A SHORT PROOF OF THE COIFMAN-MEYER MULTILINEAR THEOREM
|
|
- Ella Pope
- 6 years ago
- Views:
Transcription
1 A SHORT PROOF OF THE COIFMAN-MEYER MULTILINEAR THEOREM CAMIL MUSCALU, JILL PIPHER, TERENCE TAO, AND CHRISTOPH THIELE Abstract. We give a short proof of the well known Coifman-Meyer theorem on multilinear operators.. Introduction The main task of the present paper is to present a new proof of the classical Coifman- Meyer theorem on multilinear singular integrals, see [2], [3], [5], [6]. Let m L (IR n ) be a bounded function which is smooth away from the origin and satisfies the following Marcinkiewicz-Mihlin-Hörmander type condition α m(ξ) ξ α, () for sufficiently many multiindices α. For f,..., f n S(IR) Schwartz functions on the real line, we define the n-linear operator T m by the formula T m (f,..., f n )(x) = IR m(ξ) f (ξ )... f n (ξ n )e 2πix(ξ +...+ξ n) dξ...dξ n. (2) n The following theorem holds, see [2], [3], [5], [6]. Theorem.. As defined, the multilinear operator T m maps L p... L pn as < p i, i n, /p /p n = /p and 0 < p <. L p as long When such an n + -tuple (p,..., p n, p), has the property that 0 < p < and p j = for some j n then, for some technical reasons (see [5], [7]), by L one actually means L c the space of bounded measurable functions with compact support. The case p has been proven in [3] while the general case p > /n has been independently settled in [6] and [5]. The interesting fact that p can be a number smaller than goes back to [2]. The usual argument to prove the theorem (see [2], [3], [5], [6]) uses the celebrated T theorem of David and Journe [4] and relies on BMO theory, Carleson measures and C.Fefferman s duality theorem between the Hardy space H and BMO. As the reader will see, our proof is conceptually simpler and does not use any of the aforementioned ingredients. It is based on a careful stopping time argument involving the Hardy-Littlewood maximal function and the Littlewood-Paley square function. Throughout the paper we will write A B iff there is a universal constant C > 0 so that A CB.
2 2 CAMIL MUSCALU, JILL PIPHER, TERENCE TAO, AND CHRISTOPH THIELE 2. Model operators For simplicity we treat the n = 2 case only. It is a standard fact by now (see for instance the papers [7], [8]) that the study of our bilinear operators T m can be reduced to the study of finitely many discrete model operators Π j P, j = 0,, 2, 3 of the form Π j P (f, f 2 ) = ɛ P I P /2 f, φ P f 2, φ P2 φ P3. (3) Here P is a collection of lacunary dyadic tiles corresponding to lattice points (k, n) in Z 2 and (ɛ P ) P is an arbitrary sequence of uniformly bounded constants. More precisely, P i i =, 2, 3 are defined by P i = I P ω Pi where I P = [n2 k, (n + )2 k+ ], ω Pi = [0, 2 k ] if i = j and ω Pi = [2 k, 2 k+ ] if i j, while φ Pi i =, 2, 3 are L 2 normalized wave packets corresponding to the Heisenberg boxes P i i =, 2, 3. This means that the function φ Pi is a smooth L 2 normalized bump function adapted to the interval I P whose Fourier transform φ Pi is supported inside the interval ω Pi for i =, 2, 3. We should emphasize here that the tile P is uniquely determined by the interval I P. To explain this reduction in a few words, let Q := I J be a dyadic rectangle in the plane, having the property that diam(q) dist(q, {0}). Let also φ I, φ J be two L normalized smooth functions such that supp( φ I ) I and supp( φ J ) J. If we replace the symbol m(ξ, ξ 2 ) by φ I (ξ ) φ J (ξ 2 ) we observe that the right hand side of (2) becomes (f φ I )(x)(f 2 φ J )(x). On the other hand, inequality () implies that one can think of m as being essentially constant on each such Q and so the integral in (2) when smoothly restricted to this cube, becomes roughly ɛ Q (f φ I )(x)(f 2 φ J )(x). Then, one covers the plane by a collection of carefully selected such Whitney cubes and discretize again in the x-variable. In the end, one obtains a formula in which the general T m is written as an average of operators of the type of the model operators above. The details can be found in [7], [8]. Now our analysis of the operators Π j P will be independent on j = 0,, 2, 3 and so we can assume without loss of generality that j = and write from now on, for simplicity, Π P instead of Π P (the reader will observe that for j, the only difference is that the roles of the Hardy Littlewood maximal function M and the Littlewood Paley square function S, get permuted). It is therefore enough to prove the theorem for the bilinear operator Π P. 3. The proof First, let us observe that it is very easy to obtain the necessary L p estimates in the particular case when all the indices are strictly between and. To see this, let f L p, g L q, h L r for < p, q, r < with /p + /q + /r =. Then, IR Π P(f, g)(x)h(x) dx I P f, φ /2 P g, φ P2 h, φ P3 =
3 A SHORT PROOF OF THE COIFMAN-MEYER MULTILINEAR THEOREM 3 IR f, φ P I P /2 g, φ P2 I P /2 h, φ P3 I P /2 χ I P (x) dx (4) ( ) ( ) /2 ( /2 f, φ P sup IR I P χ g, φ P2 2 h, φ P3 2 /2 I P (x) χ IP (x) χ IP (x)) dx I P I P IR Mf(x)Sg(x)Sh(x) dx Mf p Sg q Sh r f p g q h r, where M is the maximal function of Hardy and Littlewood and S is the discrete square function of Littlewood and Paley, see [9] and [7]. This means that theorem. is nontrivial only when one index is, or less or equal than. To prove the general case we just need to show that the bilinear operator Π P maps L L L /2, because then, by interpolation and symmetry the theorem follows as in [7]. Let f, g L be such that f = g =. We now recall Lemma 5.4 in []. Lemma 3.. Let 0 < p < and A > 0. Then the following statements are equivalent up to constants: (i) f p, A. (ii) For every set E with 0 < E <, there exists a subset E E with E E and f, χ E A E /p. Here p is defined by /p + /p = (note that p can be a negative number!). Proof To see that (i) implies (ii), set E := E \ {x : f(x) CA E /p }. If C is a sufficiently large constant, then (i) implies E E and the claim follows. To see that (ii) implies (i), let λ > 0 be arbitrary E := {x : Re(f(x)) > λ}. Then by (ii) we have λ E λ E A E /p, and (i) easily follows (replacing Re by Re, Im, Im as necessary). Using this Lemma 3. in the particular case p = /2 and the scale invariance, it is enough to show that given E IR E =, one can find a subset E E with E such that I P /2 f, Φ P g, Φ P2 h, Φ P3 (5) where h := χ E. Fix such a set E with E =. To construct the subset E, we first consider Ω 0 = {x IR : M(f)(x) > C} {x IR : S(g)(x) > C} {x IR : M(g)(x) > C}. Also, define
4 4 CAMIL MUSCALU, JILL PIPHER, TERENCE TAO, AND CHRISTOPH THIELE Ω = {x IR : M( Ω0 )(x) > 00 }. Clearly, we have Ω < /2, if C is a big enough constant which we fix from now on. Then, we define E := E \ Ω = E Ω c and observe that indeed E. After this, we split our sum in (5) into two parts = I P Ω c + I P Ω c = := I + II. We also assume that the set P is finite, since our estimates do not depend on its cardinality. First, we estimate term I. Since I P Ω c, it follows that I P Ω 0 I P < or equivalently, 00 I P Ω c 0 > 99 I 00 P. We are now going to describe three decomposition procedures, one for each function f, g, h. Later on, we will combine them, in order to handle our sum. First, define then define Ω = {x IR : M(f)(x) > C 2 } T = {P P : I P Ω > 00 I P }, Ω 2 = {x IR : M(f)(x) > C 2 2 } T 2 = {P P \ T : I P Ω 2 > 00 I P }, and so on. (The constant C > 0 is the one which we fixed before). Since there are finitely many tiles, this algorithm ends after a while, producing the sets {Ω n } and {T n } such that P = n T n. Independently, define then define Ω = {x IR : S(g)(x) > C 2 } T = {P P : I P Ω > 00 I P }, Ω 2 = {x IR : S(g)(x) > C 2 2 } T 2 = {P P \ T : I P Ω 2 > 00 I P },
5 A SHORT PROOF OF THE COIFMAN-MEYER MULTILINEAR THEOREM 5 and so on, producing the sets {Ω n } and {T n } such that P = nt n. We would like to have such a decomposition available for the function h also. To do this, we first need to construct the analogue of the set Ω 0, for it. We will therefore pick N > 0 a big enough integer such that for every P P we have I P Ω c N > 99 I 00 P where we defined Ω N = {x IR : S(h)(x) > C2N }. Then, similarly to the previous algorithms, we define then define Ω N+ = {x IR : S(h)(x) > C2N 2 } T N+ = {P P : I P Ω N+ > 00 I P }, Ω N+2 = {x IR : S(h)(x) > C2N 2 2 } T N+2 = {P P \ T N+ : I P Ω N+2 > 00 I P }, and so on, constructing the sets {Ω n } and {T n } such that P = nt n. Then we write the term I as n,n 2 >0,n 3 > N P T n,n 2,n 3 I P 3/2 f, Φ P g, Φ P2 h, Φ P3 I P, (6) where T n,n 2,n 3 := T n T n 2 T n 3. Now, if P belongs to T n,n 2,n 3 this means in particular that P has not been selected at the previous n, n 2 and n 3 steps respectively, which means that I P Ω n < I 00 P, I P Ω n 2 < I 00 P and I P Ω n 3 < I 00 P or equivalently, I P Ω c n > 99 I 00 P, I P Ω c n2 > 99 I 00 P and I P Ω c n 3 > 99 I 00 P. But this implies that I P Ω c n Ω c n2 Ω c n 3 > I P. (7) In particular, using (7), the term in (6) is smaller than n,n 2 >0,n 3 > N n,n 2 >0,n 3 > N I P f, Φ 3/2 P g, Φ P2 h, Φ P3 I P Ω c n c Ω n2 c Ω n 3 = P T n,n 2,n 3 Ω c n Ω c n2 Ω c n 3 n,n 2 >0,n 3 > N P T n,n 2,n 3 I P 3/2 f, Φ P g, Φ P2 h, Φ P3 χ IP (x) dx Ω c n Ω c n2 Ω c n 3 Ω Tn,n 2,n 3 M(f)(x)S(g)(x)S(h)(x) dx
6 6 CAMIL MUSCALU, JILL PIPHER, TERENCE TAO, AND CHRISTOPH THIELE where On the other hand we can write n,n 2 >0,n 3 > N Ω Tn,n 2,n 3 := 2 n 2 n 2 2 n 3 Ω Tn,n 2,n 3, (8) P T n,n 2,n 3 I P. Ω Tn,n 2,n 3 Ω Tn {x IR : M(χ Ωn )(x) > 00 } Ω n = {x IR : M(f)(x) > C 2 } n 2n. Similarly, we have Ω Tn,n 2,n 3 2 n 2 and also Ω Tn,n 2,n 3 2 n2α, for every α, since E. In particular, it follows that Ω Tn,n 2,n 3 2 n θ 2 n 2θ 2 2 n 3αθ 3 (9) for any 0 θ, θ 2, θ 3 <, such that θ + θ 2 + θ 3 =. Now we split the sum in (8) into n,n 2 >0,n 3 >0 2 n 2 n 2 2 n 3 Ω Tn,n 2,n 3 + n,n 2 >0,0>n 3 > N 2 n 2 n 2 2 n 3 Ω Tn,n 2,n 3. To estimate the first term in (0) we use the inequality (9) in the particular case θ = θ 2 = /2, θ 3 = 0, while to estimate the second term we use (9) for θ j, j =, 2, 3 such that θ <, θ 2 < and αθ 3 > 0. With these choices, the geometric sums in (0) are finite. This ends the discussion on I. Now term II is much simpler, being just an error term. We split (0) where and easily observe that Then, term II is smaller than d>0 d ;I P Ω P := d>0 P d P d := {P P : dist(i P, Ω c ) I P d ;I P Ω I P f, Φ P I P /2 g, Φ P 2 I P /2 h, Φ P 3 I P /2 d>0 for any big number K > 0, and this ends the proof. 2 d } I P Ω. () d ;I P Ω I P 2 d 2 d 2 Kd,
7 A SHORT PROOF OF THE COIFMAN-MEYER MULTILINEAR THEOREM 7 References [] Auscher P., Hofmann S., Muscalu C., Tao T. and Thiele C. Carleson measures, trees, extrapolation and T b theorems, Publ. Mat., vol. 46, , [2002]. [2] Coifman R. and Meyer Y., On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. vol. 22, 35-33, [975]. [3] Coifman R. and Meyer Y., Ondelettes et operateurs III. Operateurs multilineaires, Hermann, Paris, [99]. [4] David G. and Journe J-L., A boundedness criterion for generalized Calderon-Zygmund operators, Ann. of Math., vol. 20, , [984]. [5] Grafakos L. and Torres R., Multilinear Calderon-Zygmund theory, Adv. Math., vol. 65, 24-64, [2002]. [6] Kenig C. and Stein E., Multilinear estimates and fractional integration, Math. Res. Lett., vol. 6, -5, [999]. [7] Muscalu C., Tao T. and Thiele C., Multilinear operators given by singular multipliers, J. Amer. Math. Soc. 5, [2002], [8] Muscalu C., Pipher J., Tao T. and Thiele, C., Bi-parameter paraproducts, Preprint, [2003]. [9] Stein, E., Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton, [993]. Department of Mathematics, Cornell University, Ithaca, NY address: camil@math.cornell.edu Department of Mathematics, Brown University, Providence, RI address: jpipher@math.brown.edu Department of Mathematics, UCLA, Los Angeles, CA address: tao@math.ucla.edu Department of Mathematics, UCLA, Los Angeles, CA address: thiele@math.ucla.edu
MULTI-PARAMETER PARAPRODUCTS
MULTI-PARAMETER PARAPRODUCTS CAMIL MUSCALU, JILL PIPHER, TERENCE TAO, AND CHRISTOPH THIELE Abstract. We prove that the classical Coifman-Meyer theorem holds on any polydisc T d of arbitrary dimension d..
More informationarxiv:math.ca/ v1 23 Oct 2003
BI-PARAMETER PARAPRODUCTS arxiv:math.ca/3367 v 23 Oct 23 CAMIL MUSCALU, JILL PIPHER, TERENCE TAO, AND CHRISTOPH THIELE Abstract. In the first part of the paper we prove a bi-parameter version of a well
More informationParaproducts in One and Several Variables M. Lacey and J. Metcalfe
Paraproducts in One and Several Variables M. Lacey and J. Metcalfe Kelly Bickel Washington University St. Louis, Missouri 63130 IAS Workshop on Multiparameter Harmonic Analysis June 19, 2012 What is a
More informationBOUNDS FOR A MAXIMAL DYADIC SUM OPERATOR
L p BOUNDS FOR A MAXIMAL DYADIC SUM OPERATOR LOUKAS GRAFAKOS, TERENCE TAO, AND ERIN TERWILLEGER Abstract. The authors prove L p bounds in the range
More informationMichael Lacey and Christoph Thiele. f(ξ)e 2πiξx dξ
Mathematical Research Letters 7, 36 370 (2000) A PROOF OF BOUNDEDNESS OF THE CARLESON OPERATOR Michael Lacey and Christoph Thiele Abstract. We give a simplified proof that the Carleson operator is of weaktype
More informationESTIMATES FOR MAXIMAL SINGULAR INTEGRALS
ESTIMATES FOR MAXIMAL SINGULAR INTEGRALS LOUKAS GRAFAKOS Abstract. It is shown that maximal truncations of nonconvolution L -bounded singular integral operators with kernels satisfying Hörmander s condition
More informationParaproducts and the bilinear Calderón-Zygmund theory
Paraproducts and the bilinear Calderón-Zygmund theory Diego Maldonado Department of Mathematics Kansas State University Manhattan, KS 66506 12th New Mexico Analysis Seminar April 23-25, 2009 Outline of
More informationClassical Fourier Analysis
Loukas Grafakos Classical Fourier Analysis Second Edition 4y Springer 1 IP Spaces and Interpolation 1 1.1 V and Weak IP 1 1.1.1 The Distribution Function 2 1.1.2 Convergence in Measure 5 1.1.3 A First
More informationHARMONIC ANALYSIS. Date:
HARMONIC ANALYSIS Contents. Introduction 2. Hardy-Littlewood maximal function 3. Approximation by convolution 4. Muckenhaupt weights 4.. Calderón-Zygmund decomposition 5. Fourier transform 6. BMO (bounded
More informationarxiv:math/ v2 [math.ca] 18 Jan 2006
COMMUTATORS WITH REISZ POTENTIALS IN ONE AND SEVERAL PARAMETERS arxiv:math/0502336v2 [math.ca] 18 Jan 2006 MICHAEL T. LACEY GEORGIA INSTITUTE OF TECHNOLOGY Abstract. Let M b be the operator of pointwise
More informationAlternatively one can assume a Lipschitz form of the above,
Dedicated to the memory of Cora Sadosky MINIMAL REGULARITY CONDITIONS FOR THE END-POINT ESTIMATE OF BILINEAR CALDERÓN-ZYGMUND OPERATORS CARLOS PÉREZ AND RODOLFO H. TORRES ABSTRACT. Minimal regularity conditions
More informationAnalytic families of multilinear operators
Analytic families of multilinear operators Mieczysław Mastyło Adam Mickiewicz University in Poznań Nonlinar Functional Analysis Valencia 17-20 October 2017 Based on a joint work with Loukas Grafakos M.
More informationLOUKAS GRAFAKOS, DANQING HE, AND PETR HONZÍK
THE HÖRMADER MULTIPLIER THEOREM, II: THE BILIEAR LOCAL L 2 CASE LOUKAS GRAFAKOS, DAQIG HE, AD PETR HOZÍK ABSTRACT. We use wavelets of tensor product type to obtain the boundedness of bilinear multiplier
More informationWeighted norm inequalities for singular integral operators
Weighted norm inequalities for singular integral operators C. Pérez Journal of the London mathematical society 49 (994), 296 308. Departmento de Matemáticas Universidad Autónoma de Madrid 28049 Madrid,
More informationOn pointwise estimates for maximal and singular integral operators by A.K. LERNER (Odessa)
On pointwise estimates for maximal and singular integral operators by A.K. LERNER (Odessa) Abstract. We prove two pointwise estimates relating some classical maximal and singular integral operators. In
More informationFRACTIONAL DIFFERENTIATION: LEIBNIZ MEETS HÖLDER
FRACTIONAL DIFFERENTIATION: LEIBNIZ MEETS HÖLDER LOUKAS GRAFAKOS Abstract. Abstract: We discuss how to estimate the fractional derivative of the product of two functions, not in the pointwise sense, but
More informationBoth these computations follow immediately (and trivially) from the definitions. Finally, observe that if f L (R n ) then we have that.
Lecture : One Parameter Maximal Functions and Covering Lemmas In this first lecture we start studying one of the basic and fundamental operators in harmonic analysis, the Hardy-Littlewood maximal function.
More information8 Singular Integral Operators and L p -Regularity Theory
8 Singular Integral Operators and L p -Regularity Theory 8. Motivation See hand-written notes! 8.2 Mikhlin Multiplier Theorem Recall that the Fourier transformation F and the inverse Fourier transformation
More informationDuality of multiparameter Hardy spaces H p on spaces of homogeneous type
Duality of multiparameter Hardy spaces H p on spaces of homogeneous type Yongsheng Han, Ji Li, and Guozhen Lu Department of Mathematics Vanderbilt University Nashville, TN Internet Analysis Seminar 2012
More informationBilinear pseudodifferential operators: the Coifman-Meyer class and beyond
Bilinear pseudodifferential operators: the Coifman-Meyer class and beyond Árpád Bényi Department of Mathematics Western Washington University Bellingham, WA 98225 12th New Mexico Analysis Seminar April
More informationRESULTS ON FOURIER MULTIPLIERS
RESULTS ON FOURIER MULTIPLIERS ERIC THOMA Abstract. The problem of giving necessary and sufficient conditions for Fourier multipliers to be bounded on L p spaces does not have a satisfactory answer for
More informationClassical Fourier Analysis
Loukas Grafakos Classical Fourier Analysis Third Edition ~Springer 1 V' Spaces and Interpolation 1 1.1 V' and Weak V'............................................ 1 1.1.l The Distribution Function.............................
More informationSCALE INVARIANT FOURIER RESTRICTION TO A HYPERBOLIC SURFACE
SCALE INVARIANT FOURIER RESTRICTION TO A HYPERBOLIC SURFACE BETSY STOVALL Abstract. This result sharpens the bilinear to linear deduction of Lee and Vargas for extension estimates on the hyperbolic paraboloid
More informationMULTILINEAR SINGULAR INTEGRALS. Christoph M. Thiele
Publ. Mat. (2002), 229 274 Proceedings of the 6 th International Conference on Harmonic Analysis and Partial Differential Equations. El Escorial, 2000. MULTILINEAR SINGULAR INTEGRALS Christoph M. Thiele
More informationTOPICS. P. Lax, Functional Analysis, Wiley-Interscience, New York, Basic Function Theory in multiply connected domains.
TOPICS Besicovich covering lemma. E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton, N.J., 1971. Theorems of Carethedory Toeplitz, Bochner,...
More informationMulti-Parameter Riesz Commutators
Multi-Parameter Riesz Commutators Brett D. Wick Vanderbilt University Department of Mathematics Sixth Prairie Analysis Seminar University of Kansas October 14th, 2006 B. D. Wick (Vanderbilt University)
More information1 I (x)) 1/2 I. A fairly immediate corollary of the techniques discussed in the last lecture is Theorem 1.1. For all 1 < p <
1. Lecture 4 1.1. Square functions, paraproducts, Khintchin s inequality. The dyadic Littlewood Paley square function of a function f is defined as Sf(x) := ( f, h 2 1 (x)) 1/2 where the summation goes
More informationMULTILINEAR SINGULAR INTEGRAL OPERATORS WITH VARIABLE COEFFICIENTS
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Volumen 50, Número 2, 2009, Páginas 157 174 MULTILINEAR SINGULAR INTEGRAL OPERATORS WITH VARIABLE COEFFICIENTS RODOLFO H. TORRES Abstract. Some recent results for
More informationMULTILINEAR SQUARE FUNCTIONS AND MULTIPLE WEIGHTS
MULTILINEA SUAE FUNCTIONS AND MULTIPLE WEIGHTS LOUKAS GAFAKOS, PAASA MOHANTY and SAUABH SHIVASTAVA Abstract In this paper we prove weighted estimates for a class of smooth multilinear square functions
More informationRecent developments in the Navier-Stokes problem
P G Lemarie-Rieusset Recent developments in the Navier-Stokes problem CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C. Table of contents Introduction 1 Chapter 1: What
More informationM ath. Res. Lett. 16 (2009), no. 1, c International Press 2009
M ath. Res. Lett. 16 (2009), no. 1, 149 156 c International Press 2009 A 1 BOUNDS FOR CALDERÓN-ZYGMUND OPERATORS RELATED TO A PROBLEM OF MUCKENHOUPT AND WHEEDEN Andrei K. Lerner, Sheldy Ombrosi, and Carlos
More informationVariational estimates for the bilinear iterated Fourier integral
Variational estimates for the bilinear iterated Fourier integral Yen Do, University of Virginia joint with Camil Muscalu and Christoph Thiele Two classical operators in time-frequency analysis: (i) The
More informationMULTILINEAR CALDERÓN ZYGMUND SINGULAR INTEGRALS
MULTILINEAR CALDERÓN ZYGMUND SINGULAR INTEGRALS LOUKAS GRAFAKOS Contents 1. Introduction 1 2. Bilinear Calderón Zygmund operators 4 3. Endpoint estimates and interpolation for bilinear Calderón Zygmund
More informationALMOST ORTHOGONALITY AND A CLASS OF BOUNDED BILINEAR PSEUDODIFFERENTIAL OPERATORS
ALMOST ORTHOGONALITY AND A CLASS OF BOUNDED BILINEAR PSEUDODIFFERENTIAL OPERATORS Abstract. Several results and techniques that generate bilinear alternatives of a celebrated theorem of Calderón and Vaillancourt
More informationarxiv: v1 [math.ca] 29 Dec 2018
A QUANTITATIVE WEIGHTED WEAK-TYPE ESTIMATE FOR CALDERÓN-ZYGMUND OPERATORS CODY B. STOCKDALE arxiv:82.392v [math.ca] 29 Dec 208 Abstract. The purpose of this article is to provide an alternative proof of
More informationA RELATIONSHIP BETWEEN THE DIRICHLET AND REGULARITY PROBLEMS FOR ELLIPTIC EQUATIONS. Zhongwei Shen
A RELATIONSHIP BETWEEN THE DIRICHLET AND REGULARITY PROBLEMS FOR ELLIPTIC EQUATIONS Zhongwei Shen Abstract. Let L = diva be a real, symmetric second order elliptic operator with bounded measurable coefficients.
More informationHardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus.
Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus. Xuan Thinh Duong (Macquarie University, Australia) Joint work with Ji Li, Zhongshan
More informationON HÖRMANDER S CONDITION FOR SINGULAR INTEGRALS
EVISTA DE LA UNIÓN MATEMÁTICA AGENTINA Volumen 45, Número 1, 2004, Páginas 7 14 ON HÖMANDE S CONDITION FO SINGULA INTEGALS M. LOENTE, M.S. IVEOS AND A. DE LA TOE 1. Introduction In this note we present
More informationATOMIC DECOMPOSITIONS AND OPERATORS ON HARDY SPACES
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Volumen 50, Número 2, 2009, Páginas 15 22 ATOMIC DECOMPOSITIONS AND OPERATORS ON HARDY SPACES STEFANO MEDA, PETER SJÖGREN AND MARIA VALLARINO Abstract. This paper
More informationV. CHOUSIONIS AND X. TOLSA
THE T THEOEM V. CHOUSIONIS AND X. TOLSA Introduction These are the notes of a short course given by X. Tolsa at the Universitat Autònoma de Barcelona between November and December of 202. The notes have
More informationDyadic structure theorems for multiparameter function spaces
Rev. Mat. Iberoam., 32 c European Mathematical Society Dyadic structure theorems for multiparameter function spaces Ji Li, Jill Pipher and Lesley A. Ward Abstract. We prove that the multiparameter (product)
More informationWeighted restricted weak type inequalities
Weighted restricted weak type inequalities Eduard Roure Perdices Universitat de Barcelona Joint work with M. J. Carro May 18, 2018 1 / 32 Introduction Abstract We review classical results concerning the
More informationSOBOLEV SPACE ESTIMATES AND SYMBOLIC CALCULUS FOR BILINEAR PSEUDODIFFERENTIAL OPERATORS
SOBOLEV SPACE ESTIMATES AND SYMBOLIC CALCULUS FOR BILINEAR PSEUDODIFFERENTIAL OPERATORS Abstract. Bilinear operators are investigated in the context of Sobolev spaces and various techniques useful in the
More informationTWO COUNTEREXAMPLES IN THE THEORY OF SINGULAR INTEGRALS. 1. Introduction We denote the Fourier transform of a complex-valued function f(t) on R d by
TWO COUNTEREXAMPLES IN THE THEORY OF SINGULAR INTEGRALS LOUKAS GRAFAKOS Abstract. In these lectures we discuss examples that are relevant to two questions in the theory of singular integrals. The first
More informationMAXIMAL AVERAGE ALONG VARIABLE LINES. 1. Introduction
MAXIMAL AVERAGE ALONG VARIABLE LINES JOONIL KIM Abstract. We prove the L p boundedness of the maximal operator associated with a family of lines l x = {(x, x 2) t(, a(x )) : t [0, )} when a is a positive
More informationGeometric-arithmetic averaging of dyadic weights
Geometric-arithmetic averaging of dyadic weights Jill Pipher Department of Mathematics Brown University Providence, RI 2912 jpipher@math.brown.edu Lesley A. Ward School of Mathematics and Statistics University
More informationCharacterization of multi parameter BMO spaces through commutators
Characterization of multi parameter BMO spaces through commutators Stefanie Petermichl Université Paul Sabatier IWOTA Chemnitz August 2017 S. Petermichl (Université Paul Sabatier) Commutators and BMO Chemnitz
More informationOn a class of pseudodifferential operators with mixed homogeneities
On a class of pseudodifferential operators with mixed homogeneities Po-Lam Yung University of Oxford July 25, 2014 Introduction Joint work with E. Stein (and an outgrowth of work of Nagel-Ricci-Stein-Wainger,
More informationJUHA KINNUNEN. Harmonic Analysis
JUHA KINNUNEN Harmonic Analysis Department of Mathematics and Systems Analysis, Aalto University 27 Contents Calderón-Zygmund decomposition. Dyadic subcubes of a cube.........................2 Dyadic cubes
More informationRESTRICTED WEAK TYPE VERSUS WEAK TYPE
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 133, Number 4, Pages 1075 1081 S 0002-9939(04)07791-3 Article electronically published on November 1, 2004 RESTRICTED WEAK TYPE VERSUS WEAK TYPE
More informationA REMARK ON AN EQUATION OF WAVE MAPS TYPE WITH VARIABLE COEFFICIENTS
A REMARK ON AN EQUATION OF WAVE MAPS TYPE WITH VARIABLE COEFFICIENTS DAN-ANDREI GEBA Abstract. We obtain a sharp local well-posedness result for an equation of wave maps type with variable coefficients.
More informationA NEW PROOF OF THE ATOMIC DECOMPOSITION OF HARDY SPACES
A NEW PROOF OF THE ATOMIC DECOMPOSITION OF HARDY SPACES S. DEKEL, G. KERKYACHARIAN, G. KYRIAZIS, AND P. PETRUSHEV Abstract. A new proof is given of the atomic decomposition of Hardy spaces H p, 0 < p 1,
More informationA PHYSICAL SPACE PROOF OF THE BILINEAR STRICHARTZ AND LOCAL SMOOTHING ESTIMATES FOR THE SCHRÖDINGER EQUATION
A PHYSICAL SPACE PROOF OF THE BILINEAR STRICHARTZ AND LOCAL SMOOTHING ESTIMATES FOR THE SCHRÖDINGER EQUATION TERENCE TAO Abstract. Let d 1, and let u, v : R R d C be Schwartz space solutions to the Schrödinger
More informationThe oblique derivative problem for general elliptic systems in Lipschitz domains
M. MITREA The oblique derivative problem for general elliptic systems in Lipschitz domains Let M be a smooth, oriented, connected, compact, boundaryless manifold of real dimension m, and let T M and T
More informationHARMONIC ANALYSIS TERENCE TAO
HARMONIC ANALYSIS TERENCE TAO Analysis in general tends to revolve around the study of general classes of functions (often real-valued or complex-valued) and operators (which take one or more functions
More informationBoundedness of Fourier integral operators on Fourier Lebesgue spaces and related topics
Boundedness of Fourier integral operators on Fourier Lebesgue spaces and related topics Fabio Nicola (joint work with Elena Cordero and Luigi Rodino) Dipartimento di Matematica Politecnico di Torino Applied
More informationContents. 1. Introduction Decomposition 5 arxiv:math/ v4 [math.ca] 9 Aug Complements The Central Lemmas 11
Contents 1. Introduction 2 2. Decomposition 5 arxiv:math/0307008v4 [math.ca] 9 Aug 2005 2.1. Complements 10 3. The Central Lemmas 11 3.1. Complements 14 4. The Density Lemma 14 4.1. Complements 15 5. The
More informationJordan Journal of Mathematics and Statistics (JJMS) 9(1), 2016, pp BOUNDEDNESS OF COMMUTATORS ON HERZ-TYPE HARDY SPACES WITH VARIABLE EXPONENT
Jordan Journal of Mathematics and Statistics (JJMS 9(1, 2016, pp 17-30 BOUNDEDNESS OF COMMUTATORS ON HERZ-TYPE HARDY SPACES WITH VARIABLE EXPONENT WANG HONGBIN Abstract. In this paper, we obtain the boundedness
More informationIn this note we give a rather simple proof of the A 2 conjecture recently settled by T. Hytönen [7]. Theorem 1.1. For any w A 2,
A SIMPLE PROOF OF THE A 2 CONJECTURE ANDREI K. LERNER Abstract. We give a simple proof of the A 2 conecture proved recently by T. Hytönen. Our proof avoids completely the notion of the Haar shift operator,
More informationChapter One. The Calderón-Zygmund Theory I: Ellipticity
Chapter One The Calderón-Zygmund Theory I: Ellipticity Our story begins with a classical situation: convolution with homogeneous, Calderón- Zygmund ( kernels on R n. Let S n 1 R n denote the unit sphere
More informationCarlos Pérez University of the Basque Country and Ikerbasque 1 2. School on Nonlinear Analysis, Function Spaces and Applications T re st, June 2014
A theory of weights for singular integral operators Carlos Pérez University of the Basque Country and Ikerbasque 2 School on Nonlinear Analysis, Function Spaces and Applications T re st, June 204 Introduction:
More informationON MAXIMAL FUNCTIONS FOR MIKHLIN-HÖRMANDER MULTIPLIERS. 1. Introduction
ON MAXIMAL FUNCTIONS FOR MIKHLIN-HÖRMANDER MULTIPLIERS LOUKAS GRAFAKOS, PETR HONZÍK, ANDREAS SEEGER Abstract. Given Mihlin-Hörmander multipliers m i, i = 1,..., N, with uniform estimates we prove an optimal
More informationWEAK TYPE ESTIMATES FOR SINGULAR INTEGRALS RELATED TO A DUAL PROBLEM OF MUCKENHOUPT-WHEEDEN
WEAK TYPE ESTIMATES FOR SINGULAR INTEGRALS RELATED TO A DUAL PROBLEM OF MUCKENHOUPT-WHEEDEN ANDREI K. LERNER, SHELDY OMBROSI, AND CARLOS PÉREZ Abstract. A ell knon open problem of Muckenhoupt-Wheeden says
More informationSharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces
Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 1/45 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces Dachun Yang (Joint work) dcyang@bnu.edu.cn
More informationL p -boundedness of the Hilbert transform
L p -boundedness of the Hilbert transform Kunal Narayan Chaudhury Abstract The Hilbert transform is essentially the only singular operator in one dimension. This undoubtedly makes it one of the the most
More informationUnimodular Bilinear multipliers on L p spaces
Jotsaroop Kaur (joint work with Saurabh Shrivastava) Department of Mathematics, IISER Bhopal December 18, 2017 Fourier Multiplier Let m L (R n ), we define the Fourier multiplier operator as follows :
More informationHAAR SHIFTS, COMMUTATORS, AND HANKEL OPERATORS
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Volumen 50, Número 2, 2009, Páginas 3 HAAR SHIFTS, COMMUTATORS, AND HANKEL OPERATORS MICHAEL LACEY Abstract. Hankel operators lie at the junction of analytic and
More informationVECTOR A 2 WEIGHTS AND A HARDY-LITTLEWOOD MAXIMAL FUNCTION
VECTOR A 2 WEIGHTS AND A HARDY-LITTLEWOOD MAXIMAL FUNCTION MICHAEL CHRIST AND MICHAEL GOLDBERG Abstract. An analogue of the Hardy-Littlewood maximal function is introduced, for functions taking values
More informationBorderline variants of the Muckenhoupt-Wheeden inequality
Borderline variants of the Muckenhoupt-Wheeden inequality Carlos Domingo-Salazar Universitat de Barcelona joint work with M Lacey and G Rey 3CJI Murcia, Spain September 10, 2015 The Hardy-Littlewood maximal
More informationMODULATION INVARIANT BILINEAR T(1) THEOREM
MODULATION INVARIANT BILINEAR T(1) THEOREM ÁRPÁD BÉNYI, CIPRIAN DEMETER, ANDREA R. NAHMOD, CHRISTOPH M. THIELE, RODOLFO H. TORRES, AND PACO VILLARROYA Abstract. We prove a T(1) theorem for bilinear singular
More informationTeoría de pesos para operadores multilineales, extensiones vectoriales y extrapolación
Teoría de pesos para operadores multilineales, extensiones vectoriales y extrapolación Encuentro Análisis Funcional Sheldy Ombrosi Universidad Nacional del Sur y CONICET 8 de marzo de 208 If f is a locally
More informationMULTIPARAMETER RIESZ COMMUTATORS. 1. Introduction
MULTIPARAMETER RIESZ COMMUTATORS MICHAEL T. LACEY 1, STEFANIE PETERMICHL 2, JILL C. PIPHER 3, AND BRETT D. WICK 4 Abstract. It is shown that product BMO of S.-Y. A. Chang and R. Fefferman, ined on the
More informationCommutators of Weighted Lipschitz Functions and Multilinear Singular Integrals with Non-Smooth Kernels
Appl. Math. J. Chinese Univ. 20, 26(3: 353-367 Commutators of Weighted Lipschitz Functions and Multilinear Singular Integrals with Non-Smooth Kernels LIAN Jia-li MA o-lin 2 WU Huo-xiong 3 Abstract. This
More informationA COUNTEREXAMPLE TO AN ENDPOINT BILINEAR STRICHARTZ INEQUALITY TERENCE TAO. t L x (R R2 ) f L 2 x (R2 )
Electronic Journal of Differential Equations, Vol. 2006(2006), No. 5, pp. 6. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) A COUNTEREXAMPLE
More informationA characterization of the two weight norm inequality for the Hilbert transform
A characterization of the two weight norm inequality for the Hilbert transform Ignacio Uriarte-Tuero (joint works with Michael T. Lacey, Eric T. Sawyer, and Chun-Yen Shen) September 10, 2011 A p condition
More informationGood Lambda Inequalities and Riesz Potentials for Non Doubling Measures in R n
Good Lambda Inequalities and Riesz Potentials for Non Doubling Measures in R n Mukta Bhandari mukta@math.ksu.edu Advisor Professor Charles Moore Department of Mathematics Kansas State University Prairie
More informationSOLUTIONS TO HOMEWORK ASSIGNMENT 4
SOLUTIONS TO HOMEWOK ASSIGNMENT 4 Exercise. A criterion for the image under the Hilbert transform to belong to L Let φ S be given. Show that Hφ L if and only if φx dx = 0. Solution: Suppose first that
More informationNotes. 1 Fourier transform and L p spaces. March 9, For a function in f L 1 (R n ) define the Fourier transform. ˆf(ξ) = f(x)e 2πi x,ξ dx.
Notes March 9, 27 1 Fourier transform and L p spaces For a function in f L 1 (R n ) define the Fourier transform ˆf(ξ) = f(x)e 2πi x,ξ dx. Properties R n 1. f g = ˆfĝ 2. δλ (f)(ξ) = ˆf(λξ), where δ λ f(x)
More informationx i y i f(y) dy, x y m+1
MATRIX WEIGHTS, LITTLEWOOD PALEY INEQUALITIES AND THE RIESZ TRANSFORMS NICHOLAS BOROS AND NIKOLAOS PATTAKOS Abstract. In the following, we will discuss weighted estimates for the squares of the Riesz transforms
More informationAlgebras of singular integral operators with kernels controlled by multiple norms
Algebras of singular integral operators with kernels controlled by multiple norms Alexander Nagel Conference in Harmonic Analysis in Honor of Michael Christ This is a report on joint work with Fulvio Ricci,
More informationSTRONG TYPE INEQUALITIES AND AN ALMOST-ORTHOGONALITY PRINCIPLE FOR FAMILIES OF MAXIMAL OPERATORS ALONG DIRECTIONS IN R 2
STRONG TYPE INEQUALITIES AND AN ALMOST-ORTHOGONALITY PRINCIPLE FOR FAMILIES OF MAXIMAL OPERATORS ALONG DIRECTIONS IN R 2 ANGELES ALFONSECA Abstract In this aer we rove an almost-orthogonality rincile for
More informationIn this work we consider mixed weighted weak-type inequalities of the form. f(x) Mu(x)v(x) dx, v(x) : > t C
MIXED WEAK TYPE ESTIMATES: EXAMPLES AND COUNTEREXAMPLES RELATED TO A PROBLEM OF E. SAWYER S.OMBROSI AND C. PÉREZ Abstract. In this paper we study mixed weighted weak-type inequalities for families of functions,
More informationWeighted Littlewood-Paley inequalities
Nicolaus Copernicus University, Toruń, Poland 3-7 June 2013, Herrnhut, Germany for every interval I in R let S I denote the partial sum operator, i.e., (S I f ) = χ I f (f L2 (R)), for an arbitrary family
More informationA NOTE ON WAVELET EXPANSIONS FOR DYADIC BMO FUNCTIONS IN SPACES OF HOMOGENEOUS TYPE
EVISTA DE LA UNIÓN MATEMÁTICA AGENTINA Vol. 59, No., 208, Pages 57 72 Published online: July 3, 207 A NOTE ON WAVELET EXPANSIONS FO DYADIC BMO FUNCTIONS IN SPACES OF HOMOGENEOUS TYPE AQUEL CESCIMBENI AND
More informationNEW MAXIMAL FUNCTIONS AND MULTIPLE WEIGHTS FOR THE MULTILINEAR CALDERÓN-ZYGMUND THEORY
NEW MAXIMAL FUNCTIONS AND MULTIPLE WEIGHTS FOR THE MULTILINEAR CALDERÓN-ZYGMUND THEORY ANDREI K. LERNER, SHELDY OMBROSI, CARLOS PÉREZ, RODOLFO H. TORRES, AND RODRIGO TRUJILLO-GONZÁLEZ Abstract. A multi(sub)linear
More informationSingular Integrals. 1 Calderon-Zygmund decomposition
Singular Integrals Analysis III Calderon-Zygmund decomposition Let f be an integrable function f dx 0, f = g + b with g Cα almost everywhere, with b
More informationarxiv: v3 [math.ca] 6 Oct 2009
SHARP A 2 INEQUALITY FOR HAAR SHIFT OPERATORS arxiv:09061941v3 [mathca] 6 Oct 2009 MICHAEL T LACEY, STEFANIE PETERMICHL, AND MARIA CARMEN REGUERA Abstract As a Corollary to the main result of the paper
More informationSHARP BOUNDS FOR GENERAL COMMUTATORS ON WEIGHTED LEBESGUE SPACES. date: Feb
SHARP BOUNDS FOR GENERAL COMMUTATORS ON WEIGHTED LEBESGUE SPACES DAEWON CHUNG, CRISTINA PEREYRA AND CARLOS PEREZ. Abstract. We show that if an operator T is bounded on weighted Lebesgue space L 2 (w) and
More informationGlobal well-posedness for KdV in Sobolev spaces of negative index
Electronic Journal of Differential Equations, Vol. (), No. 6, pp. 7. ISSN: 7-669. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Global well-posedness for
More informationBILINEAR FOURIER INTEGRAL OPERATORS
BILINEAR FOURIER INTEGRAL OPERATORS LOUKAS GRAFAKOS AND MARCO M. PELOSO Abstract. We study the boundedness of bilinear Fourier integral operators on products of Lebesgue spaces. These operators are obtained
More informationA new class of pseudodifferential operators with mixed homogenities
A new class of pseudodifferential operators with mixed homogenities Po-Lam Yung University of Oxford Jan 20, 2014 Introduction Given a smooth distribution of hyperplanes on R N (or more generally on a
More informationCarleson Measures for Besov-Sobolev Spaces and Non-Homogeneous Harmonic Analysis
Carleson Measures for Besov-Sobolev Spaces and Non-Homogeneous Harmonic Analysis Brett D. Wick Georgia Institute of Technology School of Mathematics & Humboldt Fellow Institut für Mathematik Universität
More informationBilinear operators, commutators and smoothing
Bilinear operators, commutators and smoothing Árpád Bényi Department of Mathematics Western Washington University, USA Harmonic Analysis and its Applications Matsumoto, August 24-28, 2016 Á. B. s research
More information(b) If f L p (R), with 1 < p, then Mf L p (R) and. Mf L p (R) C(p) f L p (R) with C(p) depending only on p.
Lecture 3: Carleson Measures via Harmonic Analysis Much of the argument from this section is taken from the book by Garnett, []. The interested reader can also see variants of this argument in the book
More informationPotential Analysis meets Geometric Measure Theory
Potential Analysis meets Geometric Measure Theory T. Toro Abstract A central question in Potential Theory is the extend to which the geometry of a domain influences the boundary regularity of the solution
More informationMaximal Functions in Analysis
Maximal Functions in Analysis Robert Fefferman June, 5 The University of Chicago REU Scribe: Philip Ascher Abstract This will be a self-contained introduction to the theory of maximal functions, which
More informationarxiv: v3 [math.ca] 26 Apr 2017
arxiv:1606.03925v3 [math.ca] 26 Apr 2017 SPARSE DOMINATION THEOREM FOR MULTILINEAR SINGULAR INTEGRAL OPERATORS WITH L r -HÖRMANDER CONDITION KANGWEI LI Abstract. In this note, we show that if T is a multilinear
More informationSINGULAR INTEGRAL OPERATORS WITH ROUGH CONVOLUTION KERNELS. Andreas Seeger University of Wisconsin-Madison
SINGULA INTEGAL OPEATOS WITH OUGH CONVOLUTION KENELS Andreas Seeger University of Wisconsin-Madison. Introduction The purpose of this paper is to investigate the behavior on L ( d ), d, of a class of singular
More informationOn the local existence for an active scalar equation in critical regularity setting
On the local existence for an active scalar equation in critical regularity setting Walter Rusin Department of Mathematics, Oklahoma State University, Stillwater, OK 7478 Fei Wang Department of Mathematics,
More informationHardy-Stein identity and Square functions
Hardy-Stein identity and Square functions Daesung Kim (joint work with Rodrigo Bañuelos) Department of Mathematics Purdue University March 28, 217 Daesung Kim (Purdue) Hardy-Stein identity UIUC 217 1 /
More information