 0 < r <. Case 1: 1 < r < In this case, L.M. use a duality argument. We will need the following obvious fact: If {a j,i } I D are sequences such that {a 1,I } l, {a 2,I }, {a 3,I } l 2, then a 1,I a 2,I a 3,I a 1,I a 2,I 2 a 3,I 2, and in particular, ( 3 I D j=1 I D sup I D ) f j, φ j,i 1 I (x) I f 1, φ 1,I I 1 I (x) [ 3 f j, φ j,i 2 1 I (x) I j=2 = (M 1 f 1 )(S 2 f 2 )(S 3 f 3 )(x) I D ] 1/2

15 Proof of One-Parameter Result Case 1: 1 < r < Let r be dual to r, fix f 3 L r with f 3 r = 1, and calculate: B(f 1, f 2 ), f 3 = 2 I 1/2 φ 3,I f j, φ j,i f 3 I D j=1 3 I 3/2 f j, φ j,i 1 I I D j=1 (Mf 1 )(S 2 f 2 )(S 3 f 3 ) Mf 1 p1 S 2 f 2 p2 S 3 f 3 r f 1 p1 f 2 p2, where Hölder s inequality is used. Taking the supremum over all such f 3 yields: B(f 1, f 2 ) r f 1 p1 f 2 p2.

16 Proof of One-Parameter Result Case 2: 1/2 < r < 1 M.L. prove the following weak-type estimates: λ {B(f 1, f 2 ) > λ} 1/r f 1 p1 f 2 p2, (1) and multi-linear Marcinkiewicz interpolation yields the desired strong estimates. To get the weak estimates, use the following lemma: Lemma 1 (Auscher, Hofmann, Muscalu, Tao, Thiele 02) Let 0 < r <. If for all sets E with 0 < E <, there is subset E E with E E and f, 1 E A E 1/r then f r, A. In particular, let f = B(f 1, f 2 ) and A = f 1 p1 f 2 p2.

17 The Λ operator Consider the following multi-linear operator: Λ(f 1, f 2, f 3 ) := I D 3 I 1/2 f j, φ j,i. (2) Then, M.L. shows that for each f 1, f 2, and set E, there is a set E E with E E such that j=1 Λ(f 1, f 2, f 3 ) E 1/r f 1 p1 f 2 p2, for all f 3 supported in E and bounded by 1. (particularly, f 3 = 1 E.) By multi-linearity, we can assume f 1 p1 = f 2 p2 = 1. As the class of the multi-linear forms Λ is invariant under dilations by powers of two, we can assume E = 1. Specifically, if D λ is the dilation operator defined by (D λ f )(x) = f (λ 1 x) and Λ k (f 1, f 2, f 3 ) := 2 k Λ(D 2 k f 1, D 2 k f 2, D 2 k f 3 ), then Λ k is a multi-linear form of type (2) for k Z.

18 End! The End!

19 Paraproducts in One and Several Variables (Part II) M. Lacey and J. Metcalfe Kelly Bickel Washington University St. Louis, Missouri IAS Workshop on Multiparameter Harmonic Analysis June 19, 2012 Paraproducts in One and Several Variables (Part II) M. Lacey and J. Metca

20 Bi-parameter Paraproducts- Review Recall: for I be an interval, we say φ I is a bump function adapted to I iff φ I 2 = 1 and D n φ I (x) I n 1/2 ( 1 + ) N x c(i ), n = 0, 1, I where c(i ) is the center of I and N is sufficiently large. Last time, we considered paraproducts of the form: B(f 1, f 2 ) := I D 2 I 1/2 φ 3,I f j, φ j,i, where, for each I, each φ j,i is adapted to I and two of the φ j,i have integral zero. Let R be the set of dyadic rectangles in R 2. A function φ R is adapted to the rectangle R, where R = R 1 R 2, if where each φ Rk is adapted to R k. j=1 φ R (x) = φ R1 (x 1 )φ R2 (x 2 ), Paraproducts in One and Several Variables (Part II) M. Lacey and J. Metca

21 Bi-parameter Paraproducts- Review The bi-parameter model paraproducts are of the form: B(f 1, f 2 ) = R R R 1/2 φ 3,R 2 f j, φ j,r, where each φ j,r is adapted to R and for each coordinate x k, k = 1, 2, there are two positions in j = 1, 2, 3 such that φ j,r (x 1, x 2 )dx k = 0 x i x k and R R. R Then we say B has x k zeros in the j th position (or {φ j,r } has x k zeros). Theorem 2 (Muscalu, Pipher, Tao, Thiele 04) Whenever 1 < p 1, p 2, 1/r = 1/p 1 + 1/p 2, and 0 < r <, j=1 B(f 1, f 2 ) r f 1 p1 f 2 p2. Paraproducts in One and Several Variables (Part II) M. Lacey and J. Metca

22 Variants of Square and Maximal functions Again, M.L. use variants of square and maximal functions, adapted to the specific bump functions appearing in the given paraproduct B. Consider the following iterates of one-variable square and maximal functions: f, φ R MM(f ) := sup 1 R R R R S 1 M 2 (f ) := SS(f ) := [ sup R R 2 D 1 D [ R R ] 1/2 f, φ R1 R R, R = R 1 R 2 R ] 1/2 f, φ R 2 1 R, R where we can similarly define S 2 M 1, M 1 S 2, and M 2 S 1. If a square function is applied to the set {φ R } in the x k coordinate, we require the functions {φ R } to have x k zeros. Paraproducts in One and Several Variables (Part II) M. Lacey and J. Metca

23 Biparameter Proof: Case 1 As before, the interated square and maximal functions are bounded from L p L p, for 1 < p <. Specficically, if T is an operator on the previous slide, Tf p f p for 1 < p <. As before, we define the multilinear form Λ by for f 3 L r, and have: Λ(f 1, f 2, f 3 ) = R R R 1/2 B(f 1, f 2 ), f 3 Λ(f 1, f 2, f 3 ) 3 f j, φ j,r j=1 = 3 R 1/2 f j, φ j,r 1 R, R R j=1 We will use operators to bound the sum inside the integral. The operators we choose will depend on where B has zeros in each coordinate. Paraproducts in One and Several Variables (Part II) M. Lacey and J. Metca

24 Bi-parameter Proof Case 1 To illustrate, assume B has x 2 zeros in the j = 1, 2 positions and x 1 zeros in the j = 2, 3 positions. Then we have: R R j=1 3 R 1/2 f j, φ j,r 1 R sup R R 2 D 1 D f 3, φ 3,R R 1 R 2 j=1 ( R 2 D ) 1/2 f j, φ j,r 2 1 R R sup R 1 ( R 2 ) 1 ( 2 f 1, φ 1,R 2 1 R R R ) 1 ( f 2, φ 2,R 2 2 f 3, φ 3,R 2 1 R sup R R 2 R R 1 1 R ) 1 2 = (M 1 S 2 f 1 )(SSf 2 )(S 1 M 2 f 3 ) (S 2 M 1 f 1 )(SSf 2 )(S 1 M 2 f 3 ). Paraproducts in One and Several Variables (Part II) M. Lacey and J. Metca

25 Bi-parameter Proof Case 1 In general, there are 3 iterated square/maximal operators T j for j = 1, 2, 3 with B(f 1, f 2 ), f 3 Λ(f 1, f 2, f 3 ) = 3 R 1/2 f j, φ j,r 1 R R R j=1 T 1 f 1 T 2 f 2 T 3 f 3. Case 1: For 1 < r <, let r be dual to r and choose f 3 L r with f 3 r = 1. Then B(f 1, f 2 ), f 3 T 1 f 1 p1 T 2 f 2 p2 T 3 f 3 r which gives the result for 1 < r <. f 1 p1 f 2 p2, Paraproducts in One and Several Variables (Part II) M. Lacey and J. Metca

26 Bi-parameter Proof Case 2 Case 2: 1/2 < r < 1 M.L. prove the following weak-type estimates: λ {B(f 1, f 2 ) > λ} 1/r f 1 p1 f 2 p2, and multi-linear Marcinkiewicz interpolation yields the desired strong estimates. To get the weak estimates, show, that for each f 1, f 2 with f 1 p1 = f 2 p2 = 1 and set E with E = 1, there is a set E with E E with E E and Λ(f 1, f 2, f 3 ) 1, for every f 3 supported in E and bounded by 1. Further, we can assume each f j is smooth and compactly supported. Let T j, for j = 1, 2, 3, be the operators bounding B as in the previous slide. Paraproducts in One and Several Variables (Part II) M. Lacey and J. Metca

27 O Notation We will be estimating Λ(f 1, f 2, f 3 ) = R R R 1/2 3 f j, φ j,r. In particular, we will decompose R into several classes of rectangles. Let O be a class of dyadic rectangles. Then define sum(o) = R O R 1/2 j=1 3 f j, φ j,r. Recall that for each iterated operator T j we were summing (or sup-ing) over f, φ R, for R R. Let T O denote an iterated square or maximal function restricted to the class of dyadic rectangles O. For example, if T = SS, ( ) 1/2 f, φ R 2 T O f = 1 R. R R O Before we define E, we need to establish several bounds on sum(o) and T O 2 for classes of rectangles O satisfying special properties. j=1 Paraproducts in One and Several Variables (Part II) M. Lacey and J. Metca

28 Technical Lemma 1 Lemma 2 Let O R and let µ > 1 be a constant such that supp(f ) µr = R O, for a given function f. Then T O f 2 µ N f 2, where N = N N 0, where N is the integer in the definition of adapted for the {φ R } defining T, and N 0 is the smallest integer needed to get the L p bounds on the square and maximal functions. Idea of Proof Let {φ R } be adapted with integer N > 0. One can define a new set of adapted functions { φ R } adapted with integer N 0 such that φ R (x) = µ N φ R (x) x µr. Define T with the {φ R } and T with the { φ R }. If f satisfies the assumptions of the lemma, then T O f = µ N TO f, and the result follows since T is bounded on L 2, with bounds independent of µ. Paraproducts in One and Several Variables (Part II) M. Lacey and J. Metca

29 Technical Lemma 2 Lemma 3 Let c 1, c 2, c 3 > 0 be constants and O a collection of rectangles such that R {T j f j > c j } R for R O, j = 1, 2, 3. (3) Then we have the estimate: sum(o) c 1 c 2 c 3 sho. If (3) is not known for j = 3, we have: sum(o) c 1 c 2 sho 1/2 T 3O f 3 2. Idea of Proof: Let W = sh(o) 3 j=1 {T jf j < c j }, so that R W R. Then: 3 sum(o) R 1/2 f j, φ j,r 1 R W R O j=1 W T 1 f 1 T 2 f 2 T 3 f 3 sh(o) c 1 c 2 c 3. Paraproducts in One and Several Variables (Part II) M. Lacey and J. Metca

30 Definition of E Fix f 1, f 2, E with f 1 p1 = f 2 p2 = E = 1. Define 4ν = min(p 1, p 2 ) and let T 0 be the strong maximal function (in two parameters). Define Ω j,l := {T j f j > C2 l }, l Z, j = 1, 2, Ω l := 2 j=1ω j,l, Ω := l N {T 0 1 Ωl > 2 νl /100}, Ω := {T 0 1 Ω > 1/2}. Set E = Ω c E. We can choose C so that E 1/2 by choosing C so that Ω < 1/8. Using the L 2 boundedness oft 0 and L p j boundedness of the T j for j = 1, 2, we have Ω K 1 Ω l 2 2νl K 2 l N l N j=1 2 C p j 2 l(2ν p j ), which converges, and so we can choose C >> 0 to give Ω the desired size. Paraproducts in One and Several Variables (Part II) M. Lacey and J. Metca

31 Decomposition of R Recall, we are trying to show: Λ(f 1, f 2, f 3 ) = R R R 1/2 3 f j, φ j,r 1, where f 3 is bounded by one and supported on E. Then, for 1 < p 3 <, f 3 p3 1. We consider the sum restricted to specific classes of rectangles in R and split the rectangles into classes as follows: j=1 R is in class O j,l iff l is the greatest integer so that R Ω j,l = R {T j f j > C2 l } R. As the T j f j are bounded, every rectangle R is in precisely one O j,l for each j and so we can associate to each R a tuple l = (l 1, l 2, l 3 ) of integers. Paraproducts in One and Several Variables (Part II) M. Lacey and J. Metca

32 l with l1, l 2 0 Let L denote the tuples with l 1, l 2, l 3 0. Fix such an l = (l 1, l 2, l 3 ) and define Then for each R O l, and j = 1, 2, 3, O l = 3 j=1o j,lj. R Ω j,lj +1 = R {T j f j > C2 l j +1 } < R. and so Technical Lemma 2 yields: sum(o l ) sh(o l ) 2 l1+l2+l3. The L p j -boundedness of T j implies that, for θ 1 + θ 2 + θ 3 = 1, sh(o l ) sh(o 1,l1 ) θ1 sh(o 2,l2 ) θ2 sh(o 3,l3 ) θ3 2 p1l1θ1 p2l2θ2 p3l3θ3. Then we can calculate sum(o l ) 2 l1(1 p1θ1)+l2(1 p2θ2)+l3(1 p3θ3), l L l L which converges for θ 1, θ 2, θ 3 and p 3 > 0 with 1 p j θ j > 0. Paraproducts in One and Several Variables (Part II) M. Lacey and J. Metca

33 l with l1 > 0 or l 2 > 0 Let G denote the tuples with at least one of l 1, l 2 0. Fix such an l = (l 1, l 2, l 3 ) and define O l = 2 j=1o j,lj. Fix such an l and without loss of generality, assume l 1 > 0. Let R O l. Let 2 νl1/2 R be the rectangle obtained by dilating each side of R by a factor of 2 νl1/2 and keeping the same center. Then 1 2 νl1/2 R 2 νl 1 /2 R which implies 2 µl1/2 R Ω and so Technical Lemma 1 gives: for N sufficiently large. 1 Ωl1 = 2 νl1/2 R Ω l1 / 2 νl1/2 R R Ω l1 /2 νl1 R 2 νl1 /100, 2 νl1/2 R supp(f 3 ) = R O l. T Ol,3f N νl 1/2 f l1, Paraproducts in One and Several Variables (Part II) M. Lacey and J. Metca

34 l with l1 > 0 or l 2 > 0 Actually, we showed: Now, as each R O l satisfies: T Ol,3f 3 2 min(2 10l1, 2 10l2 ). R {T j f j > C2 l j +1 } R for R O l, j = 1, 2, Technical Lemma 2 implies: sum(o l ) 2 l1+l2 sho l 1/2 T Ol,3f l1+l2 min(2 10l1, 2 10l2 ), which is clearly summable over all tuples (l 1, l 2 ) with l 1 or l 2 positive. This covers the entire class of dyadic rectangles. Thus, we have proved: as desired. Λ(f 1, f 2, f 3 ) = R R R 1/2 3 f j, φ j,r 1, j=1 Paraproducts in One and Several Variables (Part II) M. Lacey and J. Metca