Dyadic harmonic analysis and weighted inequalities

Transcription

1 Dyadic harmonic analysis and weighted inequalities Cristina Pereyra University of New Mexico, Department of Mathematics and Statistics Workshop for Women in Analysis and PDE IMA Institute for Mathematics and its Applications Minneapolis, MN May 30 to June 2, 2012 Cristina Pereyra (UNM) 1 / 64

2 General Outline General Outline Lecture 1 Lecture 2 Lecture 3 Lecture 4 Overview - with emphasis on the Hilbert transform Overview - A 2 conjecture A study case: dyadic paraproduct Loose ends: Bellman functions and others Cristina Pereyra (UNM) 2 / 64

3 Lecture 1 Outline Lecture 1 Outline Lecture 1 In this lecture we introduce the main objects of study with a focus on the classical Hilbert transform. Weighted inequalities A p weights Hilbert transform H Dyadic intervals, Haar functions Martingale transform Petermichl s Haar shift operator X H as average of Xs Why are L p estimates important? Cristina Pereyra (UNM) 3 / 64

4 Lecture 1 Weighted inequalities Weighted inequalities Question (Two-weights L p -inequalities for operator T ) Is there a constant C p (u, v) > 0 such that T f L p (v) C p (u, v) f L p (u), for all f L p (u)? The weights u, v are a.e. positive locally integrable functions on R d. f L p (u) iff f L p (u) := ( f(x) p u(x) dx) 1/p <. Operator T : L p (u) L p (v). Goal 1: given operator T, identify and classify weights u, v for whom the operator T is bounded from L p (u) to L p (v)). Goal 2: understand nature of constant C p (u, v). Cristina Pereyra (UNM) 4 / 64

5 Lecture 1 Weighted inequalities We concentrate on one-weight L 2 inequalities: u = v = w, and p = 2, for Calderón-Zygmund singular integral operators. Question (One-weight L 2 inequality for operator T ) Is there a constant C(w) > 0 such that T f L 2 (w) C(w) f L 2 (w), for all f L 2 (w)? In particular we study one-weight inequalities in L 2 (w), for the Hilbert transform T = H, and even more specifically, for simpler dyadic operators such as the martingale transform T σ, Petermichl s Haar shift operator X ( Sha ), the dyadic paraproduct π b, the dyadic square function S d. For all these operators to be bounded in L 2 (w), the weight w must belong to the Muckenhoupt A 2 -class. Cristina Pereyra (UNM) 5 / 64

6 Lecture 1 Weighted inequalities A p weights Definition A weight w is in the Muckenhoupt A p class if its A p characteristic, [w] Ap is finite, where, ( )( 1 1 p 1 [w] Ap := sup w dx w dx) 1/(p 1), 1 < p <, Q Q Q Q Q the supremum is over all cubes in R d with sides parallel to the axes. Note that a weight w A 2 if and only if ( )( 1 1 [w] A2 := sup w dx Q Q Q Example In R, w(x) := x α, w A p 1 < α < p 1. Q Q ) w 1 dx <. Cristina Pereyra (UNM) 6 / 64

7 Lecture 1 Weighted inequalities History (Hunt-Muckenhoupt-Wheeden, 1973) Hilbert transform is bounded on L p (w) if and only if w A p. (Coifman-Fefferman, 1974) Same holds for convolution type singular integral with standard kernels. (A 2 conjecture) Linear (in weight A 2 characteristic) estimate for arbitrary Calderón-Zygmund operator T. Theorem (Hytönen, Annals 2012) T f L 2 (w) C T [w] A2 f L 2 (w), Cristina Pereyra (UNM) 7 / 64

8 Lecture 1 Hilbert transform H as a Fourier multiplier Definition On Fourier side the Hilbert transform is defined by Ĥf(ξ) := i sgn(ξ) f(ξ), where sgn(ξ) = 1 if ξ > 0, sgn(ξ) = 1 if ξ < 0, and is zero at ξ = 0. The absolute value of the symbol m H (ξ) := i sgn(ξ) is 1 a.e., and Plancherel s identity used twice implies that H : L 2 (R) L 2 (R) and that it is an isometry, Hf 2 = Ĥf 2= f 2 = f 2. Recall: f(ξ) = R f(x)e 2πiξx dx for f in the Schwartz class. Cristina Pereyra (UNM) 8 / 64

9 Lecture 1 Hilbert transform H as a singular integral operator Definition Hf(x) := p.v. 1 π f(y) 1 f(y) dy := lim x y ɛ 0 π x y >ɛ x y dy. The Hilbert transform is given on Fourier side by Ĥf(ξ) = m H (ξ) f(ξ), where m H (ξ) = i sgn(ξ). Multiplication on Fourier side corresponds to convolution on space Hf(x) = K H f(x), where K H, is the inverse Fourier transform of the multiplier m H. K H (x) := (m H ) (x) := m H (ξ)e 2πixξ dξ = p.v. 1 πx. Recall: f g(x) = R f(y)g(x y)dy = g f(x). R Cristina Pereyra (UNM) 9 / 64

10 Lecture 1 Hilbert transform Boundedness properties of H Integrable kernel implies boundedness on L p (R) by Hausdorff-Young s inequality for p 1: if g L 1 (R), f L p (R) then g f p g 1 f p. But K H is not in L 1 (R), despite this fact, Properties Example H is bounded on L p (R) for all 1 < p < (Marcel Riesz 1927): Hf p C p f p. H is not bounded on L 1, is of weak-type (1,1) (Kolmogorov 1927). H is not bounded on L, is bounded on BMO (Fefferman 1971?). A calculation shows Hχ [0,1] (x) = (1/π) log x / x 1. log x is in BMO but not in L. Properties shared by all Calderón-Zygmund singular integral operators. Cristina Pereyra (UNM) 10 / 64

11 Lecture 1 Hilbert transform Calderón-Zygmund singular integral operators Definition A Calderón-Zygmund operator is an operator T defined on some dense subspace of L 2 (R d ), and that has a kernel representation T f(x) = K(x, y)f(y)dy, x / suppf, R d where K is a standard kernel (with smoothness parameter α): K is a function defined on R d R d minus the diagonal, such that (size condition) K(x, y) C 1 x y d, (cancellation) For all x, x, y R d with x y > 2 x x, K(x, y) K(x, y) + K(y, x) K(y, x ) C 2 x x α x y d+α. Cristina Pereyra (UNM) 11 / 64

12 Lecture 1 Hilbert transform T (1) Theorem Departure point for the CZ theory is boundedness on L 2. Standard estimates plus L 2 imply weak (1, 1) estimates (Calderón-Zygmund decomposition). Interpolation then gives L p for 1 < p < 2. Duality gives p > 2. To deduce boundedness on L 2 : Plancherel: bounded Fourier multiplier (convolution operators). Cotlar s lemma (almost-orthogonality). T (1) theorem: Theorem (David-Journé 1984) A CZ operator can be extended to be a bounded operator in L 2 (R d ) if and only if T (1), T (1) BMO, and T is weakly bounded. Cristina Pereyra (UNM) 12 / 64

13 Lecture 1 Hilbert transform Symmetries for H Properties Convolution H commutes with translations τ h (Hf) = H(τ h f) where τ h f(x) := f(x h). Homogeneity of kernel H commutes with dilations D δ (Hf) = H(D δ f) where D δ f(x) = f(δx). Kernel odd H anticommutes with reflections (Hf) = H( f) where f(x) := f( x). A linear and bounded operator T in L 2 (R) that commutes with translations, dilations, and anticommutes with reflections must be a constant multiple of the Hilbert transform: T = ch. Using this principle, (Stefanie Petermichl 2000) showed that we can write H as a suitable average of dyadic operators. Cristina Pereyra (UNM) 13 / 64

14 Lecture 1 Dyadic harmonic analysis on R Dyadic intervals Definition The standard dyadic intervals D is the collection of intervals of the form [k2 j, (k + 1)2 j ), for all integers k, j Z. They are organized by generations: D = j Z D j, where I D j iff I = 2 j. Each generation is a partition of R. They satisfy Properties Nested: I, J D then I J =, I J, or J I. One parent: if I D j then there is a unique interval Ĩ D j 1 (the parent) such that I Ĩ, and Ĩ = 2 I. Two children: There are exactly two disjoint intervals I r, I l D j+1 (the right and left children), such that I = I r I l, and I = 2 I r = 2 I l. Cristina Pereyra (UNM) 14 / 64

15 Lecture 1 Dyadic harmonic analysis on R Random dyadic grids on R Definition A dyadic grid in R is a collection of intervals, organized in generations, each of them being a partition of R, that have the nestedness and two children per interval properties. For example, the shifted and rescaled regular dyadic grid will be a dyadic grid. However these are not all possible dyadic grids. The following parametrization will capture all dyadic grids on R. Lemma For each scaling or dilation parameter r with 1 r < 2, and the random parameter β with β = {β i } i Z, β i = 0, 1, let x j = i< j β i2 i, the collection of intervals D r,β = j Z D r,β j is a dyadic grid. Where D r,β j := rd β j, and Dβ j := x j + D j. Cristina Pereyra (UNM) 15 / 64

16 Lecture 1 Dyadic harmonic analysis on R Random dyadic grids were introduced by Nazarov, Treil and Volberg in their study of CZ singular integrals on non-homogeneous spaces [NTV 2003], utilized by Hytönen in his representation theorem [Hytonen 2012]. The advantage of this parametrization is that there is a very natural probability space, say (Ω, P) associated to the parameters, and averaging here means calculating the expectation in this probability space, that is E Ω f = Ω f(ω) dp(ω). Cristina Pereyra (UNM) 16 / 64

17 Lecture 1 Dyadic harmonic analysis on R Haar basis Definition Given an interval I, its associated Haar function is defined to be h I (x) := I 1/2( χ Ir (x) χ Il (x) ), where χ I (x) = 1 if x I, zero otherwise. Properties h I 2 = 1, and it has zero integral h I = 0. {h I } I D is a complete orthonormal system in L 2 (R) (Haar 1910). The Haar basis is an unconditional basis in L p (R) for 1 < p <. The Haar basis is an unconditional basis in L p (w) if w A p (Treil-Volberg 1996). Cristina Pereyra (UNM) 17 / 64

18 Lecture 1 Dyadic harmonic analysis on R Unconditional basis Definition A basis {ψ n } n N in a Banach space X is an unconditional basis if for all x X, x = n N a nψ n then for all choices of signs ɛ n = ±1, the series n N ɛ na n ψ n converges in X. Example The trigonometric functions {e 2πinθ } n Z form an unconditional basis in L p (T) only if p = 2. Wavelets are unconditional bases in various functional spaces. We can define an operator, the martingale transform, whose boundedness implies the unconditionality of the Haar basis in L p, p > 1. Cristina Pereyra (UNM) 18 / 64

19 Lecture 1 Dyadic harmonic analysis on R Martingale transform Definition (The Martingale transform) T σ f(x) := I D σ I f, h I h I (x), where σ I = ±1. Martingale transform is a good model for CZ singular operators. Unconditionality of the Haar basis on L p (R) follows from boundedness of T σ on L p (R) with norm independent on the choice of signs, sup T σ f p C p f p. σ Burkholder (1984) found the optimal constant C p. Unconditionality on L p (w) when w A p follows from the boundedness of the martingale transform on L p (w) (Treil-Volberg 1996). Sharp linear bounds on L 2 (w) (Wittwer 2000). Cristina Pereyra (UNM) 19 / 64

20 Lecture 1 Petermichl s dyadic shift operator Petermichl s dyadic shift operator Definition Petermichl s dyadic shift operator X (pronounced Sha ) associated to the standard dyadic grid D is defined for functions f L 2 (R) by Xf(x) := I D f, h I H I (x), where H I = 2 1/2 (h Ir h Il ). X is an isometry on L 2 (R), i.e. Xf 2 = f 2. Notice that Xh I (x) = H I (x). The profiles of h I and H I can be viewed as a localized sine and cosine. First indication that the dyadic shift operator maybe a good dyadic model for the Hilbert transform. More evidence comes from the way X interacts with translations, dilations and reflections. Cristina Pereyra (UNM) 20 / 64

21 Lecture 1 Petermichl s dyadic shift operator Symmetries for Sha Denote by X r,β Petermichl s shift operator associated to the random dyadic grid D r,β. Properties The following symmetries for the family of shift operators {X r,β } (r,β) Ω hold, Translation τ h (X r,β f) = X r,τ h β(τ h f). Dilation D δ (X r,β f) = X Dδ (r,β)(d δ f). Reflection X r,β f = X r, β( f). where β i = 1 β i, τ h β {0, 1} Z, and D δ (r, β) [1, 2) {0, 1} Z = Ω. Cristina Pereyra (UNM) 21 / 64

22 Lecture 1 Petermichl s dyadic shift operator Petermichil s representation theorem for H Each Shift Dyadic Operator does not have the symmetries that characterize the Hilbert transform, but an average over all random dyadic grids does. Theorem (Petermichl 2000) E Ω X r,β = Ω X r,β dp(r, β) = ch, Result follows once one verifies that c 0 (which she did!). Similar representation works for the Beurling-Ahlfors (Petermichl-Volberg 2002), Riesz transforms (Petermichl 2008). There is a representation valid for all Calderón-Zygmund singular integral operators (Hytönen 2010). Cristina Pereyra (UNM) 22 / 64

23 Lecture 1 Petermichl s dyadic shift operator L p -boundedness of the Hilbert Transform Estimates for the Hilbert transform H follow from uniform estimates for Petermichl s shift operators. Theorem (Riesz 1927) Hilbert transform is bounded on L p for 1 < p <. Hf p C p f p. Proof. Suffices to check that sup X r,β f p C p f p. (r,β) Ω Case p = 2 follows from orthonormality of the Haar basis. Cristina Pereyra (UNM) 23 / 64

24 Lecture 1 Petermichl s dyadic shift operator Proof (continuation). First rewrite Petermichl s shift operator in the following manner, where Ĩ is the parent of I in the dyadic grid D r,β, X r,β f = I D r,β 1 2 sgn(i, Ĩ) f, h Ĩ h I, where sgn(i, Ĩ) = 1 if I = Ĩr, and 1 if I = Ĩl. By Plancherel (remember that each parent has two children), X r,β f 2 2 = f, hĩ 2 2 I D r,β = f 2 2. Minkowski Integral Inequality then shows that EX r,β f 2 E X r,β f 2 = f 2. Case p 2 follows from L p estimates for the dyadic square function. Cristina Pereyra (UNM) 24 / 64

25 Lecture 1 Petermichl s dyadic shift operator Why are we interested in these estimates? Boundedness of H on L p (T) implies convergence on L p (T) of the partial Fourier sums. Hf is the boundary value of the harmonic conjugate of the Poisson extension of a function f L p (R). To show that wavelets are unconditional bases on several functional spaces. Boundedness properties of Riesz transforms (singular operators on R d ) have deep connections to PDEs. Boundedness of the Beurling transform (a singular integral operator in R 2 ) on L p (w) for p > 2 and with linear estimates on [w] Ap, has important consequences in the theory of quasiconformal mappings (Astala-Iwaniecz-Saksman 01, Petermichl-Volberg 02). Cristina Pereyra (UNM) 25 / 64

26 Lecture 2 Outline Outline Lecture 2 In this lecture we discuss: Weighted inequalities for the Hilbert transform Linear estimates on L 2 (w) for dyadic operators Estimates in L p (w) via sharp extrapolation theorem Dyadic square function Maximal function Haar shift operators of complexity (m, n) Dyadic paraproducts Boundedness properties for Haar shift operators Hytönen s theorem (A 2 conjecture). Cristina Pereyra (UNM) 26 / 64

27 Lecture 2 Boundedness on weighted L p Boundedness of H on weighted L p Theorem (Hunt-Muckenhoupt-Wheeden 1972) w A p Hf L p (w) C p (w) f L p (w). Dependence of the constant on [w] Ap was found 30 years later. Theorem (Petermichl 2007) Sketch of the proof. Hf L p (w) C p [w] max {1, 1 A p p 1 } f L p (w). For p = 2 suffices to find uniform (on the grids) linear estimates for Petermichl s shift operator on L 2 (w). For p 2 a sharp extrapolation theorem, that we will discuss later, automatically gives the result from the linear estimate on L 2 (w). Cristina Pereyra (UNM) 27 / 64

28 Lecture 2 Chronology of first Linear Estimates on L 2 (w) Chronology of first Linear Estimates on L 2 (w) Maximal function (Steve Buckley 1993) Martingale transform (Janine Wittwer 2000) Dyadic square function (Hukovic,Treil,Volberg; and Wittwer 2000) Beurling transform (Petermichl, Volberg 2002) Hilbert transform (Stephanie Petermichl 2003, published 2007) Riesz transforms (Stephanie Petermichl 2008) Dyadic paraproduct in R (Oleksandra Beznosova 2008) Estimates based on Bellman functions and bilinear Carleson estimates (except for maximal function). The Bellman function method was introduced to harmonic analysis by Nazarov, Treil and Volberg. With their students and collaborators have been able to use this method to obtain a number of astonishing results not only in this area see [VasV, V] and references. Cristina Pereyra (UNM) 28 / 64

29 Lecture 2 Estimates in L p (w) via sharp extrapolation Sharp extrapolation L p (w) inequalities are deduced from linear bounds on L 2 (w): Rubio de Francia extrapolation theorem. Theorem (Sharp Extrapolation Theorem) If for all w A r there is α > 0, and C > 0 such that [T f L r (w) C[w] α A r f L r (w) for all f L r (w). then for each 1 < p < and for all w A p, there is C p,r > 0 r 1 α max {1, [T f L p p 1 (w) C p,r [w] } A p f L p (w) for all f L p (w). Dragicevic-Grafakos-P-Petermichl 05, new proof Duoandikoetxea 11. Example Sharp extrapolation from r = 2, α = 1, is sharp for the martingale, Hilbert, Beurling-Ahlfors and Riesz transforms for all 1 < p <. Cristina Pereyra (UNM) 29 / 64

30 Lecture 2 Estimates in L p (w) via sharp extrapolation Dyadic square function It is sharp for the dyadic square function only when 1 < p 2 ("sharp" DGPPet, "only" Lerner 07). Definition (The dyadic square function) (S d f) 2 (x) := m I f mĩf 2 χ I (x) = f, h I 2 χ I (x), I I D I D where m I f = (1/ I ) I f, Ĩ is the parent of I. S d is an isometry on L 2 (R). S d is bounded on L p (R) for 1 < p < furthermore S d f p f p. This plays the role of Plancherel s theorem in L p (Littlewood-Paley theory). It implies boundedness of T σ (and X) on L p T σ f p S d (T σ f) p = S d f p f p. Cristina Pereyra (UNM) 30 / 64

31 Lecture 2 Estimates in L p (w) via sharp extrapolation Dyadic square function - weighted estimates S d is bounded on L 2 (w) if w A 2 S d f L 2 (w) C[w] A2 f L 2 (w) (HukTV, Witt 2000). Extrapolation will give boundedness on L p and on weighted-l p. S d is bounded on L p (w) if w A p S d f L p (w) C p [w] max{ 1 2, 1 p 1 } A p f L p (w) (CrMPz 2010). This power is optimal. It corresponds to sharp extrapolation starting at r = 3 with square root power instead of starting at r = 2 with linear power. Cristina Pereyra (UNM) 31 / 64

32 Lecture 2 Estimates in L p (w) via sharp extrapolation The Hardy-Littlewood maximal function Definition The Hardy-Littlewood maximal function 1 Mf(x) := sup I x I M is bounded on L p (R) for 1 < p. I f(y) dy. M is not bounded on L 1 (R), but it is of weak-type (1, 1) (Hardy-Littlewood 1930). M is bounded on L p (w) if and only if w A p (Muckenhoupt 1972). Cristina Pereyra (UNM) 32 / 64

33 Lecture 2 Estimates in L p (w) via sharp extrapolation Buckley s Theorem The optimal dependence on the A p -characteristic of the weight was discovered 20 years later. Theorem (Buckley 1993) Let w A p and 1 < p then Mf L p (w) C p [w] 1 p 1 A p f L p (w). This estimate is key in the proof of the sharp extrapolation theorem. Observe that if we start with Buckley s estimate for M on L r (w) then sharp extrapolation will give the right power for all 1 < p r, 1 however for p > r it will simply give r 1 which is bigger than the 1 correct power p 1. Cristina Pereyra (UNM) 33 / 64

34 Lecture 2 Haar shift operators Haar shift operators Definition A Haar shift operator of complexity (m, n) is X m,n f(x) := L D I D m(l),j D n(l) c L I,J f, h I h J (x), where the coefficients c L I,J I J L, and D m (L) denotes the dyadic subintervals of L with length 2 m L. The cancellation property of the Haar functions and the normalization of the coefficients ensures that X m,n f 2 f 2. T σ is a Haar shift operator of complexity (0, 0). X is a Haar shift operator of complexity (0, 1). The dyadic paraproduct π b is not one of these. Cristina Pereyra (UNM) 34 / 64

35 Lecture 2 The dyadic paraproduct The dyadic paraproduct Definition A locally integrable function b BMO d iff for all J D there is a constant C > 0 such that for all J D b(x) m J b 2 dx = b, h I 2 C J. J I D(J) (b BMO d iff { b, h I 2 } I D is a Carleson sequence.) Definition The dyadic paraproduct associated to b BMO d is π b f(x) := I D m I f b, h I h I (x), where m I f = 1 I I f(x) dx = f, χ I/ I. Cristina Pereyra (UNM) 35 / 64

36 Lecture 2 The dyadic paraproduct The adjoint of the dyadic paraproduct Definition The adjoint of the dyadic paraproduct is πb f(x) = f, h I b, h I χ I(x). I I D These are bounded operators in L p (R) (if and only if b BMO d ). Formally, fb = π b f + π b f + π f b. In T (1) theorem: T = T 0 + π T 1 + π T 1. Cristina Pereyra (UNM) 36 / 64

37 Lecture 2 Estimates for Shift Operators Estimates for Shift Operators Michael Lacey, Stefanie Petermichl and Mari Carmen Reguera (2010) proved the A 2 conjecture for the Haar shift operators of arbitrary complexity (with constant depending exponentially in the complexity). They don t use Bellman functions. They use a corona decomposition and a two-weight theorem for well localized operators of Nazarov, Treil and Volberg. Cruz-Uribe, Martell, and Pérez (2010) recover all results for Haar shift operators. No Bellman functions, no two-weight results. Instead they use a local median oscillation introduced by Lerner. The method is very flexible, they get new results such as the sharp bounds for the square function for p > 2, for the dyadic paraproduct, also for vector-valued maximal operators, and two-weight results as well. Dependence on complexity is exponential. Cristina Pereyra (UNM) 37 / 64

38 Lecture 2 The A 2 conjecture (now Theorem) The A 2 conjecture (now Theorem) Theorem (Hytönen 2010) Let 1 < p < and let T be any Calderón-Zygmund singular integral operator in R n, then there is a constant c T,n,p > 0 such that Sketch of the proof. T f L p (w) c T,n,p [w] max{1, 1 A p p 1 } f L p (w). Enough to show p = 2 thanks to sharp extrapolation. Prove a representation theorem in terms of Haar shift operators of arbitrary complexity and paraproducts on random dyadic grids. Prove linear estimates on L 2 (w) with respect to the A 2 characteristic for Haar shift operators and with polynomial dependence on the complexity (independent of the dyadic grid). Cristina Pereyra (UNM) 38 / 64

39 Lecture 2 Hytönen s Representation theorem Hytönen s Representation theorem Theorem (Hytönen s Representation Theorem 2010) Let T be a Calderón-Zygmund singular integral operator, then T f = E Ω a m,n X r,β m,nf + π r,β T 1 f + (πr,β T 1 ) f, (m,n) N 2 with a m,n = e (m+n)α/2, α is the smoothness parameter of T. X r,β m,n are Haar shift operators of complexity (m, n), π r,β T 1 a dyadic paraproduct, (π r,β T 1 ) the adjoint of the dyadic paraproduct, All defined on random dyadic grid D r,β. Cristina Pereyra (UNM) 39 / 64

40 Lecture 2 Recent Progress Recent progress Active area of research! There is a nice survey by Lacey up to Since the appearance of Hytönen s theorem several simplifications of the argument have appeared [HytPzTV,NV,T,L2,T,Hyt ]. There are already extension to metric spaces with geometric doubling condition [NRezV 2011], and to the solution of the two weight bump conjecture [NRTV, CrRV, HytPz, Lerner ]. Also mixed A p A estimates. Different attempts to get rid of one or more components of the proofs: randomness, Bellman functions, Haar shift operators [LHyt, HytLPz, Le ]. Cristina Pereyra (UNM) 40 / 64

41 Lecture 3 Outline Outline Lecture 3 In this lecture we discuss: Ingredients in the proof of the A 2 conjecture for the dyadic paraproduct. Weighted or disbalanced Haar basis. Carleson and w-carleson sequences. The Little Lemma The α Lemma Weighted maximal function. Weighted Carleson Lemma. The Proof Cristina Pereyra (UNM) 41 / 64

42 Lecture 3 Weighted or disbalanced Haar basis Weighted or disbalanced Haar basis Definition Given weight v and interval I, the weighted Haar function h v I is h v I(x) := 1 v(i ) v(i) v(i + ) χ I + (x) 1 v(i + ) v(i) v(i + ) χ I (x). {h v I } I D is an orthonormal system in L 2 (v). There exist sequences α v I, βv I such that h I (x) = α v Ih v I(x) + β v I χ I (x) I (i) αi v m I v, (ii) βi v I v m I v, and Iv := m I+ v m I v. Cristina Pereyra (UNM) 42 / 64

43 Lecture 3 Carleson and w-carleson sequences Carleson and w-carleson sequences Definition Let λ I, σ I > 0, and v a weight. {λ I } I D is a Carleson sequence with intensity Q > 0, if I D(J) λ I Q J, for all J D. {σ I } I D is a v-carleson sequence with intensity Q > 0, if I D(J) σ I Qv(J) for all J D. Example b BMO d, then { b, h I 2 } I D is Carleson with intensity b 2 BMO d. Cristina Pereyra (UNM) 43 / 64

44 Lecture 3 Carleson and w-carleson sequences Beznosova s Little Lemma To create v-carleson sequences from a given Carleson sequences we have the following lemma. Lemma (Beznosova 2008) Let v be a weight, such that v 1 is a a weight as well. Let {λ I } I D be a Carleson sequence with intensity Q, then for all J D λ I I D(J) λ I 4Q v(j). m I v 1 "The sequence { m I } v 1 I D is a v-carleson sequence with intensity 4Q." The proof uses a Bellman function argument. Example ( Sigma1 ) b BMO d, b I = b, h I then {b 2 I /m Iv 1 } I D is a v-carleson sequence. Cristina Pereyra (UNM) 44 / 64

45 Lecture 3 Carleson and w-carleson sequences Algebra of Carleson sequences Lemma Given a weight v. Let λ I and γ I be two v-carleson sequences with intensities A and B respectively then for any c, d > 0 cλ I + dγ I is a v-carleson sequence with intensity at most ca + db. λi γ I is a v-carleson sequence with intensity at most AB. The proof is a simple exercise. Sigma2 Cristina Pereyra (UNM) 45 / 64

46 Lecture 3 Carleson and w-carleson sequences The α-lemma Lemma (Beznosova 2008) If w A 2 and 0 < α < 1/2, then the sequence ( µ I := (m I w) α (m I w 1 ) α I w 2 I m 2 I w + Iw 1 2 ) m 2 I w 1 I D is a Carleson sequence with Carleson intensity at most C α [w] α A 2 α (0, 1/2). for any By α- Lemma, and algebra of Carleson sequences Sigma2, if w A d 2 : {ν I := I w 1 2 (m I w) 2 I } I D is Carleson with intensity [w] 2 A 2. If b BMO d, and b I := b, h I, then {b I νi } I D is Carleson with B = C[w] A2 b BMO. Cristina Pereyra (UNM) 46 / 64

47 Lecture 3 Weighted maximal function Weighted maximal function Definition The weighted dyadic maximal function Mv d, v a weight, is defined by Mv d 1 f(x) := f(y) v(y) dy. v(i) Lemma sup I:x I D For all f L p (v) there is a constant C(p) > 0 such that M d v f L p (v) C(p) f L p (v). Moreover C(p) p at least for p near 1. Let I be a dyadic interval, then m I ( f v) m I v I inf x I M d v f(x). Cristina Pereyra (UNM) 47 / 64

48 Lecture 3 Weighted Carleson Lemma Weighted Carleson Lemma Lemma If v is a weight, {σ L } L D is a v-carleson sequence with intensity Q, and F is a positive measurable function on R, then σ L inf F (x) Q F (x)v(x) dx. x L L D In applications F involves the dyadic weighted maximal function M d v F (x) = [M d v f(x)] 2 : the RHS becomes M d v f 2 L 2 (v) C f 2 L 2 (v). F (x) = [Mv d f p (x)] 2/p : the RHS becomes Mv d f p 2/p, this is L 2/p (v) useful if 1 < p < 2 so that 2/p > 1. In that case: Sigma1 M d v f p 2/p L 2/p (v) c(2/p) f p 2/p L 2/p (v) = c 2 2 p f 2 L 2 (v). Sigma2 Cristina Pereyra (UNM) 48 / 64 R

49 Lecture 3 A 2 conjecture for dyadic paraproduct A 2 conjecture for the dyadic paraproduct We present a proof of Beznosova s theorem, for b BMO d, w A d 2, The proof uses π b f L 2 (w) C b BMO d[w] A d 2 f L 2 (w). Same ingredients introduced by Oleksandra. An argument by Nazarov and Volberg that yields polynomial in the complexity bounds for Haar shift operators. In joint work with Jean Moraes (2011) we extended their result to paraproducts with arbitrary complexity. Suffices by duality to prove: π b (fw), gw 1 C b BMO d[w] A2 f L 2 (w) g L 2 (w 1 ) Cristina Pereyra (UNM) 49 / 64

50 Lecture 3 A 2 conjecture for dyadic paraproduct Proof of A 2 conjecture for dyadic paraproduct π b (fw), gw 1 I D b I m I ( f w) gw 1, h I Σ 1 + Σ 2, where we replace h I = αi w 1 h w 1 I + β w 1 I χ I I, to get the two sums: Σ 1 := I D Σ 2 := I D b I m I ( f w) gw 1, h w 1 I m I w 1 b I m I ( f w)m I ( g w 1 ) Iw 1 m I w 1 I Sigma1 Sigma2 Cristina Pereyra (UNM) 50 / 64

51 Lecture 3 A 2 conjecture for dyadic paraproduct First sum proof Σ 1 b I m I ( f w) g, I D mi w 1 hw 1 I m I w w 1 m I w m I w 1 b I [w] A2 inf I D mi w M wf(x) g, h w 1 1 I w 1 x I ( ) b I 2 1 ( ) 1 2 [w] A2 inf m I w 1 M wf(x) 2 g, hi w 1 w x I I D I D Use Weighted Carleson Lemma with F (x) = Mwf(x) 2 and v = w, and w-carleson sequence b 2 I /m Iw 1 by Little Lemma. ( ) 1 Σ 1 [w] A2 b BMO d Mwf(x)w(x)dx 2 2 g L 2 (w 1 ) R C[w] A2 b BMO d f L 2 (w) g L 2 (w 1 ) Cristina Pereyra (UNM) 51 / 64

52 Lecture 3 A 2 conjecture for dyadic paraproduct Second sum proof Using similar arguments that we used for Σ 1 Σ 2 I D I D b I m I( f w) m I ( g w 1 ) I m I w m I w 1 w 1 2 (m I w) 2 I b I ν I inf x I M wf(x) M w 1g(x), where b I 2 and ν I are Carleson sequences with intensities b 2 and BMO d [w] 2 Alpha Lemma A 2 then by algebra CS the sequence bi ν I is Carleson sequence with intensity b BMO d[w] A2. Using Weighted Carleson Lemma with F (x) = M w f(x) M w 1g(x), v = 1, Σ 2 [w] A2 b BMO d M w f(x)m w 1g(x)dx. R Cristina Pereyra (UNM) 52 / 64

53 Lecture 3 A 2 conjecture for dyadic paraproduct To finish use Cauchy-Schwarz and w 1/2 (x) w 1/2 (x) = 1, Σ 2 [w] A2 b BMO d M w f(x)m w 1g(x)dx R ( ) 1 [w] A2 b BMO d Mwf(x)w(x)dx 2 R = [w] A2 b BMO d M w f L 2 (w) M w 1g L 2 (w 1 ) C[w] A2 b BMO d f L 2 (w) g L 2 (w 1 ). ( 2 Mw 2 g(x)w 1 (x)dx 1 R ) 1 2 We are done!! Cristina Pereyra (UNM) 53 / 64

54 Lecture 4 Outline Outline Lecture 4 Bellman function proof of the Little Lemma Induction on scales The Bellman function Bellman function proof of the α Lemma Proof of the Weighted Carleson Lemma Cristina Pereyra (UNM) 54 / 64

55 Lecture 4 Outline Beznosova s Little Lemma Lemma (Beznosova 2008) Let v be a weight, such that v 1 is a a weight as well. Let {λ I } I D be a Carleson sequence with intensity Q, then for all J D λ I I D(J) λ I 4Q v(j). m I v 1 "The sequence { m I } v 1 I D is a v-carleson sequence with intensity 4Q." The proof uses a Bellman function argument, which we now describe. Cristina Pereyra (UNM) 55 / 64

56 Lecture 4 Proof of the Little Lemma Proof of the Little Lemma The first lemma encodes what now is called an induction on scales argument. If we can find a Bellman function with certain properties, then we will solve our problem by induction on scales. Lemma (Induction on scales) Suppose there exists a real valued function of 3 variables B(x) = B(u, v, l), whose domain D contains points x = (u, v, l) D = {(u, v, l) R 3 : u, v > 0, uv 1 and 0 l 1}, whose range is given by 0 B(x) u, and such that the following convexity property holds: for all x ± D such that x x ++x 2 = (0, 0, α) we have B(x) B(x +) + B(x ) 1 2 4v α. Then the Little Lemma holds. Cristina Pereyra (UNM) 56 / 64

57 Lecture 4 Proof of the Little Lemma Induction on scales Proof. Fix a dyadic interval J. Let u J = m J w, v J = m J (w 1 ) and l J = 1 J Q I D(J) λ I, then x J := (u J, v J, l J ) D. Let x ± := x J ± D. x J x J + + x J 2 = (0, 0, α J ), where α J := 1 J Q λ J. Then, by the size and convexity conditions, and J + = J = J /2, J m J w J B(x J ) J + B(x J +) + J B(x J ) + λ J 4Qm J (w 1 ). Repeat for J + B(x J +) and J B(x J ), use that B 0 on D to get: m J w 1 4 J Q I D(J) λ I m I (w 1 ). Cristina Pereyra (UNM) 57 / 64

58 Lecture 4 Proof of the Little Lemma The Bellman function Lemma (Beznosova 2008) The function B(u, v, l) := u 1 v(1 + l) is defined on D, 0 B(x) u for all x = (u, v, l) D and on D: ( B/ l)(u, v, l) 1/(4v), (du, dv, dl) d 2 B(u, v, l) (du, dv, dl) t 0, where d 2 B(u, v, l) denotes the Hessian matrix of the function B evaluated at (u, v, l). Moreover, these imply the dyadic convexity condition B(x) B(x +)+B(x ) 2 α/(4v). Cristina Pereyra (UNM) 58 / 64

59 Lecture 4 Proof of the Little Lemma Differential convexity implies dyadic convexity Proof. Differential conditions can be check by direct calculation. By the Mean Value Theorem and some calculus, B(x) B(x +) + B(x ) 2 = B l (u, v, l )α 1 1 (1 t )b (t)dt v α. where b(t) := B(x(t)), x(t) := 1 + t 2 x t 2 x, 1 t 1. Note that x(t) D whenever x + and x do, since D is a convex domain and x(t) is a point on the line segment between x + and x, and l is a point between l and l ++l 2. Cristina Pereyra (UNM) 59 / 64

60 Lecture 4 Proof Alpha Lemma Proof Alpha Lemma Beznosova Use the Bellman function method. Figure out the domain, range and convexity conditions needed to run an induction on scale arguments that will yield the inequality. Verify that the Bellman function B(u, v) = (uv) α satisfies those conditions (or at least a differential version). Cristina Pereyra (UNM) 60 / 64

61 Lecture 4 Weighted Carleson Lemma Weighted Carleson Lemma Lemma Let v be a weight, {α L } L D a v-carleson sequence with intensity Q, and F a positive measurable function on R, then α L inf F (x) Q F (x)v(x) dx. x L Proof. L D Assume that F L 1 (v) otherwise the first statement is automatically true. Setting γ L = inf F (x), we can write x L L D γ L α L = L D 0 R χ(l, t) dt α L = 0 ( χ(l, t) α L )dt, L D where χ(l, t) = 1 for t < γ L and zero otherwise, and by the MCT. Cristina Pereyra (UNM) 61 / 64

62 Lecture 4 Weighted Carleson Lemma Proof Weighted Carleson Lemma Proof (continuation). Define E t = {x R : F (x) > t}. Since F is assumed a v-measurable function then E t is a v-measurable set for every t. Since F L 1 (v) we have, by Chebychev s inequality, that the v-measure of E t is finite for all real t. Moreover, there is a collection of maximal disjoint dyadic intervals P t that will cover E t except for at most a set of v-measure zero. L E t if and only if χ(l, t) = 1. All together we can rewrite the integrand in previous page as χ(l, t)α L = α L α I B v(l) = Bv(E t ), L D L E t L P t L P t I D(L) Cristina Pereyra (UNM) 62 / 64

63 Lecture 4 Weighted Carleson Lemma Proof Weighted Carleson Lemma Proof (continuation). χ(l, t)α L = L D L E t α L L P t I D(L) α I B L P t v(l) = Bv(E t ), we used in the second inequality the fact that {α J } I D is a v-carleson sequence with intensity B. Thus we can estimate γ L α L B v(e t )dt = B F (x) v(x) dx. 0 R L D where the last equality follows from the layer cake representation. Cristina Pereyra (UNM) 63 / 64

64 Lecture 4 Weighted Carleson Lemma Thanks ;-) ;-) Thanks for your patience!!!! Cristina Pereyra (UNM) 64 / 64