Abstract interpolation of topological vector spaces see also. A multivariable version of the strong maximal function is introduced and a sharp distributional estimate for this operator in the spirit of the jessen, marcinkiewicz, and zygmund theorem. Lebesgue di erentiation theorem is an analogue, and a generalization, of the fundamental theorem of calculus in higher dimensions. 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. Here is a quick summary of the topics will study study. Some extensions of the marcinkiewicz interpolation theorem in terms of modular inequalities j. Operations on measurable functions sums, products, composition realvalued measurable functions. Interpolation 123 final 1 finite difference functions. Noncommutative hardylorentz spaces associated with semifinite subdiagonal algebras. Interpolation we can often deduce that t is bounded for intermediate values of p just by considering end points. This approach was used for example in the classical book of. Marcinkiewicz interpolation our aim is to prove a result, the marcinkiewicz interpolation theorem, that allows one to bound the norms of operators acting on l p spaces. Pdf marcinkiewicz interpolation theorems for orlicz and.
It has been chopped into chapters for conveniences sake. 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. Assume that v pb1 is realvalued and that h v admits. Introduction a generalization of the marcinkiewicz interpolation theorem to orlicz spaces contains the conditions. We focus on the study of realvariable methods and we start with one of the cornerstones of the classical interpolation theory. Interpolation theorems and applications contents 1. 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. Girardi rieszthorin interpolation theorem notatationbasics. Before, we do this, we recall a fact about how to compute the norm of a. A n t with p 1 n1 jc nj 1 pa n interpolation of operators. Zuoqin wang prepared lecture notes 0 through 11 in latex, and yanir rubinstein prepared lectures 12 through 24 in tex. We will study variablecoe cient elliptic pde, looking at. We study a generalization of the classical marcinkiewicz zygmund inequalities.
Rieszthorin, marcinkiewicz matt ziemke ifwenoticethat,for1 imbedding theorem in the limiting case with lorentz space and bmo kozono, hideo, minamidate, kouei, and wadade, hidemitsu, 2007. 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. We provide a selfcontained proof of the multilinear extension of the. Theorem weak p0,p0 and p1,p1 strong p,p, p0 theorem on the pointwise convergence of the truncated hilbert transform contents 1. Atomic decomposition of martingale weighted lorentz spaces with twoparameter and applications. Lp spaces provides, in a way, a refinement of complex interpolation, this will be. Notes on lorentz spaces and interpolation guillermo rey. Interpolation theorems play a very important role in harmonic analysis. Combinatorics with emphasis on the theory of graphs. Function spaces and interpolation for more background on the material in this section cf.
Marcinkiewicz interpolation theorem and marcinkiewicz spaces. The marcinkiewicz interpolation theorem 253 distribution functions let ft be a from math 222b at university of california, berkeley. H is a linear isometry on a hilbert space h, then there is an orthogonal direct sum 14. 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. I am trying to prove a form of the marcinkiewicz interpolation theorem. In memory of polish mathematicians murdured by the soviets and the nazis. For the case whose bidual is a marcinkiewicz sequence space mw, we describe some sufficient conditions for a set to be a boundary of either or. In quoting the result of marcinkiewicz which has been extended by a. The main purpose of this paper is to construct a rear. Marcinkiewicz interpolation we now turn to proving a general theorem about interpolation of sublinear operators.
R n, is not sufficient to describe the fine properties of functions and operators. Mohsenipour, maryam and sadeghi, ghadir, rocky mountain journal of mathematics, 2017. A general interpolation theorem of marcinkiewicz type. Fredholm alternative, unbounded operators, riesz representation theorem, haar. Marcinkiewicz interpolation theorem lp estimate for the newtonian potential w1,p estimate for n. He used professor viaclovskys handwritten notes in producing them. The former allows us to show that a linear operator that is bounded on two lp spaces is bounded on every lp space in between the two. Interpolation 123 final 1 free download as powerpoint presentation. A question about general marcinkiewicz interpolation theorem. Rieszthorin theorem and interpolation inequalities this is one of the oldest and most important interpolation theorem, which plays a key role in analysis in general and pdes in particular. The total record of accomplishments of marcinkiewicz in.
We can obviously assume f to be positive so, by definition, we only have to. The purpose of this appendix is to present for the readers convenience statements and proofs of the theorems involved and. The marcinkiewicz theorem is extended to function spaces that include both the lebesgueorlicz and lorentz spaces, namely the. Marcinkiewicz interpolation theorem, marcinkiewicz spaces, the marcin. For terminology not otherwise explained we refer to kps. The total record of accomplishments of marcinkiewicz in his short life, his talent. B s, 4jiii, 1 interpolation theorem of marcinkiewicz is very useful in this connection. The marcinkiewicz interpolation theorem in this setting states the following, see bel o, theorem. The interpolation theory was aimed in the two classical theorems. 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. In this part we give the first example of a nonmonotone couple in which are described the interpola tion orbits of all elements. Lp estimates of certain rough parametric marcinkiewicz functions.
An essay on the interpolation theorem of j\ozef marcinkiewicz. The lecture notes were prepared by two former students in the class. We prove l p estimates of a class of parametric marcinkiewicz integral operators. The dyadic maximal operator and calder onzygmund decomposition 5 5. This paper is devoted to the interpolation principle between spaces of. If you have a disability and are having trouble accessing information on this website or need materials in an alternate format, contact web. This shows how to interpolate between bounds obtained in di erent lebesgue spaces for a linear operator. Let t he an operator whose domain and range are functions on. Generalized derivatives and approximation in weighted lorentz spaces akgun, ramazan and yildirir, yunus emre, bulletin of the belgian mathematical society simon stevin, 2017. Marcinkiewicz theorem for lorentz gamma spaces arxiv.
The proof provided the basic ideas for the socalled aler interpolation method. Introduction in this paper we present two main classical results of interpolation of operators. It was observed already in the first half of the last century e. A new approach to general interpolation formulae for bivariate interpolation zou, le and tang, shuo, abstract and applied analysis, 2014. Chaudhury our goal is to prove the following interpolation theorem for any subadditive and, of course, linear operator acting between lp spaces. Marcinkiewicz real method interpolation theorem with initial assumptions a set of restricted. An extension of the marcinkiewicz interpolation theorem to. Hahnbanach theorem, extreme points, kreinmilman and caratheodory. The real and the complex interpolation methods are the most important methods to construct interpolation spaces.
Course description for real analysis, math 156 in this class, we will study elliptic pde, fourier analysis, and dispersive pde. One parameter maximal functions and covering lemmas. On the volume of unit balls of finitedimensional lorentz. The marcinkiewicz multiplier theorem 3 assumes the multiplier. The complex interpolation method is the more commonly used one and it also has a more natural and easier statement it does not involve the weak type spaces. Marcinkiewicz interpolation theorem see 8 which we state for completeness of presentation. In mathematics, the marcinkiewicz interpolation theorem, discovered by jozef marcinkiewicz 1939, is a result bounding the norms of nonlinear operators. This chapter is devoted to the proofs and some of the applications of the theorems of rieszthorin and of marcinkiewicz, each of which is concerned with.
Rieszthorin, marcinkiewicz matt ziemke and marcinkiewicz interpolation theorem in a number of places in chapters 2, 3 and 5, we employ simple forms of the marcinkiewicz interpolation theorem. Pdf marcinkiewicz interpolation theorem and marcinkiewicz spaces. For example, the function 1x belongs to l1,w but not to l1. On an exam, you will not be responsible for the proof of the rieszthorin interpolation.
For the sake of simplicity we shall limit ourselves to the special case that is needed. A generalization of the marcinkiewicz interpolation theorem to orlicz spaces contains the conditions 1. We relate this problem to the sampling sequences in the paleywiener space and. An extension of the marcinkiewicz interpolation theorem to lorentz spaces. Marcinkiewicz theorem is similar to the rieszthorin theorem about linear operators, but also applies to nonlinear operators. This thesis is primarily devoted to the study of the marcinkiewicz interpolation theorem and its applications. Let1 p marcinkiewicz interpolation theorem to orlicz spaces contains the conditions 1. The marcinkiewicz interpolation theorem 253 distribution. Lecture notes differential analysis mathematics mit. The presentation should present the theorem, include a detailed presentation of one of its proofs, convey some ideas of the others if possible. Pdf on jun 11, 2012, lech maligranda and others published marcinkiewicz interpolation. Compact operators, spectrum and spectral theorem for compact operators on hilbert spaces.