Lecture notes on singular integrals, projections, multipliers and rearrangements. Johannes Kepler University Linz

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1 Lecture notes on singular integrals, projections, multipliers and rearrangements Paul F.X. Müller Johanna Penteker Johannes Kepler University Linz

2 Contents Preface ii 1 Singular integral operators: Basic examples Hilbert transformation The Riesz transforms Beurling transformation Cauchy integrals on Lipschitz domains Double layer potential Fourier multipliers Atomic decomposition and multiplication operators 13.1 Dyadic intervals BMO and dyadic Hardy spaces Atomic decomposition Multiplication operators Rearrangement operators Know your basis and its rearrangement operators Haar rearrangements Postorder rearrangements of the Haar system Singular integral operators: Central estimates The theorem of David and Journé Operators on L p (L 1 ) Reflexive subspaces of L 1 and vector valued SIOs Haar projections Interpolatory estimates Interpolatory estimates are useful Bibliography 46 i

3 Preface In March 011, Michal Wojciechowski (IMPAN Warszawa) organized a winter school in Harmonic Analysis at the Bedlewo Conference Center of the Institute for Mathematics PAN. The lectures at the winter school were primarily addressed at graduate students of Warsaw University who had significant knowledge in Harmonic Analysis. The programme consisted of four mini-courses covering the following subjects - Non commutative harmonic analysis (M. Bozeko, Wroclaw) - The Heisenberg group (J. Dziubanski, Wroclaw ) - The Banach space approximation property (A. Szankowski, Jerusalem) - Singular integral operators (P. Müller, Linz) With considerable delay - for which the first named author assumes responsibility - the present lecture notes grew out of the minicourse on singular integral operators. Our notes reflect the original selection of classical and recent topics presented at the winter school. These were, - H p atomic decomposition and absolutely summing operators - Rearrangement operators, interpolation, extrapolation and boundedness - Calderón-Zygmund operators on L p (L q ), 1 < p, q <. - Calderón-Zygmund operators on L p (L 1 ), 1 < p <. - Interpolatory estimates, compensated compactness, Haar projections. In the present notes we worked out Figiel s proof of the David and Journé theorem, Kislyakov s proof of Bourgain s theorem on operators in L p (L 1 ), the proof of Kislyakov s embedding theorem, and our recent, explicit formulas for the Pietsch measure of absolutely summing multiplication operators. Where appropriate, we emphasized the role played by dyadic tools, such as averaging projections, rearrangements and multipliers acting on the Haar system. The authors acknowledge financial support during the preparation of these notes provided by the Austrian Science Foundation under Project Nr. P549, P3987. Linz, July 015 Paul F. X. Müller Johanna Penteker ii

4 1 Singular integral operators: Basic examples We start these notes with a short list of important singular integral operators, together with brief remarks concerning their usage and motivation. The classic texts by L. Ahlfors [Ahl66], M. Christ [Chr90] R. Coifman and Y. Meyer [CM78], V.P. Khavin and N. K. Nikolski [KN9], P. Koosis [Koo98], E. M. Stein [Ste70] or A. Zygmund [Zyg77], provide suitable references to the material reviewed in this section. The exception to this rule is the last paragraph on Fourier multipliers where we discuss the comparatively recent Marcinkiewicz decomposition of the Banach couple (H p (T), SL H (T)), for which the reference is [JM04]. 1.1 Hilbert transformation Here we recall the Hilbert transform, its basic L p estimates and the link to the convergence of the partial sums of Fourier series. Let f L (] π, π[). The Hilbert Transform H(f) of f is defined by H(f)(θ) = lim ɛ 0 { t >ɛ} f(θ t) dt tan t π, where we take f(θ + π) = f(θ). The integral on the right-hand side exists a.e. f L 1 (] π, π[) and defines a construction on L (] π, π[), if H(f) L f L. (See P. Koosis [Koo98]) Motivation We recall the use of the Hilbert transform in the study of holomorphic functions in the unit disk, and cite the book of Koosis [Koo98] as a classic introductory text. Let g be defined on the unit circle by g(e iθ ) = f(θ) with Fourier expansion g(e iθ ) = n= A n e inθ 1

5 1 Singular integral operators: Basic examples and harmonic extension to the unit disk given by u(re iθ ) = The harmonic conjugate of u is then since ũ(re iθ ) = n= n= u(re iθ ) + iũ(re iθ ) = A 0 + is analytic. By Fatou s theorem the L function g(e iθ ) = is the a.e. boundary value of ũ(re iθ ), i.e. and r n A n e inθ, r < 1. isign(n)r n A n e inθ, r < 1, n= r n A n e inθ n=1 isign(n)a n e inθ, g(e iθ ) = lim r 1 ũ(re iθ ) for a.e θ, g(e iθ ) = lim ɛ 0 { t >ɛ} f(θ t) dt tan t π = H(f). Consequently Plancherel s theorem gives L estimates for the Hilbert transform Hf L f L. (1.1) Inequality (1.1) continues to hold true in the reflexive L p spaces. Theorem 1.1 (M. Riesz). Let f L p (] π, π[), 1 < p <. Then (taking f(θ + π) = f(θ)) f(θ t) dt H(f)(θ) = lim ɛ 0 { t >ɛ} tan t π exists a.e. and where C p p /(p 1). Hf L p C p f L p, By algebraic manipulations the Theorem of Riesz implies the norm-convergence of Fourier series in L p. Let f L p (] π, π[), 1 < p < with formal Fourier series n= A n e inθ.

6 1 Singular integral operators: Basic examples Let and N S N (f)(θ) = A n e inθ n= N P (f)(θ) = 1 ( Id + ih)f(θ) + 1 A 0. By the above computation we have P (f)(θ) = The partial sums S N (f) coincide with A n e inθ. n=0 e inθ P (e in( ) f( )) e i(n+1)θ P (e i(n+1)( ) f( )). This identity links the Hilbert transform to partial sums of Fourier series. By the Riesz theorem we have convergence of the partial sums S N (f) in L p, 1 < p <, S N (f) f L p 0 for N, and C p p /(p 1). S N (f) L p 4 Hf L p C p f L p, (1 < p < ), 1. The Riesz transforms We recall the Riesz transforms, their definition as principal value integral, the L p norm estimate and basic identities relating Riesz transforms to the Laplacian and we work out the identities relating Riesz transforms to the Helmholtz projection onto gradient vector fields The Riesz transforms Let f L p (R n ), 1 p <. The Riesz transform of f is defined by with c n = Γ( n+1 ) π (n+1)/. R j (f)(x) = c n lim ɛ 0 t >ɛ t j f(x t) dt, j = 1,..., n, (1.) t n+1 Theorem 1. (M. Riesz). The integral on the right-hand side in (1.) exists a.e. and defines a bounded linear operator for which R j p C p, 1 < p <. 3

7 1 Singular integral operators: Basic examples Let F denote the Fourier transform normalized so that Ff(ξ) := 1 e i x,ξ f(x)dx. (π) n R n We have then the following identities identifying the Fourier multiplier of Riesz transform. F( j f) = iξ j F(f) (1.3) R j (f))(x) = if 1 ( y j F(f)(.))(y) (1.4) y Note that (1.3) is a direct consequence of the Fourier transform. Equation (1.4) however, is not at all trivial, its verification requires a delicate calculation carried out for instance in the book of Stein [Ste70]. Motivation Standard applications of Riesz transforms, their relation to the Fourier transform and its L p estimates pertain to regularity of solutions to the Poisson problem and to the Helmholtz projection onto gradient vector fields. Poisson problems. We start with the regularity estimates for solutions of the Poisson problem following Stein [Ste70]. Let u Cc (R n ) and We show that u(x) = f(x). f L p = u x j x i L p. (1.5) The Fourier Transformation F provides the link to Riesz transforms: F(f)(y) = F( u)(y) = y F(u)(y) F(u xj,x k )(y) = y j y k F(u)(y). Hence, On the other hand Summarizing we have and therefore, F(u xj,x k )(y) = y jy k y F(f)(y). F(R j (f))(y) = i y j y F(f)(y). u xj,x k = R j (R k (f)) u x j x i R j p R k p f p = Cp f p. p 4

8 1 Singular integral operators: Basic examples Gradient vector fields. Here we show how to identify the Helmholtz projections using Riesz transforms. In [Bou9] this identity was the starting point for the proof that the Banach couple of Sobolev spaces (W 1,, W 1,1 ) is K closed in (L, L 1 ). Let S p = {(f x1,..., f xn ) f W 1,p (R n )}. Then and especially S p L p... L p = L p S L... L = L. Since S is a closed subspace there exists a projection P : L S, P = P, onto S The next result gives the singular integral representation of the projection P. Theorem 1.3. Let f = (f 1,..., f n ) L. Then ( ( n )) n P (f) = R i R j f j j=1 where R k is the Riesz transform. Proof. Use the identity P (f) = (F 1 ( to show first that the image of P is in S : ξ i n j=1 ξ j ξ F(f j) f L : P (f) S, i=1, )) n i=1 i.e. If we take h W 1, (R n ) : P (f) = (h x1,..., h xn ). h = 1F 1 ( n j=1 ξ j ξ Ff j ), which is in W 1, (R n ) as f j L (R n ), then the following identity holds (P (f)) i = 1F 1 (ξ i Fh) = h xi. Therefore, P (f) = (h x1,..., h xn ), i.e. the image of f is the gradient of an element of W 1, (R n ). Next we show that g S : P (g) = g. 5

9 1 Singular integral operators: Basic examples Take any g S. Then, by definition, there is f W 1, (R n ) such that f xi = g i and Hence, we have (P (g)) i = F 1 ( (Ff xi )(ξ) = 1ξ i (Ff)(ξ) = (Fg i )(ξ). ξ i n j=1 ξ j ξ Fg j = 1 F 1 (ξ i Ff) = g i. ) = 1 F 1 ( ξ i n j=1 ξ j ξ ξ jff ) In summary, f L : P (P (f)) = P (f). Divergence free vector fields by Indeed Q is a projection, since Projections onto divergence free vector fields are given Q(f) = f P (f). and its range is divergence free, 1.3 Beurling transformation Q = ( Id P ) = Id P + P = Id P divp (f) = divf. Let f : C C, f Cc 1 (C). We define the Beurling transform as follows Bf(z) = 1 π pv f(w) (z w) dλ(w), C where λ denotes the Lebesgue measure on C. We identify first the the Fourier multiplier of the Beurling transform. View R as C with coordinates z = x + iy. Let ζ = ξ + iη. The Fourier transform of f : C C is written by ˆf(ξ, η) = e πi(ξx+ηy) f(x + iy) dx dy. R Then the following identity holds which can also be written as Bf(ξ, η) = ξ iη ξ + iη ˆf(ξ, η) Bf(ζ) = ζ ζ ˆf(ζ). 6

10 1 Singular integral operators: Basic examples Motivation The Beurling transform is an important tool in obtaining existence and regularity results for the Beltrami equation and for quasi conformal mappings. Let µ L, with µ < 1 with compact support. We want to determine f : C C (with high regularity) that solves the elliptic system f = µ f, where = 1 ( x i y ), and = 1 ( x + i y ). Solution: Try to find f of the form f(z) = z + h(z), and h(z) = 1 π C g(w) (z w) dλ(w). With this setting equation f = µ f translates into conditions on h: h = µ(1 + h). Let h = g. Then h = B(g). Hence, we require for g g = µ(1 + h) = µ(1 + B(g)). This is equivalent to ( Id µb)g = µ. A solution g is given by g(w) = ( Id µb) 1 (µ)(w) = (µb) j (µ)(w). j=1 Beurling [Beu89, Ahl66] proves the essential estimates: B = 1 and lim ɛ 0 B +ɛ = 1. Hence, we get g L 1+ɛ with compact support. h is therefore continuous and tends to 0 at infinity. 1.4 Cauchy integrals on Lipschitz domains Let A : R R be a Lipschitz function so that A <. Put C A f(x) = 1 f(y) πi R (x + ia(x)) (y + ia(y)) dy, 7

11 1 Singular integral operators: Basic examples formally. This is strongly singular for Lipschitz Domains and weakly singular when A : R R is C 1+η, η > 0, see [Chr90]. Why then is it important to have estimates for Lipschitz domains? The answer is two fold: First the only the class of Lipschitz domains is invariant under and also dilations: Put δ r f(x) = f(rx), A r (x) = ra(r 1 x) r > 0. Then C A δ r = δ r C Ar. And now, A r has exactly the same Lipschitz norm as does A, i.e. A r = A. Second if one takes two perfectly smooth domains, such as two half planes, then their intersection is at best a Lipschitz domain. Theorem 1.4 (Coifman-McIntosh-Meyer, [CMM8b]). Let 1 < p <. Then Motivation C A p C p (C 1 + C A ). The main reason for considering the operators C A is their connection to the Cauchy integral on Lipschitz graphs. Let Γ = {x + iy : y = A(x)} C and g L (Γ, ds), where ds denotes the arc length. Let z = x + ia(x). Then the Cauchy integral of g is given by 1 πi Γ g(w) (z w) dw = 1 πi R g(u + ia(u)) (z (u + ia(u))) (1 + ia (u))du = C A G(x), where G(u) = g(u + ia(u))(1 + ia (u)). Thus the L p estimates for C A translate immediately into estimates for Cauchy integrals on Lipschitz graphs. Let E Γ, with E > 0. The L estimate for the Cauchy integral on Lipschitz curves are now a corner stone in the proof of the the fact that E has then positive analytic capacity, i.e. f E : C \ E C : f E < and f E( ) 0. See [Chr90]. 1.5 Double layer potential Let A : R n 1 R be a Lipschitz function and x, y R n 1. We define K(x, y) = A(x) A(y) n 1 j=1 (x j y j ) A y j (y) ( x y + A(x) A(y) ) n/. K is weakly singular if A C 1+δ. It is a strongly singular if A C 1 or A L. Let S A (g)(x) = g(y)k(x, y)dy. R n 1 8

12 1 Singular integral operators: Basic examples Coifman-McIntosh-Meyer [CMM8b, CMM8a] proved that Motivation S A p C p (C 1 + C A ). The solution to the Dirichlet problem by means of double layer potentials provides the main motivation for the integral operators we consider here. Let Ω R n be open, bordered (locally ) by a Lipschitz surface. That is to say that for every x Ω there exists a δ > 0 and a Lipschitz function A: R n 1 R such that (possibly after rotation of the coordinate axes) B(x, δ) Ω = {x B(x, δ) : x n > A(x 1,..., x n 1 )}. (1.6) The method of layer potentials provides solutions for the Dirichlet problem: u = 0 u Ω = f, on Ω (1.7) where f is prescribed on Ω. Let y Ω and ν(y) the exterior unit normal to Ω at y. We set N(x) = 1 c n x n, c n = (n )ω n, where ω n = π n Γ( n ) is the volume of the surface of the unit ball in Rn. Then N(x) is the fundamental solution for the Laplacian in R n, n 3. One does not know the Green s function for Ω, so one hopes that ν(z), z N(x z) (1.8) is a good approximation, as is the case for smooth domains. Note that up to a constant factor K(x, y) = ν(z), y N(x z), where z = (y, A(y)), y R n 1, cf. [Tor86]. To x Ω and f form the harmonic function u f (x) = ν(y), y N(x y) f(y)dσ(y), Ω where σ denotes the surface measure on Ω. This is called the double layer potential of the function f, cf. [Ver84]. According to a theorem of [Ver84] boundary values of u f exist for a.e. p Ω and ( ) 1 lim u f(x) = x p Id + S A f(p). More precisely, if p = (y, A(y)), y R n 1 then by [Ver84, Theorem 1.10], 9

13 1 Singular integral operators: Basic examples lim u f(x) = 1 x p Ω f(p) + ν(z), y N(p z) f(y)dσ(z) = 1 f(p) + R n 1 f((y, A(y))K(p, y)d(y) = ( Id + S A )(f)(p). In summary, u = u f is a solution to the Dirichlet problem u = 0 u Ω = g, on Ω (1.9) where g = (Id + S A )(f)(p). In order to get a solution for (1.7), by this procedure we would like to know that the operator (Id + S A ) is invertible. Should that be the case we put h = (Id + S A ) 1 (f), and observe that u = u h solves the Dirichlet problem with boundary values (Id + S A )(h) = (Id + S A )(Id + S A ) 1 (f) = f. It remains to point out Verchota proved in [Ver84] that ( Id + S A ) is invertible whenever A is a Lipschitz function. 1.6 Fourier multipliers A considerable portion of present day Harmonic Analysis, Probability and PDE has its origin in the Fourier multiplier theorems of Littlewood-Paley. The classical reference to Fourier multipliers is Volume II in Zygmund [Zyg77]. Let u L p ([0, π]) with Fourier series u(θ) = a k cos kθ + b k sin kθ. We form the dyadic blocks and define the transform n (u)(θ) = v(θ) = k=1 n+1 1 k= n a k cos kθ + b k sin kθ ɛ n n (u)(θ), ɛ n { 1, 1}. n=0 Theorem 1.5 (Littlewood-Paley). For every 1 < p < there exists a constant C p > 0 so that for all ɛ n { 1, 1}, v L p C p u L p, and C p for p or p 1. Use [Ste70] and [Zyg77] for reference. We next turn to describing one of the most useful tools in treating Fourier multiplier problems. 10

14 Littlewood-Paley Function For z D, let 1 Singular integral operators: Basic examples P θ (z) = 1 z e iθ z denote the Poisson kernel for the unit disk. Let u(z), z D denote the harmonic extension of u L p ([0, π]) obtained by integration against the Poisson kernel P θ, i.e. u(z) = 1 π u(e iθ )P θ (z)dθ π 0 The Littlewood Paley function gd(u)(θ) = D u(z) log 1 z P θ(z)da(z) plays a central role in proving the multiplier theorem (Theorem 1.5). Its proof (see [Ste70, p.96, p.106] or [Zyg77] for reference) consists of basically two independent components 1. Pointwise estimates between the g functions g D (v)(θ) Cg D (u)(θ),. L p -integral estimates (see again [Ste70], [Zyg77] or [Bañ86]), c p v L p C p g D (v) L p C p v L p, p <. Uniformly bounded Littlewood Paley functions SL (T) denotes the space of all functions u with uniformly bounded Littlewood Paley Function, i.e. SL (T) = {u : T R (or C) : g D L (T)}, with norm u SL (T) = g D (u) + d dm. If we consider the definition of the Littlewood Paley function g D, then we see that the condition g D (u) < controls the growth of u and also its oscillations. Chang, Wilson, Wolff [CWW85] proved that there exists c > 0 so that π 0 exp(cu (θ))dθ g D (u) <. Hence, by [Bañ86, Theorem 1] we have SL BMO. On the other hand there exists E [0, π[ so that g D(1 E ) =, 11

15 1 Singular integral operators: Basic examples i.e. there exists functions 1 E L such that g D (u) / L. We get two different endpoints of the L p scale. L L p BMO { L SL (T) The relation of the endpoint SL to the L p scale is clarified by a Marcinkiewicz decomposition and by pointwise multipliers with values in SL (T). Multipliers into SL (T) and Marcinkiewicz-decomposition Definition 1.6. Let 1 p. Define the subspace H p (T) to consist of those f L p (T) with Fourier expansion of the following form f(θ) = c n e inθ. The harmonic expansion of f H p (T) to the disk D is then given by and hence, for z D, f(re iθ ) = f(z) = n=0 c n r n e inθ n=0 c n z n. n=0 Thus demanding that f H p (T) is the same as demanding that f L p (T) and that the harmonic extension of f to the unit disc is analytic. The next theorem gives the Marcinkiewicz decomposition for H p (T) spaces with SL (T) H (T) as endpoint. Theorem 1.7 ([JM04]). To f H p (T) and λ > 0 there exists g SL (T) H (T) so that g SL + g C 0 λ, f g 1 λ 1 p f p p Non trivial pointwise multipliers. Theorem 1.8 ([JM04]). To each E [0, π[ there exists 0 m(θ) 1 so that m1 E SL < C 0 and π 0 m1 E dθ E /. 1

16 Atomic decomposition and multiplication operators In this section we review Hardy spaces and BMO. We present Fefferman s inequality and the atomic decomposition in the dyadic setting, using the Haar system as our basic tool. Given u H p with Haar expansion u = I D x Ih I we find explicit weights w = (ω I ) such that ( ) 1 x I ϕ I h I C p u H p ϕ I ω I, H p I D whenever (ϕ I ) is a bounded sequence of scalars. This recent application of the atomic decomposition is motivated by problems on absolutely summing operators acting on l..1 Dyadic intervals Definition.1. An interval I [0, 1] is called a dyadic interval, if there exist nonnegative integers l and k with 0 k l 1 such that [ k I = I l,k =, k + 1 [. l l The length of a dyadic interval I l,k is given by I l,k = l. We denote by D the set of dyadic intervals, i.e. D = {I [0, 1] : I is dyadic interval}. For every N N 0 let D N be the set of dyadic intervals with length greater than or equal to N, i.e. D N = {I D : I N } (.1) or equivalently Carleson constant I D D N = {I l,k : 0 l N, 0 k l 1}. (.) Let C D. We define the Carleson constant of C as follows 1 C = sup J. (.3) I C I J I,J C If C is non-empty, then C 1, otherwise C = 0. 13

17 Atomic decomposition and multiplication operators Blocks of dyadic intervals Let L be a collection of dyadic intervals. We say that C(I) L is a block of dyadic intervals in L if the following conditions hold: 1. The collection C(I) has a unique maximal interval, namely the interval I.. If J C(I) and K L, then The Haar system J K I implies K C(I). We define the L -normalised Haar system (h I ) I D indexed by the dyadic intervals D as follows: 1 on the left half of I, h I = 1 on the right half of I, 0 otherwise.. BMO and dyadic Hardy spaces Let (x I ) I D be a real sequence. We define f = (x I ) I D to be the real vector indexed by the dyadic intervals. The space BMO consist of vectors f = (x I ) I D for which ( 1 f BMO = sup I D I J I We define the square function of f as follows I D ) 1 x J J <. (.4) ( ) 1 S(f)(t) = x I 1 I (t), t [0, 1]. (.5) The space H p, 0 < p <, consists of vectors f = (x I ) I D for which f H p = S(f) L p ([0,1]) <. (.6) For 1 p <, (.6) defines a norm and H p is a Banach space. For 0 < p < 1, (.6) defines a quasi-norm and the resulting Hardy spaces H p are quasi-banach spaces, cf.[woj97]. For convenience we identify f = (x I ) I D BMO resp. f = (x I ) I D H p with its formal Haar series f = I D x I h I. 14

18 Atomic decomposition and multiplication operators Paley s theorem ([Pal3], see also [Mül05]) asserts that for all 1 < p < there exists a constant A p such that for all f L p ([0, 1]) given by f = I D x Ih I the following holds 1 A p f L p S(f) L p A p f L p. (.7) This theorem identifies H q as the dual space of H p, where = 1 and 1 < p <. p q Fefferman s inequality ([FS7], see also [Mül05]) fh f H 1 h BMO, (.8) and a theorem to the effect that every continuous linear functional L : H 1 R is necessarily of the form L(f) = fϕ with ϕ BMO L identify BMO as dual space of H 1, (see [FS7],[Gar73],[Mül05]). The Haar support Let f BMO resp. f H p be given by its formal Haar series f = I D x I h I, where (x I ) I D is a real sequence. The set {I D : x I 0} is called Haar support of f..3 Atomic decomposition The following theorem states the decomposition of an element in H p into absolutely summing elements with disjoint Haar support and bounded square function. The decomposition is done by a stopping time argument that may be regarded as a constructive algorithm. The decomposition originates in the work of S. Janson and P.W. Jones [JJ8]. Theorem. (Atomic decomposition). For all 0 < p there exists a constant A p such that for every u H p with Haar expansion u = J D x J h J, there exists an index set N N and a sequence (G i ) i N such that for u i = x J h J, i N J G i of blocks of dyadic intervals the following holds: i) (G i ) i N is a disjoint partition of D. 15

19 Atomic decomposition and multiplication operators ii) I i := J G i J is a dyadic interval iii) u p H u p i p H I p i S(u i ) p A p u p H. (.9) p i N i N The family (u i, G i, I i ) i N is called the atomic decomposition of u H p. Remark.3. The scalar-valued decomposition procedure can be found in [Tor86]. However, note that the left-hand side inequality of (.9) depends only on the fact that (G i ) i N is a sequence of disjoint blocks of dyadic intervals. Therefore, for ϕ = (ϕ I ) I D l (D) we have p p ϕ I x I h I ϕ I x I h I. (.10) I D H p i N I G i H p.4 Multiplication operators Given a Banach space X with an unconditional basis {x n } multiplication operators are certainly among the most natural and important operators to consider. Their action is given by x n µ n x n where (µ k ) l. We investigate here multiplication operators acting on the Haar system in the Hardy spaces H p, 0 < p. We fix u H p with Haar expansion u = x I h I and define M u : l (D) H p by M u (ϕ) = ϕ I x I h I. (.11) I D We frequently use the lattice convention ϕ u = M u (ϕ) (.1) to emphasize that ϕ l (D) is acting as a multiplier on u H p. Since the Haar basis is 1-unconditional in H p we have M u (ϕ) u H p sup ϕ I. (.13) I D Here we show in detail that these estimates are self improving in the following sense: If (.13) holds true then there exists a probability measure ω on D such that ( 1/ M u (ϕ) H p C p u H p ϕ I dω(i)). Expressed in the language of Grothendieck we prove now that every bounded multiplier M u : l (D) H p is already absolutely summing. The starting point of our proof is the atomic decomposition of u H p. It gives us the basic ingredients from which we build explicit formulas defining the probability measure ω. D 16

20 Atomic decomposition and multiplication operators Theorem.4 ([MP15]). Let 0 < p. Let u H p with Haar expansion u = I D x I h I and atomic decomposition (u i, G i, I i ) i N. Then the sequence (ω I ) I D, defined by satisfies ω I = 1 I i 1 p x I I A p u i p u p, I G i, (.14) H p ω I 1 (.15) I D and there exists a constant C p > 0 such that for each ϕ l (D) ( ) 1 x I ϕ I h I C p u H p ϕ I ω I. (.16) H p Proof. From I D u p H p i N I D u i p H p we get the estimate u p H p i N u i p I i 1 p. (.17) We get from that This follows from I i S(u i ) p A p u p H p i N u i p I i 1 p Ap u p H. (.18) p i N u i p I i 1 p Sui p I i. By (.17) and equation.10 in Remark.3 we get for ϕ l (D) p p ϕ I x I h I = ϕ I x I h I I D H p i N I G i H p p ϕ I x I h I I i 1 p i N I G i = p x I h I u i u i p I i 1 p. i N I G i ϕ I (.19) 17

21 Atomic decomposition and multiplication operators With we get I G i ϕ I p ϕ I x I h I I D x I h I u i H p p ( i N ( = I G i ϕ I I G i ϕ I x I u i Applying Hölder s inequality with p + 1 p = 1 to ( i N we get p ϕ I x I h I I D I G i ϕ I H p x I u i ( ϕ I i N I G i I u i p I i 1 p x I u i p x I u i I ) p ) p I u i p I i 1 p. ) p ( ) u i p I i 1 p 1 p. I I i 1 p ) p ( ) u i p I i 1 p 1 p. Applying (.18) to the second term on the right-hand side we obtain the estimate p ϕ I x I h I A 1 p ( p u p(1 p ) H ϕ x ) I p I I D H p i N I G i u i p I I i 1 p p ( = A p u p x ) p I H p u i p u p I I i 1 p. H A p p ϕ I i N I G i i N Recall u i = I G i x I I. (.0) By (.18) and (.0) we obtain for the sequence (ω I ) I D, defined by the following estimate ω I = I i 1 p I x A p u p I H p u i p, I G i, 1 1 I i p I x ω I = I A p u p I D H p i N I G i u i p 1 = A p u p I i 1 p ui p H p 1. i N 18

22 Atomic decomposition and multiplication operators Remark. Finding the explicit formula for the weight (ω I ) I D (and proving the required estimates) was the main problem solved in [MP15]. This paper describes also the mathematical and historical background as well as the connections to Pisier s extrapolation lattices. Remark. The theorem admits extensions to the class of Triebel-Lizorkin spaces and also vector-valued Hardy spaces, cf. [MP15]. 19

23 3 Rearrangement operators 3.1 Know your basis and its rearrangement operators We begin with general observations describing the -potential- use of rearrangement operators acting on an unconditional bases. Let {x n } n= be a unconditional basis in a Banach space E. Its biorthogonal functionals are denoted by {x n} n=. Fix a linear operator T : E E. Let f = x n, f x n and a rearrangement. Then n= U d (x n ) = x n+d T f = n = d = d x n, f T x n x n, f x n+d, T x n x n+d n ( ) U d x n, f x n+d, T x n x n. n Hence, T f C ( d U d sup x n+d, T x n n ) f Suppose we are now given an operator T which is adapted to the basis {x n } n= in the sense that the off diagonal terms in its matrix representation x n+d, T x n decay much faster than U d 1. Then the above estimate estimate provides a bound for T. Thus rearrangement operators control the influence of the off diagonals to the operator norm. In the next section we will see that the above guidelines provide the framework for the proof of the the David and Journé theorem on Calderón-Zygmund operators. There we use rearrangement operators acting on the Haar system {h I / I 1/p : I D} normalized in L p. Recall that the Theorem of R.E.A.C. Paley asserts that {h I / I 1/p : I D} is actually an unconditional basis in L p. 0

24 3. Haar rearrangements 3 Rearrangement operators Here we consider general rearrangement operators of the Haar system defined by an injective map τ : D D. Let 1 < p, and let the rearrangement operator T p be given by T p : h I I 1/p h τ(i) τ(i) 1/p. Clearly T p acts on linear combinations of the Haar systems: If 1/p + 1/q = 1 then T p (f) = h I hτ(i) f, I 1/q τ(i). 1/p I D We give now an intrinsic characterisation of those injections τ for which the the associated rearrangement operators are bounded T p in L p. This is done in several independent steps: First we characterize the case p = with T (h I ) = h τ(i) in BMO. Recall the that we use dyadic BMO norm given by f BMO = sup I D ( 1 I ) 1/ f, h J J 1. J I Theorem 3.1 ([Mül05]). Let τ : D D be injective. T : BMO BMO defined by T (h I ) = h τ(i) satisfies: T BMO [ sup E D ] 1/ τ(e), E where E is the Carleson constant, cf. Section, i.e. E = sup I E J I,J E J I. The rearrangement operator We extend the above characterisation across the L p, (1 < p < ) and H p (0 < p < ) scales using interpolation and extrapolation of rearrangement operators. Note that the dependence on p is not trivial, since obviously we may rewrite the operators as T p (h I ) = h τ(i) I 1/p τ(i) 1/p. Theorem 3. ([Mül05]). Let < p <. Let τ : D D be injective. The rearrangement operator T p : L p L p defined by satisfies: T p : c p T 1/(1 /p) BMO h I I 1 p h τ(i) τ(i) 1 p T p p C p T 1/(1 /p) BMO. 1

25 3 Rearrangement operators Let T be given by the rearrangement τ then its transposed is determined by the inverse τ 1 : τ(d) D. Thus, by Hölder s inequality the extrapolation theorem above contains enough information to characterise the bounded rearrangement operators on L q with 1 < q <, and by Fefferman s inequality (.8) and transposition we have T BMO T 1 1 H 1. Combining our theorems we get therefore the complete description of the bounded rearrangements in H p when 1 p <, and on BMO. However, the spaces H p when 0 < p < 1 cannot be reached by duality arguments. In [GMP05] we use different ideas exploiting again the atomic decomposition for H p spaces to obtain the extrapolation law for rearrangement operators on the scale H p (0 < p < ). Theorem 3.3 ([GMP05]). For all 0 < s < p < there exists a constant c p,s > 0 such that 1 T s s p s H c T p p s H c p,s T p s s s H. s p,s Remark. We saw that rearrangement operators satisfy the same extrapolation and interpolation properties as as singular integral operators. Can that be a coincidence?? The answer is No and the reason for that will become apparent in the section on Calderón-Zygmund operators. 3.3 Postorder rearrangements of the Haar system Before we turn to giving application of rearrangement operators we study the postorder rearrangement in detail. In Computer Science and the Theory of Algorithms its origin dates back, at least, to J. v. Neumann s treatment of the merge-sort algorithm. We refer to [Pen14] for historic information and examples of algorithms based on the postorder rearrangement. For us it is important that the postorder induces the extremal rearrangements of the Haar system acting on finite dimensional BMO spaces. This is the topic we discuss next. Finite dimensional dyadic spaces Let N N 0. We define the finite dimensional BMO space denoted by BMO N and the finite dimensional Hardy spaces H p N, 0 < p < as follows. ( ) BMO N = span {h I : I D N }, BMO, and H p N = ( span {h I : I D N }, H p ), where BMO is given by equation (.4) and H p by equation (.5) and (.6).

26 3 Rearrangement operators Rearrangements on the finite dimensional spaces Recall that D N is the set of dyadic intervals with length greater than or equal to N. Let τ be a bijective map defined on the set D N. On BMO N we study rearrangements of the L -normalised Haar system (h I ) I DN given by the rearrangement operator T τ : h I h τ(i), and on H p N, 0 < p <, rearrangements of the Lp -normalised Haar system given by the rearrangement operator T τ,p : h I I 1 p h τ(i). τ(i) 1 p A standard argument (given below) yields the following norm estimates for rearrangement operators on BMO N (cf. Theorem 3.1): τ(c) 1 sup C D N C 1 non-empty T τ BMO (N + 1) 1. (3.1) Note that the lower bound in (3.1) is always greater than or equal to one. Let x = I D N x I h I. Then T τ x BMO = sup 1 I D N I J I xτ 1 (J) J sup x I D N x BMO D N. I D N Definition (.3) yields D N = N +1. Let C D N be any non-empty collection of dyadic intervals. Let x = I C h I. Then x BMO = C 1 and T τ x BMO = τ(c) 1. Let x = I C x Ih I for some non-empty collection of dyadic intervals C D N. The above argument provides the following rough upper bound T τ x BMO x BMO τ(c) 1. (3.) The adjoint operator of a rearrangement operator is again a rearrangement operator induced by the inverse rearrangement. By the duality of H 1 and BMO we have that the operator T τ on BMO N is the adjoint operator of T τ 1,1 on H 1 N with 1 C F T τ BMON T τ 1,1 H 1 N C F T τ BMON, (3.3) where C F = is the constant appearing in Fefferman s inequality (.8). We will use the following abbreviation for equation 3.3 T τ BMON CF T τ 1,1 H 1 N. (3.4) 3

27 3 Rearrangement operators Remark 3.4. Observe that by Theorem 3. and Theorem 3.3 rearrangement operators T τ,p on H p N, 0 < p <, induced by any bijective map τ acting on D N, have the norm estimate T τ,p H p c p (N + 1) 1 p 1. N Postorder rearrangements Postorder on the set D N See [MS97], [BP05] and [Knu05] Figure 3.1: Postorder on the set D Figure 3.: Lexicographic order on the set D 4. Definition 3.5 (Postorder). Let I, J D N. We say I J if either I and J are disjoint and I is to the left of J, or I is contained in J. 4

28 3 Rearrangement operators We call the postorder on D N. The natural order on the set D N is the lexicographic order, l, on the set {(l, k)}, cf. figure 3.. The postorder on D N, in contrast to the lexicographic order depends on the depth N. The rearrangements We denote by τ N the bijective map on the dyadic intervals that associates to the n th interval in postorder the n th interval in lexicographic order, cf. figure 3.3. This rearrangement is called postorder rearrangement. Its inverse, which associates to the n th interval in lexicographic order the n th interval in postorder, is denoted by σ N. The rearrange- lexicographic order of the dyadic tree D τ 4 σ postorder of the dyadic tree D 4 Figure 3.3: Lexicographic order and postorder of the dyadic tree D 4. Postorder rearrangement τ 4 on D 4 and its inverse σ 4. ments τ N and σ N induce rearrangement operators on BMO N and on the H p N-spaces. On 5

29 3 Rearrangement operators BMO N we consider the rearrangement operators T τn : h I h τn (I) and T σn : h I h σn (I) and obtain the following norm estimates for these rearrangement operators applied to functions with Haar support in the sets and D N l. We use the abbreviations and T N l,0 = {I D N : I I l,0 } M(T N l,0) = span {h I : I T N l,0} M(D N l ) = span {h I : I D N l }. Theorem 3.6 ([Pen14]). Let N N 0 and 0 l N. Let T = T τn M(T N l,0 ) or T = T σn M(DN l ). Then 1 (N l + 1) 1 T BMO (N l + 1) 1. (3.5) This theorem in combination with the general upper bound in (3.1) reveals the extremal nature of the rearrangements τ N and σ N in the sense that for T = T τn resp. T = T σn we have 1 R(BMO N ) T BMO R(BMO N ), where { } R(BMO N ) = sup T τ : BMO N BMO N : τ : D N D N bijective. Obviously, the lower bound in (3.5) is the important one for this result and the statement of Theorem 3.6. The upper bound in (3.5) is the trivial one that originates from the depth (in the sense of dyadic trees) of the sets D N l resp. T N l,0. The duality relation of H 1 N and BMO N, in particular the norm equivalence in equation (3.3), and the interpolation resp. extrapolation statements in Theorem 3. and Theorem 3.3 give equivalent norm estimates as in Theorem 3.6 for the rearrangement operators on H p N, 0 < p <, given by T τn,p : h I I 1 p h τ N (I) τ N (I) 1 p resp. T σn,p : h I I 1 p h σ N (I). σ N (I) 1 p Corollary 3.7. For all 0 < p < there exists a constant C p such that for all N N 0, 0 l N and T = T τn,p or T = T M(T N l,0 ) σ N,p M(DN l the following holds ) 1 p 1 C p (N l + 1) 1 p 1 T H p N C p (N l + 1) 1 p 1. (3.6) 6

30 3 Rearrangement operators Remark 3.8. By the convexification method for the concrete specialisation to Hardy spaces (see [MP15]) one obtains the same result as in Corollary 3.7 for the more general Triebel-Lizorkin spaces. Corollary 3.7 gives, considering the general upper bound in Remark 3.4, the same extremality statement for the rearrangement operators T = T τn,p resp. T = T σn,p on the spaces H p N, 0 < p <. For all 0 < p < there exists a constant B p such that 1 p 1 R N (H p N ) T H p R N (H p N ), N B p } where R N (H p N { T ) = sup τ : H p N Hp N : τ : D N D N bijective. 7

31 4 Singular integral operators: Central estimates We give two independent proofs for the L p estimates for scalar valued singular integral operators. The first one extends to L p (X) when X is a UMD spaces, whereas the second one to operators on L p (L 1 ) with values in L p (L α ) where α < The theorem of David and Journé In this section we discuss the T (1) theorem of David and Journé, asserting that singular integral operators are bounded in L p for 1 < p < provided that T (1), T (1) BMO. We present Figiel s proof of the T (1) theorem. His idea was to expand the integral kernels along the isotropic Haar basis and thereby obtain representations of the associated integral operator as a series rearrangement operators, Haar multipliers and paraproducts. For sake of notational simplicity we consider integral operator on R and kernels in R. We assume that the kernel K : R R satisfies the following properties 1. K(x, y) A x y 1 and K(x, y) A x y,. K, h I h I + K, h I 1 I + K, 1 I h I A I, 3. K C c (R ), and consider the integral operator T K (u) = R K(x, y)u(y)dy. The singular integral operators introduced in Section 1 can be approximated by the kernel operators defined above. Hence the L p (1 < p < ) estimates in Section 1 are a consequence of the following theorem. Theorem 4.1 (G. David, J.L. Journé [DJ84]). Let 1 < p <. Let T (1)(x) = K(x, y)dy, T (1)(y) = K(x, y)dx. Then R T K p C p (A + T (1) BMO + T (1) BMO ). R 8

32 4 Singular integral operators: Central estimates Proof. (T. Figiel. [Fig90]) Step 1: (The kernel-expansion.) We define a basis in L (R ). Let A = {0, 1} \{(0, 0)} and ε = (ε 1, ε ) A. Let I, J D with I = J and define The family of functions {h (ε) I J h (ε) I J (x, y) = hε 1 I (x)hε J (y). : ε A; I, J D, I = J } forms a complete orthogonal system in L (R ). Hence, K = K, h (ε) I J h (ε) I J I J 1, and TK(u) = K, h (ε) I J u, h ε 1 I hε J I J 1, where the index set is {ε A; I, J D, I = J }. Step : (Decomposition.) We decompose TK as follows: T K = T (1,1) K + T (1,0) K Let I, J D with I = J. Then there is m Z, so that J = I + m I or I = J m J. We define T (1,1) K (u) = K, h (1,1) I (I+m I ) u, h I h I+m I, I I m Z I D T (1,0) K T (0,1) K (u) = m Z (u) = m Z K, h (1,0) I D K, h (0,1) J D I (I+m I ) I (J m J ) J J u, h I 1 I+m I, I u, 1 J m J h J. J Step 3. Here we identify the Haar coefficients of T (1) and T (1) and we isolate the rearrangement operators hidden in the above decomposition of operators Remarks: I, J D fix. K, h (1,0) I (I+m I ) = K(x, y)h I (x)dydx m Z = h I, T (1). K, h (0,1) (J m J ) J = K(x, y)h J (y)dxdy m Z = T (1), h J. 9

33 4 Singular integral operators: Central estimates The relevant operators are then this: T m (h I ) = h I+m I, U m (h I ) = 1 I+m I 1 I, P a (u) = I D P a (u) = I D a, h I I a, h I I (Permutation operator) (Shift operator) u, h I 1 I, (Paraproduct) I u, 1 I h I. I (Transposed) Step 4. We rewrite the expansion of the operator using T m U m and paraproducts. T (1,1) K T (1,0) K T (0,1) K (u) = m Z (u) = m Z (u) = m Z K, h (1,1) I D K, h (1,0) I D K, h (0,1) J D I (I+m I ) I I (I+m I ) I J (J m J ) J u, h I T m (h I ) I u, h I U m (h I ) + P T (1) (u) I u, U m (h J ) h J + PT J (1)(u). Step 5: (Paley, Triangle inequality.) The off- diagonal decay of the kernel K translates into decay estimates for its matrix representation along the Haar system. The matrix K, h (ε) I (I+m I ) is almost diagonal: K, h (ε) I (I+m I ) CA I (1 + m ). As the Haar system is an unconditional basis in L p (Paley s inequality (.7)) we obtain with the triangle inequality that T (1,1) K (u) p m Z(1 + m ) T m (u) p T (1,1) K (u) p m Z(1 + m ) T m (u) p T (1,0) K (u) p C m Z(1 + m ) U m (u) p + P T (1) (u) p. Step 6: It remains to bound the paraproducts and to find estimates for the growth of the rearrangement operators T m, U m such that (1 + m ) ( T m + U m ) remains summable, For scalar valued paraproducts we have the estimates of R. Coifman, and Y. Meyer, [CM78]) P a (u) p C p a BMO u p. and for permutation operators T. Figiel [Fig88] proved that T m p + U m p C p log(1 + m ). 30

34 4 Singular integral operators: Central estimates Remark. Figiel s proof outlined above applies to vector valued L p spaces. provided the target space X is in the UMD class. Recall that X is a UMD space if there exists a constant C > 0 such that for every finite choice of vectors x I X and signs ɛ I {+1, 1} 1 0 ɛ I h I (t)x I Xdt C 1 0 h I (t)x I Xdt. We refer to Burkholder [Bur83] or Figiel [Fig90, Fig88] for for more information and references to Banach spaces with the UMD property. Now, if X is a UMD space and L p (X) the Bochner space of X valued p integrable random variables then Figiel [Fig90] proves that T K Id X : L p (X) L p (X) is bounded if X is a UMD space, T (1) BMO, T (1) BMO and 1 < p <. (A detailed exposition of Figiel s treatment of vector valued paraproducts is contained in [Mül05].) Remark. The Lebesgue spaces L q and the sequence spaces l q when 1 < q < are UMD spaces. The spaces L 1, l 1 and c 0 are not. 4. Operators on L p (L 1 ) In this section we prove Bourgain s estimates for singular integral operators on L p (L 1 ). We present Kislyakov s approach based on good-lambda-inequalities [Kis91] and encourage the reader to consult [Kis91] for numerous deep applications of Bourgain s estimates. For sake of simplicity we further specialize the discussion to the case of the Hilbert transform. Recall from Section 1.1 that it is given as a Fourier multiplier, by H(y n ) = ( i)sign(n)y n, y T. Kolmogorov s theorem ([Koo98]) asserts that the Hilbert transform maps L 1 to weak L 1, that is m{ Hf > λ} C f 1. λ For g L p (T) we let Mg denote its Hardy Littlewood maximal function, thus Mg(x) = sup{ g dm/ I } where the supremum is extended over all intervals in T containing I x. By the maximal function theorem of Hardy and Littlewood, M preserves the spaces L p (1 < p ) with Mg p Cp/(p 1) g p. See also [Koo98]. In this Section we are concerned with the vector-valued interpretation of Kolmogorov s theorem. Let (Ω, µ) be a probability space. For 0 < α < 1 let J Ω : L 1 (Ω, µ) L α (Ω, µ) denote the inclusion J Ω (f) = f. On the algebraic tensor product L p (T) L 1 (Ω, µ) define H = H J Ω by putting H(f ϕ) = (Hf) ϕ, f L p (T), ϕ L 1 (Ω, µ) 31

35 4 Singular integral operators: Central estimates and linear extension. Put L p (L 1 ) = L p (T, L 1 (Ω)) and L p (L α ) = L p (T, L α (Ω)). The proof presented in the previous section shows that the operator H = H J Ω extends as follows: H: L p (L q ) L p (L q ) < C p,q, where C p,q as p 1 or q 1. The same for p or q. (See [Bur83]. See also [Fig90].) In the case q = 1 the following theorem of J. Bourgain [BD86, DG85] provides a powerful vector valued extension of Kolmogorov s inequality. Theorem 4.. Let 1 < p < and 0 < α < 1. There exists A = A(p, α) so that for any probability space (Ω, µ) the operator extends to a bounded operator with H = H J Ω H : L p (L 1 ) L p (L α ) A. We present here S. Kislyakov s [Kis91] proof of Theorem 4.. Establishing the following Good Lambda Inequality is the central idea of Kislyakov s approach, which led to striking simplification over the arguments in [BD86] or [DG85]. Theorem 4.3. Let F L 1 (T) L 1 (Ω) and G = HF. For z T and 0 < α < 1 we write f(z) = F (z) L 1 and g(z) = G(z) L α, where L α = L α (Ω, µ) and L 1 = L 1 (Ω, µ). Then there exists A = A(α) such that sup λm{g > λ} A λ>0 f(y)dm(y), (4.1) and A 1 = A 1 (α, A) such that for any η > 0 and λ > 0, T m{g > A 1 λ, Mf < ηλ} ηm{mg > λ}, (4.) where M f denotes the Hardy Littlewood maximal function of f. Proof. The proof consists in extrapolating the weak type estimates for the Hilbert transform. Step 1. (Weak Type estimates.) Fix λ > 0. We prove the following weak type (1, 1) estimate: λm({g > λ}) A f(y)dm(y). (4.3) T 3

36 4 Singular integral operators: Central estimates Recall the convention L 1 = L 1 (Ω, µ). Let B = {g > λ}. By arithmetic and Fubini s theorem λm(b) λ (1 α) g(y) α dm(y) B ( ) (4.4) λ (1 α) G(y) α dm(y) L 1. B For ω Ω fixed, the weak type estimate for the Hilbert transform yields G(y, ω) α dm(y) ( F (y, ω) dm(y)) α (m(b)) 1 α (4.5) B T Inserting (4.5) to (4.4) and applying Hölder s inequality gives λ (1 α) G(y) α dm(y) L 1 B ( ) α (λm(b)) (1 α) F (y) L 1dm(y) T ( α = (λm(b)) f(y)dm(y)) (1 α). T (4.6) It remains to combine (4.4) and (4.6) to obtain (4.3) and hence, (4.1). Step. (Reduction.) Since M g is lower semicontinuous, {M g > λ} is open. There exists a sequence of disjoint intervals I j so that {Mg > λ} = I j. Fix j and put I = I j. The expanded interval (I) intersects {Mg λ}, hence gdm m(i) inf Mg(x) m(i)λ. (4.7) x I We prove that for any η > 0 and λ > 0, I m(i {g > A 1 λ, Mf < ηλ}) ηm(i). (4.8) In order to show (4.8) we will verify the following implication: If I {Mf < ηλ} then m(i {g > A 1 λ}) ηm(i). (4.9) Step 3. Put F = F 1 T\I, G = HF, and g (x) = G (x) L α. We show the following implication: If there exists x 0 I with Mf(x 0 ) > ηλ, then there exists x 1 I so that g (x 1 ) C 1 λ. To this end let F 1 = F F, G 1 = HF 1, and f 1 (x) = F 1 (x) L 1, g 1 (x) = G 1 (x) L 1. 33

37 4 Singular integral operators: Central estimates Since F 1 = F 1 I, we have f 1 dm = T T\I fdm m(i) inf x I Mf(x) m(i)mf(x 0) ηλ. The weak type inequality (4.1) applied to f 1, g 1 yields m(i {g 1 > 6Aηλ}) A f 1 dm 1 m(i). (4.10) 6Aηλ 3 Consequently by (4.10) and (4.7), m(i {g 1 6Aηλ}) 3 m(i), and m(i {g λ}) 3 m(i). Comparing the lower measure estimates shows that their intersection is non-empty. Pick x 1 I {g λ} {g 1 > 6Aηλ}. By the quasi triangle inequality in L α = L α (Ω, µ), we get g α (x 1 ) g α (x 1 ) + g α 1 (x 1 ) (6λ) α (1 + (Aη) α ), and hence T g (x 1 ) C 1 λ, where C 1 = 6(1 + (Aη) α ) 1/α. (4.11) Step 4. Here we show that there exists C > 0 so that for any x I, g α (x) λ α (C α 1 + C α ). We first use the kernel estimates for the Hilbert transform to show that there exists C > 0 so that for any x I, G (x) G (x 1 ) L α C ηλ. (4.1) As α < 1 the L α (Ω, µ) quasi-norm on the left-hand side of (4.1) is bounded by the L 1 (Ω, µ). In the latter space we use the triangle inequality. Thus, the kernel estimates for the Hilbert transform yield the following G (x) G (x 1 ) L 1 = T\I T\I (K(x, y) K(x 1, y))f (y)dm(y) x x 1 x y f(y)dm(y) C inf x I Mf(x). L 1 (4.13) Since inf x I Mf(x) Mf(x 0 ) ηλ, (4.1) follows directly from (4.13). In summary for x I we have G (x) α L α G (x 1 ) α L α + G (x) G (x 1 ) α L α, 34

38 4 Singular integral operators: Central estimates and obtain g α (x) λ α (C α 1 + C α ), x I (4.14) Step 5. (Putting it together). Put C 3 = (C α 1 + C α + α A α ) 1/α. By (4.14) g α (x) g α 1 (x) + g α (x) g α 1 (x) + λ α C α 3 (4.15) With A 1 = α C 3 we get A α 1 C α 3 = C α 3 and with (4.15), I {g > A 1 λ} I {g α > A α 1 λ α } I {g α 1 > (A α 1 C α 3 )λ α }. Applying again the weak type estimate (4.7) and (4.10) to f 1, g 1 and using that A C 3 gives m(i {g 1 > C 3 λ}) ηm(i). This completes the verification of the implication (4.9). Finally we invoke that {M g λ} {g > A 1 λ} = and obtain for any η > 0 and λ > 0, m{g > A 1 λ, Mf < ηλ} = m(i j {g > A 1 λ, Mf < ηλ}) η j=1 m(i j ) j=1 ηm{mg > λ}. We finally describe the well known path from good lambda inequalities to L p estimates. Proof of Theorem 4.. Starting from Kislyakov s distributional estimate (4.), we obtain m{g > A 1 λ, Mf < ηλ} ηm{mg > λ}, m{g > A 1 λ} m{mf > ηλ} + ηm{mg > λ}, which holds uniformly in λ and η. Multiplying both sides by pλ p 1 and integrating over 0 < λ < gives g p dm Ap 1 (Mf) p dm + ηa p T η p 1 (Mg) p dm. T T Invoking the Hardy-Littlewood maximal function estimates we obtain C > 0 so that ( CpA p ) ( g p 1 CpA p ) dm f p 1 dm + η g p dm. T (p 1)η p T p 1 T The above estimate holds uniformly over 0 < η < 1. Given 1 < p <, we select η > 0 so small that the factor in front of the second integral on the right hand side is bounded by 1/. Thus we obtain, A = A(p, A 1 ) so that g p dm A f p dm. T T 35

39 4 Singular integral operators: Central estimates 4.3 Reflexive subspaces of L 1 and vector valued SIOs We obtain here Kislyakov s embedding using Bourgain s estimate for singular integrals on L p (L 1 ), and finally we deduce Bourgain s theorem on the extension operators from reflexive subspaces of L 1. Throughout this section we follow Kislyakov [Kis91] Uniform integrability For f L 1 (T) we denote [f] = {f + h : h H 1 (T)} and we call q : L 1 L 1 /H 1, f [f] the canonical projection onto the quotient space L 1 /H 1 where L 1 /H 1 = L 1 (T)/H 1 (T) = {[f] : f L 1 (T)}. Recall the fact that the unit ball of a reflexive subspaces in L 1 is equi-integrable. The following result is a simple and useful consequence thereof. Theorem 4.4. If a closed linear subspace Y L 1 (Ω) is reflexive, then for any 0 < α < 1, there exists C = C(Y, α) so that y L 1 C y L α, y Y. (4.16) If a closed linear subspace Y H 1 (T) is reflexive, then the restriction of q to Y is invertible on its range, and there exists C = C(Y ) so that y L 1 C q(y) L 1 /H1, y Y. (4.17) Proof. If the assertion (4.16) fails to hold then there exists a sequence y n Y so that y n L 1 = 1 and y n L α 0. For any ɛ > 0 we have µ{ y n > ɛ} ɛ α y n α Lα. Thus we obtained a sequence y n Y so that y n L 1 = 1 and µ{ y n > ɛ} 0. (4.18) On the other hand a bounded set in a reflexive subspace of L 1 (Ω) is equi-integrable. Hence (4.18) contradicts the reflexivity of Y. Next we turn to proving (4.17). If the assertion would not hold true then we could find a sequence y n Y and h n H 1 (T) so that y n L 1 = 1 and y n h n L 1 0. (4.19) Let 0 < α < 1. By Kolmogorov s theorem the analytic Riesz-projection P : L (T) H (T) extends boundedly as follows, P : L 1 (T) L α (T) A α. (4.0) By assumption y n H 1 (T) and h n H 1 (T), hence P (y n h n ) = y n and by (4.0) y n L α = P (y n h n ) L α A α y n h n L 1. (4.1) By (4.19) and (4.1) we obtained a sequence y n Y satisfying y n L 1 = 1 and y n L α 0. In view of (4.16) this contradicts the reflexivity of Y. 36