The exception to this rule is the last paragraph on fourier multipliers where we discuss the comparatively recent marcinkiewicz decomposition of the banach couple hpt. We introduce the following selection criterion for a cube r. It exactly corresponds to the calder onzygmund decomposition in a context of sobolev spaces. Two others studied under zygmunds student elias stein. Decomposition book notebooks are made in the usa using 100% recycled paper, soy ink and biogas. Let us brie y recall the ideas of such decomposition. This article is a subsequent article of 7 and is concerned with the boundedness of multilinear calderon zygmund operators on products of hardy spaces. Stein stated this decomposition for lebesgue spaces as. There is an open set which is a disjoint union of open cubes and its complement f a closed set such that.
The classical proof of calderonzygmund estimates, uses the singular integrals. Citeseerx document details isaac councill, lee giles, pradeep teregowda. In this section, we shall state the higher order poincaretype inequalities for commutator of calderonzygmund singular integral operator acting on the solutions of the diracharmonic equations. We dominate nonintegral singular operators by adapted sparse operators and derive optimal norm estimates in weighted spaces. Let s be the collection of those subcubes k for which 6 z k f t. The book details tolsas solution of painleves and vitushkins. I, the cubes occurring in the calderonzygmund decomposition of f at the. W w a n t t o o b t a i n a calderbnzygmund e decomposition f o r f and. The higher integrability of commutators of calderon. In this section we prove a very useful decomposition theorem for functions. The basic statement of the lemma is that all integrable functions can be decomposed into a good part, where the function is bounded by a small. These are frequently used in mathematics particularly harmonic analysis as a way of discretizing objects in order to make computations or analysis easier.
Multilinear calderonzygmund operators on hardy spaces, ii. For a direct proof, we refer to the book of stein, chapter 5. New calder onzygmund decomposition for sobolev functions. This leads to the associated calderonzygmund decomposition of f, wherein f is written as the sum. The first result generalizes the calderonzygmund decomposition by replacing the. A geometric approach to the calderonzygmund estimates 3 corollary 1. A note on dyadic coverings and nondoubling calderon.
Our dyadic nondoubling calderon zygmund decomposition is partly formulated in terms of these cubes. Calderonzygmund operators, pseudodifferential operators. Calderonzygmund operators, pseudodifferential operators and the cauchy integral of calderon j. Given a banach function space xrn and the maximal caldero. Disjoint union of dyadic cubes in calderonzygmund decomposition.
Calderonzygmund capacities and operators on nonhomogeneous. Classical calderonzygmund decomposition and real interpolation. The theory developed includes strong type and endpoint weak type estimates, interpolation, the multilinear t1 theorem, and a variety of results regarding multilinear multiplier operators. The calderonzygmund decomposition for a function is well known, which says for any, then we can decompose into a good term and a bad term, such that for any, we can make and for each, supported in nonoverlapping cubes and we have the cancellation property, furthermore. Given a general dyadic grid d and a sparse family of cubes s qk j. In mathematics, the calderonzygmund lemma is a fundamental result in fourier analysis. Simple modifications include cubic centered maximal operator. Also, i f q 6 dk is in ct and q i s t h e only cube i n dk, containing q, we. Calderon zygmund cubes which is also the basis of the calderonzygmund decomposition. Our goal in this article is to prove that the action of a sufficiently smooth m linear calderon zygmund operator can be interchanged with infinite sums of atoms. Our assumptions on the operators are minimal and our result applies to an array of. The following are excellent books for an introduction to much of the harmonic analysis.
The calderonzygmund decomposition on product domains. Calderonzygmund decomposition on product domains calderonzygmund lemma chang and fe erman. The calderonzygmund decomposition possesses a series of simple but important properties. We have taken the liberty to do as we do because, in fact, in this book we write. The calderonzygmund decomposition as metaphor otium. A geometric approach to the calderon zygmund estimates 3 corollary 1. Replacing dyadic cubes by dyadic calderonzygmund cubes.
In mathematics, the dyadic cubes are a collection of cubes in r n of different sizes or scales such that the set of cubes of each scale partition r n and each cube in one scale may be written as a union of cubes of a smaller scale. Its also a part of a reframe of how i think since i started being immersed in this field. The calderonzygmund decomposition is a classic tool in harmonic analysis. R d c, where r d denotes euclidean space and c denotes the complex numbers, the lemma gives a precise way of partitioning r d into two sets. Omega be the calderonzygmund singular integral operator on differential forms and b be sufficiently smooth and bounded. New maximal functions and multiple weights for the. A systematic treatment of multilinear calder onzygmund operators is presented. Calderonzygmund capacities and operators on nonhomogeneous spaces.
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