A crash course in calderonzygmund singular integrals and. Calder onzygmund theory of singular integrals handout. For any 0 pseudodifferential operators and the cauchy integral of calderon lecture notes in mathematics on free shipping on qualified orders. Calderon zygmund regularity estimates for weak solutions of singular and degenerate elliptic equations. Pseudolocalization of singular integrals and noncommutative calderonzygmund theory. The zygmundcalderon lectures are named after antoni zygmund 19001992 and alberto calderon 19201998. Chapter xi calderon zygmund singular integral operators 1. Bilinear integral operators appear naturally in many contexts and have been a source of many challenging problems in harmonic analysis. Huixia mo, hongyang xue submitted on 10 jun 2015 v1, last revised 16 nov 2015 this version, v2.
A multisublinear maximal operator that acts on the product of m lebesgue spaces and is smaller that the mfold product of the hardylittlewood. Zygmund, an example in the theory of singular integrals, studia math. Extremal problems in interpolation theory, whitney. Calderonzygmund theory for secondorder elliptic systems on. I am going to survey a number of recent nonlinear regularity results that, put together, outline what might be considered a nonlinear version of caldero. Singular integrals and differentiability properties of. A characterization of compactness for singular integrals. There are many other textbooks on measure theory that cover most or all of the material in the present book, as well as much more, perhaps from somewhat different view points. Boundedness of commutators generated by mth calderonzygmund type singular integrals and local campanato functions on generalized local morrey spaces authors. The benedek calderon panzone principle in this chapter we extend the continuity results we discussed for the riesz transforms and more general odd kernels in chapter x to arbitrary calderon zygmund singular integral kernels. Pdf in this paper, we study some weighted norm inequalities related to the calderonzygmund singular integrals. Calderonzygmund regularity estimates for weak solutions of. Variants of the calderonzygmund theory for lpspaces.
In many cases the minimizers constructed in the book are stable i. Full text full text is available as a scanned copy of the original print version. Fourier integrals, plancheral, parceval identities. Sobolev spaces, distributions, interpolation inequalities 7. Fractional derivatives and pseudo differential operators 8. Commutators of singular integral operators satisfying a. The main goal here is to develop a calderonzygmund theory for singular integral operators of boundary layer potential type naturally associated with secondorder elliptic systems on riemannian manifolds which is effective in the treatment of boundary value problems in rough domains. Moreover, in the following, when treating equations as in 1. Stein 1 n otices of the ams v olume 45, number 9 t he subject matter of this essay is alberto calderons pivotal role in the creation of the modern theory of singular integrals. On weighted hadamardtype singular integrals and their applications xu, yong jia, abstract and applied analysis, 2007. Several tools and techniques for the study of such operators are discussed. References i have used in preparing these notes include steins singular integrals and differentiability properties of. Pseudolocalization of singular integrals and noncommutative calderonzygmund theory javier parcet contents introduction 1 1. In this talk, i would like to report several results on sobolev regularity theory of calderon zygmund type for weak solutions of elliptic equations in divergence form with degenerate, singular coefficients.
Cordes boundary value problems for secondorder parabolic equations 82 e. Introduction to calderonzygmund operators in this lecture we will start our study of calder onzygmund operators in the oneparameter. Buy singular integrals and differentiability properties of functions. Finally, some applications of the main results are demonstrated by examples. In this dissertation we further develop the theory of commutators of multilinear singular integral operators with pointwise multiplication. The theory of singular integrals started with the study of singular. The theory developed includes strong type and endpoint weak type estimates, interpolation, a multilinear t1 theorem, and a variety of results regarding multilinear multiplier operators. Weak estimates, marcinkiewicz interpolation theorem lorentz spaces 3. To that aim, we have developed a pseudolocalization principle for singular integrals which is of independent interest, even in the classical theory. In the third of the distinguished lecture series given by eli stein here at ucla, eli presented a slightly different topic, which is work in preparation with alex nagel, fulvio ricci, and steve wainger, on algebras of singular integral operators which are sensitive to multiple different geometries in a nilpotent lie group. The main result of this paper is the following weak type llog. The hilbert transform on the real line l is a typical example. Generally speaking, the commutator of two operators is itself an operator that measures the changes which occur when switching the order in which the commuted operators are being applied.
While the new method is presented in great detail in the second part, the first is mainly devoted to the prerequisites needed for a selfcontained. Introduction in the last ten years, it has been of great interest to obtain optimal operator. Zygmundcalderon lectures in analysis department of. Rd c, where rd denotes euclidean space and c denotes. In order to state the result we must first define some terms. We develop a wide general theory of bilinear biparameter singular integrals t. Lecture notes on singular integrals, projections, multipliers and.
Singular integrals 1 calderonzygmund decomposition people. This is the first part, covering some motivation, the case of a single frequency projection and its vectorvalued generalisation. It is named for the mathematicians alberto calderon and antoni zygmund. Calderonzygmund capacities and operators on nonhomogeneous spaces cbms regional conference series in mathematics 9780821832523. In the 1980s and early 1990s, coifman, weiss, and christ noticed that the theory of calderonzygmund operators can be generalized from euclidean spaces to spaces of homogeneous type. Kx,yfydy, whose associated kernel function k is singular along the diagonal x y.
Pdf a survey of known results in the theory of convolution type. Stefanov, lp bounds for singular integrals and maximal singular integrals with rough kernels, indiana univ. In this paper we obtain the weak type 1,1 boundedness of calderonzygmund operators acting over operatorvalued functions. The big contribution to the general theory of operators that share similar properties with the hilbert transform is due to the work of. Calderonzygmund capacities and operators on nonhomogeneous. Boundedness of maximal singular integral operators on spaces of homogeneous type and its applications hu, guoen, yang, dachun, and yang, dongyong, journal of the mathematical society of japan, 2007. Pdf convolution calderonzygmund singular integral operators. This article deals with the boundedness properties of singular integrals that generalize the double hilbert transform on product domains. Towards a nonlinear calderon zygmund theory giuseppe mingione to tadeusz iwaniec on his 60th birthday contents 1. Pdf bilinear calderonzygmund theory on product spaces. The theory has had quite a success in the solution of many problems in both real. New maximal functions and multiple weights for the.
In that great enterprise calderon had the good fortune of working with antoni zygmund, who was. A third type of integrals suggested by the theory of spherical summa. Questions of this type have been traditionally answered using the theory of singular integrals and using harmonic analysis. First, we prove a dyadic representation theorem starting from t1assumptions and apply it to show many estimates. The canonical example of such an operator is the hilbert transform, which is given by.
In their pioneering papers which appeared in american journal of mathematics1956 a. This created the chicago school of hard analysis sometimes simply. A variety of results regarding multilinear calderonzygmund sin gular integral operators is systematically presented. Calderonzygmund theory, multilinear operators, pseudodifferential op erators, interpolation, t1 theorem, littlewoodpaley theory, wavelets, schurs test. The calderonzygmund theory for elliptic problems with. Bilinear littlewoodpaley square functions and singular.
Contents 1 a brief introduction to the a p theory of weights 4 2 lp properties of functions with controlled oscillation 9. Singular integrals nowadays appear in partial di erential equations, several complex variables, operator theory, and other areas of analysis. Motivation and description of the main results the calderonzygmund theory of singular integral operators was carried over to the vectorvalued setting by several authors, but we have been especially influenced in writing this paper by the work of benedek, calderon, and panzone 2. For every 1 a p a 8 one can extend any singular integral operator t to an operator bounded from lp n lp. Jul 04, 2007 pdf file 4047 kb article info and citation.
Pseudolocalization of singular integrals and noncommutative. Some aspects of vectorvalued singular integrals citeseerx. In mathematics, the calderonzygmund lemma is a fundamental result in fourier analysis, harmonic analysis, and singular integrals. The seminal work on singular integrals by calder on and zygmund, as originated in 5, and the real variable methods later developed have played a crucial and in uential role in modern harmonic analysis.
Zygmund defined a class of kernels that generalized the hilbert transform and riesz transform at the same time to higher dimensional settings and they studied the lp boundedness of these kernels. In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Instead, an endpoint theory was provided for this operator, such as the weak type llog. A crash course in calderonzygmund singular integrals and applications joaquim bruna february 4, 2016 part i. It includes some important research results by the authors and their cooperators, such as singular integrals with rough kernels on block spaces and hardy spaces, the criterion on boundedness of oscillatory singular integrals, and boundedness of the rough marcinkiewicz integrals. Calderonzygmund estimates for parabolic measure data equations. This work presents full proofs of vitushkins conjecture and of the semiadditivity of analytic capacity, both of which remained open problems until very recently. This book introduces some important progress in the theory of calderonzygmund singular integrals, oscillatory singular integrals, and littlewoodpaley theory over the last decade.
An example in the theory of singular integrals, studia math. Interpolation of operators riesz theorem on interpolation in lp spaces. New maximal functions and multiple weights for the multilinear calderonzygmund theory andrei k. A systematic treatment of multilinear calderonzygmund operators is presented. Calderon had the good fortune of work ing with antoni zygmund, who was at first his. A geometric approach to the calderonzygmund estimates abstract.
Arc spaces, motivic integration and stringy invariants veys, willem, 2006. A course in calderonzygmund singular integrals and applications. Calderonzygmund theory of singular integrals handout. Zygmund remarks about theory of singular integral operators 74 h. Our assumptions on the operators are minimal and our result applies to an array of. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. These results are new even in the euclidean space and are the. Multilinear calderonzygmund singular integrals 3 further use of interpolation provides an improved version of 4 in which lp. This volume of essays on fourier analysis and its applications arose from a conference held at princeton university in may 1991 in honour of the mathematician elias steins 60th birthday. Gilbargtrudinger 5, geometric measure theory, geometric group theory, ergodic theory and dynamical systems, and differential topology and geometry. The final chapter of the book develops a general theory of singular integrals that generalizes and unifies these examples.
Broadly speaking a singular integral is an integral operator. Revista matematica iberoamericana, issn 022230, vol. Many important properties of singular integrals have been thoroughly studied for calderonzygmund operators. This is one of the first general theories of multiparameter singular integrals that goes beyond the product theory of singular integrals and their analogs. Lecture notes on singular integrals, projections, multipliers. These results have frequently been cited in many published papers. Giuseppe mingione the standard calderonzygmund theory deals with optimal integrability properties of. A geometric approach to the calderonzygmund estimates 3 corollary 1. The tb theorem provides sufficient conditions for a singular integral operator to be a calderonzygmund operator, that is for a singular integral operator associated to a calderonzygmund kernel to be bounded on l 2. Other related questions are also discussed, such as the relationship between rectifiability and the existence of principal values for the cauchy transforms and other singular integrals. We dominate nonintegral singular operators by adapted sparse operators and derive optimal norm estimates in weighted spaces.
H is a limiting case just outside the scope of that theory, for. L estimate and the weak type h 1, l 1 estimate see 20, 24. It includes some important research results by the authors and their co. The theory of calder onzygmund singular integrals has expanded and interacted with many areas of mathematics and, as several articles in these proceed ings show, it continues to be a very strong and active area of research. Some better than expected estimates for certain multilinear operators are presented too. Commutators of multilinear singular integral operators. Coifman algebras of singular integral operators 18 a. I have written some notes on littlewoodpaley theory for a masterclass, which i thought i would share here as well. Introduction to calderonzygmund operators in this lecture we will start our study of calder onzygmund operators in the oneparameter setting.
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