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. Citeseerx document details isaac councill, lee giles, pradeep teregowda. An optimal interpolation theorem of marcinkievicz type in. We characterise interpolation spaces between two marcinkiewicz spaces in terms of hardy type operators involving suprema. Marcinkiewicz deal with estimates of the lpnorms of an operator, knowing its behavior at the endpoints of the interval of the exponents p, where the operator is still. Chaudhury our goal is to prove the following interpolation theorem for any subadditive and, of course, linear operator acting between lp spaces. Pdf on jun 11, 2012, lech maligranda published marcinkiewicz interpolation theorem and marcinkiewicz. Mar 16, 2015 recently, liu and wang obtained a marcinkiewicz type interpolation theorem for weak orlicz spaces.
Interpolation theorem for marcinkiewicz spaces with. The real and the complex interpolation methods are the most important methods. We study general properties of such operators and their behavior on lorentz gamma spaces. Marcinkiewicz interpolation theorem lp estimate for the newtonian potential w1,p estimate for n. Some extensions of the marcinkiewicz interpolation theorem in terms of modular inequalities. Now we need to take t to be linear rather than sublinear.
Also, the boundedness of hardylittlewood maximal function is much more straightforward than the general marcinkiewicz interpolation theorem. A general interpolation theorem of marcinkiewicz type. Modify, remix, and reuse just remember to cite ocw as the. The advantage of this proof is that it yields a logarithmically. Mar 10, 1998 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. Department of mathematics the university of chicago.
Pdf some extensions of the marcinkiewicz interpolation. Marcinkiewicz interpolation theorem, marcinkiewicz spaces, the marcin. Pdf multilinear interpolation between adjoint operators. Pdf an extension of the marcinkiewicz interpolation theorem. Intuition behind the rieszthorin interpolation theorem. Hunt and guido weiss we show that the marcinkiewicz theorem on the interpolation of operators acting on lp spaces see 3, pp. Pdf a sharp form of the marcinkiewicz interpolation theorem for. V j o v j t fy 7 v j o third, previously open questions concerning the marcinkiewicz theorem are settled by showing our result is best possible. The proof provided the basic ideas for the socalled aler interpolation method. Interpolation theorem of craig 1957 seems a rather technical result for connoisseurs inside logical metatheory.
Pdf an extension of the marcinkiewicz interpolation. Olof thorin extended the interpolation theorem to the entire square, removing the lowertriangle restriction. The purpose of this appendix is to present for the readers convenience statements and proofs of the theorems involved and to explain the concepts associated with them. Marcinkiewicz theorem is similar to the rieszthorin theorem about linear operators, but also applies to nonlinear operators. Marcinkiewicz real method interpolation theorem with initial assumptions a set of restricted weak. We provide a selfcontained proof of the multilinear extension of the marcinkiewicz real method interpolation theorem with initial assumptions a set of restricted weak type estimates, considering possible degenerate situations that may arise.
Lp0 lp 1 nl0 be sublinear and set, for q pr0,1s, 1 pq. A proof of the marcinkiewicz interpolation theorem in the special case p 0 q 0 1. Marcinkiewicztype multiplier theorem for qvariation q 1. Hot network questions is it weird to display ads on an academic website. A sharp form of the marcinkiewicz interpolation theorem for orlicz. A generalization of the marcinkiewicz interpolation theorem to orlicz spaces contains the conditions 1. The interpolation theory was aimed in the two classical theorems. In this paper, i discuss my own current favourite views of interpolation. Marcinkiewicztype multiplier theorem for qvariation q. Lebesgues differentiation theorem the lebesue set of an l1 function fundamental theorem of calculus i. An extension of the marcinkiewicz interpolation theorem is proved, yielding a necessary and. Since t is bounded from l p 0 to l q 0 andfroml p 1 tol q 1,wethenhavethatktg n tgk q 0 of q q 0 q.
A course on integration theory including more than 150. Pdf the brothers riesz theorem for conjugate differential. The interpolation theorem was originally stated and proved by marcel riesz in 1927. May 30, 2019 not long ago we discussed one of the main direct applications of the littlewoodpaley theory, namely the marcinkiewicz multiplier theorem. Versions of this theorem have been obtained in the literature by strichartz 11, sharpley 9,10, zafran, christ 1, janson 5, grafakos and kalton 3, and grafakos and tao 4. Let t be a linear operator such that the form 1 is wellde. For example, the function 1x belongs to l 1,w but not to l 1. Maximal operator maps lp to lp for 1 theorem of marcinkiewicz concerning interpolation of operations. But over the past decades, its broader importance has become clear from many angles. We do not give the proof of theorem 2 since it is similar but more complicated than that of theorem 1. The marcinkiewicz multiplier theorem 3 assumes the multiplier.
Marcinkiewicz interpolation theorem with initial restricted weaktype conditions and multiplicative bounds for the intermediate spaces. In this article we give a version of marcinkiewicz s real interpolation theorem for multilinear operators starting from a. The claim a marcinkiewicz interpolation theorem see e. Interpolation theorems play a very important role in harmonic analysis.
Generalized minkowski inequality another proof of youngs inequality distribution functions marcinkiewicz interpolation. Maximal operator maps lp to lp for 1 feb, 2017 this paper is devoted to the interpolation principle between spaces of weak type. Marcinkiewicz interpolation theorem from wikipedia, the free encyclopedia in mathematics, the marcinkiewicz interpolation theorem, discovered by jozef marcinkiewicz 1939, is a result bounding the norms of nonlinear operators acting on l p spaces. Pdf interpolation theorems and applications semantic scholar. In a subsequent paper i shall prove that for every point group there exists a point xo 1 xo 1 so that for infinitely many n 1t 9 y lkxa.
Oct 11, 2018 interpolation theorem for marcinkiewicz spaces with applications to lorentz gamma spaces. According to the theory presented, we must increase the number of nodes to obtain better results in this region. Rieszthorin interpolation theorem and its extension by stein, both of which are presented in the. The former allows us to show that a linear operator that is bounded on. The marcinkiewicz interpolation theorem and its extensions. In mathematics, the marcinkiewicz interpolation theorem, discovered by jozef marcinkiewicz.
Let1 p marcinkiewicz interpolation theorem with initial restricted weaktype conditions and multiplicative bounds for the intermediate spaces. Lecture notes measure and integration mathematics mit. Mathematics free fulltext classical lagrange interpolation. Feb 01, 2011 the importance of marcinkiewicz s short paper is reflected in the myriad applications and generalizations which earns the right to be called marcinkiewicz interpolation theory marcinkiewicz interpolation theorem came after the celebrated convexity theorem of m. The radonnikodym theorem, notions of harmonic analysis, classical inequalities and interpolation theorems, including marcinkiewicz s theorem, the definition of lebesgue points and lebesgue differentiation theorem are further topics included. The proof of thorin was originally published in 1938 and was subsequently. Lecture notes differential analysis mathematics mit. The 1927 paper establishes the theorem only for the lower triangle of the riesz diagram, viz. Recall that the singlevariable version of this theorem can be formulated as follows.
An essay on the interpolation theorem of jozef marcinkiewicz. The reason we consider this topic is mainly based on two facts. Pdf on jun 11, 2012, lech maligranda published marcinkiewicz interpolation theorem and marcinkiewicz spaces find, read and cite all the research you need on researchgate. Marcinkiewicz interpolation theorem, full version theorem let p 0,p 1,q 0,q 1 pr1,8sobey p 0. 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. More general versions of the mar cinkiewicz theorem can be found in 9, section. Interpolation theory and applications to the boundedness of. Mathematics and its applications east european series, vol 41.
Chapter 4 gives a gentle introduction to these results, using the rieszthorin theorem and the marcinkiewicz interpolation formula. The theorem holds precisely as stated, except with. Hardylittlewoodsobolev inequality without marcinkiewicz. Rieszthorin, marcinkiewicz matt ziemke alsohavethatkg n gk p 0. One of the longtime users of fourier analysis is probability theory. The importance of marcinkiewicz s short paper is reflected in the myriad applications and generalizations which earns the right to be called marcinkiewicz interpolation theory marcinkiewicz interpolation theorem came after the celebrated convexity theorem of m. Pdf interpolation theorems and applications semantic. Pdf marcinkiewicz interpolation theorem and marcinkiewicz spaces.
A particular emphasis is placed on elementary and comprehensive proofs. The case of the chebyshev abscissas shows that theorem 2 is bestpossible. We discuss and prove two major theorems, the rieszthorin interpolation theorem and the marcinkiewicz interpolation theorem. The marcinkiewicz interpolation theorem and its extensions author.
A question about general marcinkiewicz interpolation theorem. An extension of the marcinkiewicz interpolation theorem to lorentz spaces. An optimal interpolation theorem of marcinkiewicz type in. Marcinkiewicz interpolation theorem, marcinkiewicz spaces, the marcin kiewicz integral. Generally an interpolation method is an functor ffrom the cathegory of admissible. On a theorem of marcinkiewicz concerning interpolation of. Hunt, journalbulletin of the american mathematical society, year1964, volume70, pages803807 richard a. An extension of marcinkiewicz interpolation theorem, allowing intermediate spaces of orlicz type, is proved. Upper bound for operator norm in marcinkiewicz interpolation theorem. Pdf marcinkiewicz interpolation theorem and marcinkiewicz. The first one is a wellknown result of hardy see 1, pp.
This paper is devoted to the interpolation principle between spaces of weak type. The purpose of this paper is to study marcinkiewicz type interpolation theorem for weak orlicz martingale spaces and give its applications. Marcinkiewicz interpolation theorem and marcinkiewicz spaces. Let1 p marcinkiewicz theorems for orlicz and lorentz gamma spaces 9 corresponding to an r. It begins with the standard material, but also includes chapters dealing with distributions and weak derivatives, special topics e. In mathematics, the marcinkiewicz interpolation theorem, discovered by jozef marcinkiewicz, is a result bounding the norms of nonlinear operators acting on l p spaces. Interpolation theorems and applications contents 1. Feb 01, 2011 marcinkiewicz interpolation theorem came after the celebrated convexity theorem of m.
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