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Lecture Notes on Singular Integrals, Projections, Multipliers and Rearrangements

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Lecture Notes on Singular Integrals, Projections, Multipliers and Rearrangements Lecture notes on singular integrals, projections, multipliers and rearrangements Paul F.X. M¨uller Johanna Penteker Johannes Kepler University Linz Contents Preface ii 1 Singular integral operators: Basic examples 1 1.1 Hilbert transformation . .1 1.2 The Riesz transforms . .3 1.3 Beurling transformation . .6 1.4 Cauchy integrals on Lipschitz domains . .7 1.5 Double layer potential . .8 1.6 Fourier multipliers . 10 2 Atomic decomposition and multiplication operators 13 2.1 Dyadic intervals . 13 2.2 BMO and dyadic Hardy spaces . 14 2.3 Atomic decomposition . 15 2.4 Multiplication operators . 16 3 Rearrangement operators 20 3.1 Know your basis and its rearrangement operators . 20 3.2 Haar rearrangements . 21 3.3 Postorder rearrangements of the Haar system . 22 4 Singular integral operators: Central estimates 28 4.1 The theorem of David and Journ´e. 28 4.2 Operators on Lp(L1)............................. 31 4.3 Reflexive subspaces of L1 and vector valued SIOs . 36 5 Haar projections 42 5.1 Interpolatory estimates . 42 5.2 Interpolatory estimates are useful . 43 Bibliography 46 i Preface In March 2011, 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 pro- gramme 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. M¨uller,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, - Hp atomic decomposition and absolutely summing operators - Rearrangement operators, interpolation, extrapolation and boundedness - Calder´on-Zygmund operators on Lp(Lq); 1 < p; q < : 1 - Calder´on-Zygmund operators on Lp(L1); 1 < p < : 1 - Interpolatory estimates, compensated compactness, Haar projections. In the present notes we worked out Figiel's proof of the David and Journ´etheorem, Kislyakov's proof of Bourgain's theorem on operators in Lp(L1), the proof of Kislyakov's embedding theorem, and our recent, explicit formulas for the Pietsch measure of abso- lutely 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 pro- vided by the Austrian Science Foundation under Project Nr. P22549, P23987. Linz, July 2015 Paul F. X. M¨uller Johanna Penteker ii 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 [KN92], 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 (Hp(T); SL1 H1(T)), for which the reference is [JM04]. \ 1.1 Hilbert transformation Here we recall the Hilbert transform, its basic Lp estimates and the link to the conver- gence of the partial sums of Fourier series. Let f L2(] π; π[): The Hilbert Transform H(f) of f is defined by 2 − Z f(θ t) dt H(f)(θ) = lim − t ; !0 fjtj>g 2 tan 2 π where we take f(θ + 2π) = f(θ): The integral on the right-hand side exists a.e. if f L1(] π; π[) and defines a construction on L2(] π; π[); 2 − − H(f) 2 f 2 : k kL ≤ k kL (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(eiθ) = f(θ) with Fourier expansion 1 iθ X inθ g(e ) = Ane n=−∞ 1 1 Singular integral operators: Basic examples and harmonic extension to the unit disk given by 1 iθ X jnj inθ u(re ) = r Ane ; r < 1: n=−∞ The harmonic conjugate of u is then 1 iθ X jnj inθ u~(re ) = isign(n)r Ane ; r < 1; − n=−∞ since 1 iθ iθ X jnj inθ u(re ) + iu~(re ) = A0 + 2r Ane n=1 is analytic. By Fatou's theorem the L2 function 1 iθ X inθ g~(e ) = isign(n)Ane ; − n=−∞ is the a.e. boundary value ofu ~(reiθ); i.e. g~(eiθ) = lim u~(reiθ) for a.e θ; r!1 and Z iθ f(θ t) dt g~(e ) = lim − t = H(f): !0 fjtj>g 2 tan 2 π Consequently Plancherel's theorem gives L2 estimates for the Hilbert transform Hf 2 f 2 : (1.1) k kL ≤ k kL Inequality (1.1) continues to hold true in the reflexive Lp spaces. Theorem 1.1 (M. Riesz). Let f Lp(] π; π[), 1 < p < . Then (taking f(θ + 2π) = f(θ)) 2 − 1 Z f(θ t) dt H(f)(θ) = lim − t !0 fjtj>g 2 tan 2 π exists a.e. and Hf Lp Cp f Lp ; k k ≤ k k 2 where Cp p =(p 1): ∼ − By algebraic manipulations the Theorem of Riesz implies the norm-convergence of Fourier series in Lp. Let f Lp(] π; π[), 1 < p < with formal Fourier series 2 − 1 1 X inθ Ane : n=−∞ 2 1 Singular integral operators: Basic examples Let N X inθ SN (f)(θ) = Ane n=−N and 1 1 P (f)(θ) = ( Id + iH)f(θ) + A : 2 2 0 By the above computation we have 1 X inθ P (f)(θ) = Ane : n=0 The partial sums SN (f) coincide with e−iNθP (eiN(·)f( )) ei(N+1)θP (e−i(N+1)(·)f( )): · − · This identity links the Hilbert transform to partial sums of Fourier series. By the Riesz p theorem we have convergence of the partial sums SN (f) in L ; 1 < p < ; 1 SN (f) f Lp 0 for N ; k − k ! ! 1 and SN (f) Lp 4 Hf Lp Cp f Lp ; (1 < p < ); k k ≤ k k ≤ k k 1 2 Cp p =(p 1): ∼ − 1.2 The Riesz transforms We recall the Riesz transforms, their definition as principal value integral, the Lp 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 Lp(Rn), 1 p < . The Riesz transform of f is defined by 2 ≤ 1 Z tj Rj(f)(x) = cn lim f(x t) dt; j = 1; : : : ; n; (1.2) !0 − t n+1 jtj> j j n+1 Γ( 2 ) with cn = π(n+1)=2 : Theorem 1.2 (M. Riesz). The integral on the right-hand side in (1:2) exists a.e. and defines a bounded linear operator for which Rj p Cp; 1 < p < : k k ≤ 1 3 1 Singular integral operators: Basic examples Let denote the Fourier transform normalized so that F Z 1 −ihx,ξi f(ξ) := n e f(x)dx: 2 F (2π) Rn We have then the following identities identifying the Fourier multiplier of Riesz trans- form. (@jf) = iξj (f) (1.3) F F −1 yj Rj(f))(x) = i ( (f)(:))(y) (1.4) F y F j j 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 Lp 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 2 n problem following Stein [Ste70]. Let u Cc (R ) and 2 ∆u(x) = f(x): We show that @2u f Lp = Lp: (1.5) 2 ) @xj@xi 2 The Fourier Transformation provides the link to Riesz transforms: F (f)(y) = (∆u)(y) = y 2 (u)(y) F F −| j F (ux ;x )(y) = yjyk (u)(y): F j k − F Hence, yjyk (ux ;x )(y) = (f)(y): F j k y 2 F j j On the other hand yj (Rj(f))(y) = i (f)(y): F y F j j Summarizing we have ux ;x = Rj(Rk(f)) j k − and therefore, 2 @ u 2 Rj p Rk p f p = Cp f p: @xj@xi p ≤ k k k k k k k k 4 1 Singular integral operators: Basic examples Gradient vector fields. Here we show how to identify the Helmholtz projections using Riesz transforms. In [Bou92] this identity was the starting point for the proof that the Banach couple of Sobolev spaces (W 1;1;W 1;1) is K closed in (L1;L1). 1;p n − Let Sp = (fx1 ; : : : ; fxn ) f W (R ) . Then f j 2 g p p M p Sp L ::: L = L ⊂ ⊕ ⊕ and especially M S L2 ::: L2 = L2: 2 ⊂ ⊕ ⊕ Since S2 is a closed subspace there exists a projection M P : L2 S ;P 2 = P; ! 2 onto S2 The next result gives the singular integral representation of the projection P .
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