The marcinkiewicz interpolation theorem and its extensions author. In this paper we focus on the two standard such methods, the kmethod and the jmethod, which are complementary, dual constructions due to peetre and lions see e. Some extensions of the marcinkiewicz interpolation theorem in terms of modular inequalities article pdf available in journal of the mathematical society of japan 552003 april 2003 with 58 reads. In fact such techniques as the interpolation theorem of marcinkiewicz, the application of harmonic functions, and the littlewoodpaleygfunction, are essential parts of the theory detailed below. In this paper, we prove sharp lp estimates of marcinkiewicz integral operators with rough kernels on product spaces. We treat two essentially different interpolation methods. Special cases of the marcinkiewicz interpolation theorem. An optimal interpolation theorem of marcinkiewicz type in. We continue the study with the kmethod of interpolation, which it may be regarded as a lifting of the marcinkiewicz interpolation theorem from its classical context in spaces of. On a theorem of marcinkiewicz and zygmund springerlink. An extension of the grunwaldmarcinkiewicz interpolation theorem article pdf available in bulletin of the australian mathematical society 6302 april 2001 with 44 reads how we measure reads.
We provide a selfcontained proof of the multilinear extension of the marcinkiewicz real method interpolation theorem with initial assumptions a set. Click download or read online button to get discrete cosine and sine transforms book now. An extension of the marcinkiewicz interpolation theorem to lorentz spaces. We show that the marcinkiewicz theorem on the interpolation of operators acting on. There is a direct and selfcontained proof of hls inequality in analysis by lieb and loss, theorem 4. The aim of this article is to obtain two anisotropic interpolation theorems of musielakorlicz type, which are weighted anisotropic extension of marcinkiewicz interpolation theorems. Both theorems have their own advantages and disadvantages, and will be used frequently.
Interpolation we can often deduce that t is bounded for intermediate values of p just by considering end points. An extension of the marcinkiewicz interpolation theorem is proved, yielding a necessary and sufficient condition for every quasilinear operator, satisfying given endpoint estimates of weak type, to be bounded from an orlicz space into another. We continue the study with the kmethod of interpolation, which it may be regarded as a lifting of the marcinkiewicz interpolation theorem from its classical context in spaces. If a linear or sublinear mapping tis of weak type 1,1. A question about general marcinkiewicz interpolation theorem. Click download or read online button to get introduction to topology and modern analysis book now. 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. In this paper, we discuss the boundedness of the fractional type marcinkiewicz integral operators associated to surfaces, and we extend a result given by chen et al. Using a simple method, we generalize marcinkiewicz interpolation theorem to operators on orlicz space and apply it to several important theorems in harmonic analysis. Abstractan extension of the marcinkiewicz interpolation theorem is proved, yielding a necessary and sufficient condition for every quasilinear operator, satisfying given endpoint estimates of weak type, to be bounded from an orlicz space into another. Introduction to topology and modern analysis download ebook. Our treatment of the rieszthorin interpolation theorem follows 8. Let t be a linear operator such that the form 1 is wellde.
The advantage of this proof is that it yields a logarithmically convex bound for the norm of the operator on the intermediate spaces in terms of the initial restricted. I am trying to prove a form of the marcinkiewicz interpolation theorem. Pdf marcinkiewicz interpolation theorems for orlicz and. Anisotropic interpolation theorems of musielakorlicz type. The theorem will be used for obtaining important inequalities in harmonic analysis, so the working method uses ele. Marcinkiewicz wrote 55 scientific papers in six years 19331939 and his name in mathematics is connected with the marcinkiewicz interpolation theorem, marcinkiewicz spaces, the marcinkiewicz integral and marcinkiewicz function, the marcinkiewicz multiplier theorem, the jessen marcinkiewicz zygmund strong differentiation theorem, marcinkiewicz. The fourier transform and convolution 12 acknowledgments 14 references 14 1.
Marcinkiewicz integrals on product spaces and extrapolation. In 1937 marcinkiewicz and zygmund obtained an estimate of the norm inl q 0,1 of the sum of the series. In this paper, we will apply the technique of atomic decompositions to show two marcinkiewicz type interpolation theorems for weak orlicz martingale spaces, one of which can be regarded as martingale version of marcinkiewicz type interpolation theorem obtained in. The marcinkiewicz multiplier theorem 3 assumes the multiplier. An essay on the interpolation theorem of jozef marcinkiewicz.
Discrete cosine and sine transforms download ebook pdf. Interpolation theorems play a very important role in harmonic analysis. On a theorem of marcinkiewicz concerning interpolation of. Introduction the use of interpolation theorems is the rst step in proving.
Intensive program on interpolation and noncommutative. A sharp form of the marcinkiewicz interpolation theorem for orlicz. The marcinkiewicz theorem is extended to function spaces that include both the lebesgueorlicz and lorentz spaces, namely the rearrangement invariant function spaces. Approximation theory deals with two types of problems. The rieszthorin interpolation theorem that motivates the complex interpolation and was proved by riesz in 1927 but only for the lowertriangle case, and the general case by thorin in 1938. Marcinkiewicz interpolation theorem lp estimate for the newtonian potential w1,p estimate for n. This thesis is primarily devoted to the study of the marcinkiewicz interpolation theorem and its applications. Calderon the purpose of the present paper is to prove the following result. Chaudhury our goal is to prove the following interpolation theorem for any subadditive and, of course, linear operator acting between lp spaces. In memory of polish mathematicians murdured by the soviets and the nazis. Marcinkiewicz interpolation theorems for orlicz and lorentz gamma spaces ron kerman, colin phipps, and lubo s pick abstract. Marcinkiewicz interpolation theorem to get the above bound for all 1 ft1y. The theorem will follow from the results of section 2 below.
The interpolation theory was aimed in the two classical theorems. We find necessary density conditions for marcinkiewiczzygmund inequalities and interpolation for spaces of spherical harmonics in s d with respect to the l p norm. Notice that our above example shows why we could not expect an. The marcinkiewicz interpolation theorem and its extensions. This paper is devoted to the interpolation principle between spaces of. Mathematics and its applications east european series, vol 41. Fractional type marcinkiewicz integral operators associated. Real analysis, problem set 4 in this problem set, we study the proofs of sierpinskis estimate for the gauss circle problem, decay estimates for pde, and the marcinkiewicz interpolation theorem. Moreover, we prove that there are no complete interpolation families for p.
These estimates are used in an extrapolation arguments to obtain some new improved and extended results in marcinkiewicz integrals. In general this does not hold true in two and more variables. Marcinkiewicz interpolation theorems for orlicz and lorentz gamma spaces. Every quasilinear operator t of weaktypes p0,p0 and p1,p1. Mfx gis open, because if x2e, there exists a ball bsuch that x2band 1 jbj z b jfyjdy the hardylittlewood maximal function is a tool which can be used to study the identity operator. Interpolation theorems and applications contents 1. These two methods are modelled on the proofs of the marcinkiewicz theorem and the rieszthorin theorem respectively, as they are given in chapter 1. The lecture notes were prepared by two former students in the class. Theorem weak p0,p0 and p1,p1 strong p,p, p0 marcinkiewicz in his short life, his talent. Here, we prove a different interpolation theorem using complex analytic methods. A course in calderonzygmund singular integrals and applications. Marcinkiewiczzygmund inequalities and interpolation by.
A course in calderonzygmund singular integrals and. In this paper it is shown that this condition cannot be discarded. A weighted hardy space variant of the marcinkiewicz interpolation theorem by d. Pdf marcinkiewicz interpolation theorem and marcinkiewicz spaces. Generally an interpolation method is an functor ffrom the cathegory of admissible. They showed that under certain conditions the fractional type marcinkiewicz integral operators are bounded from the triebellizorkin spaces f. A introduction to interpolation theory, interpolation spaces and interpolation groups. Marcinkijevi ceva teorema interpolacije sa primjenom the. Rieszthorin, marcinkiewicz matt ziemke ifwenoticethat,for1 theorem on the almost everywhere convergence of fourier series of lp functions. In mathematics, the marcinkiewicz interpolation theorem, discovered by jozef marcinkiewicz, is a result bounding the norms of nonlinear operators acting on l p spaces marcinkiewicz theorem is similar to the rieszthorin theorem about linear operators, but also applies to nonlinear operators. In mathematics, the marcinkiewicz interpolation theorem, discovered by jozef marcinkiewicz 1939, is a result bounding the norms of nonlinear operators.
Introduction the marcinkiewicz interpolation theorem states that, if 1. Introduction in this paper we present two main classical results of interpolation of operators. Also, the boundedness of hardylittlewood maximal function is much more straightforward than the general marcinkiewicz interpolation theorem. The total record of accomplishments of marcinkiewicz in his short life, his talent.
This approach was used for example in the classical book of. Marcinkiewicz interpolation theorem see 8 which we state for completeness of presentation. The marcinkiewicz interpolation theorem 253 distribution. Marcinkiewicz interpolation theorem 3 corresponding to the multi linear riesz. Pdf some extensions of the marcinkiewicz interpolation.
In mathematics, the marcinkiewicz interpolation theorem, discovered by jozef marcinkiewicz, is a result bounding the norms of nonlinear operators acting on l p spaces. As an application, we prove some martingale inequalities with weak orlicz space. Marcinkiewicz type interpolation theorems for weak orlicz. Marcinkiewicz interpolation theorem theorem marcinkiewicz interpolation theorem assume that t is a linear operator with the following properties p 0 6 p 1 i t. Classical and modern results on interpolation of operators. The behavior of the constant loukas grafakos, liguang liu, shanzhen lu, and fayou zhao abstract.
Marcinkiewicz theorem for lorentz gamma spaces arxiv. It has been chopped into chapters for conveniences sake. A new approach to general interpolation formulae for bivariate interpolation zou, le. Let fp, p xi, x2, x, be a function harmonic for xn 0 and such that for every point q of a set. Marcinkiewicz interpolation theorem i woke up in a strange. Marcinkiewicz interpolation theorem, which states that under certain conditions one can deduce the strong boundedness of a sublinear operator t on the interpolation spaces provided that the operator is weakly bounded at the endpoints of the interpolation pair. Special cases of the marcinkiewicz interpolation theorem in a number of places in chapters 2, 3 and 5, we employ simple forms of the marcinkiewicz interpolation theorem. Hardylittlewoodsobolev inequality without marcinkiewicz. The proof now proceeds as the proof of the rieszthorin theorem, except that.
A generalized marcinkiewicz interpolation theorem and its. The above results are new even for the isotropic weighted settings. Marcinkiewicz interpolation theorem and marcinkiewicz spaces. An optimal interpolation theorem of marcinkievicz type in. The idea of hermite interpolation is clear in the univariate case, namely, when some of the interpolation points coalesce, the interpolating polynomials converge to the hermite interpolation polynomial which interpolates function values and derivatives.
Finally, we study multilinear interpolation theory, exposing the proof a version of marcinkiewiczs interpolation theorem for bisublinear operators. We focus on the study of realvariable methods and we start with one of the cornerstones of the classical interpolation theory. The purpose of this appendix is to present for the readers convenience. This completes the proof of the marcinkiewicz interpolation theorem when r. Atomic decomposition of martingale weighted lorentz spaces with twoparameter and applications mohsenipour, maryam and sadeghi, ghadir, rocky mountain journal of mathematics, 2017. Singular integrals and differentiability properties of. Zuoqin wang prepared lecture notes 0 through 11 in latex, and yanir rubinstein prepared lectures 12 through 24 in tex. Noncommutative hardylorentz spaces associated with semifinite subdiagonal algebras han, yazhou, banach journal of mathematical analysis, 2016. This site is like a library, use search box in the widget to get ebook that you want. Now we need to take t to be linear rather than sublinear. Marcinkiewicz theorem is similar to the rieszthorin theorem about linear operators, but also applies to nonlinear operators. Some extensions of the marcinkiewicz interpolation theorem in terms of modular inequalities j. A multilinear version of the marcinkiewicz interpolation theorem. Lecture notes differential analysis mathematics mit.
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