Intuition behind the rieszthorin interpolation theorem. Pdf marcinkiewicz interpolation theorem and marcinkiewicz. Compact operators, spectrum and spectral theorem for compact operators on hilbert spaces. The case w 2 of the theorem was established by marcinkiewicz and zygmund l2 using methods of conformai mapping.
In general this does not hold true in two and more variables. The operatorvalued marcinkiewicz multiplier theorem and. The marcinkiewicz interpolation theorem 253 distribution. Stein, allowing for interpolation of families of operators. A first example of banach spaces are the lebesgue spaces lp, for 1. In the note, we devote ourselves to establish the following boundedness of the commutators of marcinkiewicz integral of schrodinger operator type. Actually, theorem 1 can be obtained by extrapolation arguments see or and the following result. Marcinkiewicz interpolation theorem i woke up in a. Marcinkiewicz interpolation theorem with initial restricted weaktype conditions and multiplicative bounds for the intermediate spaces. The rst two sections of the chapters are devoted to developing the nec. Giovanni leoni, spring 2015 monday, january 12, 2015 1 lp spaces and interpolation 1. Marcinkiewicz type interpolation theorems for weak orlicz martingale spaces. A introduction to interpolation theory, interpolation spaces and.
Giovanni leoni, spring 2015 monday, january 12, 2015. As an application, we prove some martingale inequalities with weak orlicz space norm. The rieszthorin interpolation theorem in this notes we give a proof of the rieszthorin interpolation theorem. We provide a complete proof of an optimal version of the marcinkiewicz multiplier theorem and discuss a relevant example.
A generalization for fourier transforms of a theorem due. A generalized marcinkiewicz interpolation theorem and its. Introduction and statement of results we revisit a producttype sobolev space version of the marcinkiewicz multiplier the orem. The marcinkiewicz interpolation theorem and its extensions author. One may also notice that we sometimes avoid stating the results in the full generality. The convergence of series estabalished in section 1. The factor theorem states that ifpnn 0, then x n is a factor of px, which means that nis a root, or zero, ofpnx. Marcinkiewicz type interpolation theorems for weak orlicz. The marcinkiewicz interpolation theorem and its extensions. Introduction in his classical paper 11 1927 marcel riesz proved a theorem on linear operators mapping lp spaces on one measure space onto lq spaces on another measure space. On the other hand, are bounded on and are bounded from to.
Two new interpolation methods based on the duality map by jaak peetre university of land, sweden 0. Notice that our above example shows why we could not expect an inequality. The boundedness of marcinkiewicz integral associated with. One of the longtime users of fourier analysis is probability theory. Introduction to fourier analysis and wavelets brooks cole series in advanced mathematics mark a. The operators are bounded on, for, and are bounded from to weak. Approximation theory deals with two types of problems. For example, the function 1x belongs to l1,w but not to l1. 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. Pdf some extensions of the marcinkiewicz interpolation theorem. The nature of the general derivation suggests denoting the family of material point method discretization schemes developed here the generalized interpola.
As the difference illustrated by the xaxis, is closer to 0, the more similar the overall interpolated datasets fit the measured. In 1937 marcinkiewicz and zygmund obtained an estimate of the norm inl q 0,1 of the sum of the series. Pinsky written by a successful author and respected mathematician, this book emphasizes a concrete and computational approach to the subject of fourier analysis and wavelet theory while maintaining a balance between theory and applications. For those of you who followed my harmonic analysis notes a couple of years back, here is an updated and slightly more polished version in pdf form, from an advanced course i gave at aalto university. Now we need to take t to be linear rather than sublinear. In the past 30 years marcinkiewicz zygmund type inequalities for trigonometric and algebraic polynomials with various weights were widely used in the study of the convergence of fourier series, lagrange and hermite interpolation, positive quadrature formulas, scattered data interpolation, see 9 for a survey on the univariate marcinkiewicz zygmund. Marcinkiewiczs name in mathematics is connected with the marcinkiewicz interpolation theorem, marcinkiewicz spaces, the marcinkiewicz integral and function, marcinkiewiczzygmund inequalities. Classical and modern results on interpolation of operators.
We then address several important applications of these theorems, including the boundedness of the hardylittlewood maximal operator, the boundedness of the fourier transform, youngs inequality for convolution, and the boundedness of the hilbert transform. It is shown that the maximal operator of the marcinkiewicz means of a tempered distribution is bounded from h p r 2 to l p r 2 for all p 0 0, let u,t x luxl t, we define the distribution function of u to be ut s ftu,t, wherez is the lebesgue measure on i n. This completes the proof of the marcinkiewicz interpolation theorem when r. Two new interpolation methods based on the duality map. We treat two essentially different interpolation methods. Using a simple method, we generalize marcinkiewicz interpolation theorem to operators on orlicz space and apply it to several important theorems in harmonic analysis. 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. That is, x n 0, and the derivative of a polynomial px can be obtained from eq. Let1 p 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. Dec 15, 2000 as a consequence we obtain a generalization for fourier transforms of a summability result due to marcinkiewicz and zhizhiashvili, more exactly, the marcinkiewicz means of a function f. In mathematics, the marcinkiewicz interpolation theorem, discovered by jozef marcinkiewicz, is a result bounding the norms of nonlinear operators acting on l p spaces.
This result, known as the stein interpolation theorem, was included in his 1955 doctoral dissertation and was subsequently published in ste56. In the case when the underlying measure spaces areinite sets it can. We revisit the real and complex interpolation theory for hardy spaces by examining our newly found formulae. Elevation difference between the measured points lidar and sonar dataset to the esri interpolated surfaces. Fredholm alternative, unbounded operators, riesz representation theorem, haar.
Marcinkiewicz theorem is similar to the rieszthorin theorem about linear operators, but also applies to nonlinear operators. Akfk1, where ais a constant which depends only on the dimension i. In chapter 5 the central limit theorem, iterated log theorem, and berryesseen theorems are developed using the suitable fourieranalytic tools. First, the theorem of marcinkiewicz on interpolation of operators see 9, pp. Weiss, an extension of a theorem of marcinkiewicz and some of its applications, j. Hardylittlewoodsobolev inequality without marcinkiewicz interpolation. Introduction the use of interpolation theorems is the rst step in proving carlesons theorem. The marcinkiewicz multiplier theorem revisited loukas grafakos and lenka slav ikova abstract. Rieszthorin, marcinkiewicz matt ziemke and marcinkiewicz interpolation theorem to. The proof of the marcinkiewicz interpolation theorem is based on an. For t 0, let u,t x luxl t, we define the distribution function of u to be ut s ftu,t, wherez is the lebesgue measure on i n. Complex interpolation ucla department of mathematics. A proof of the marcinkiewicz interpolation theorem in the special case p0 q0 1,p1.
Rieszthorin interpolation theorem was then generalized by elias m. Rieszthorin interpolation theorem and its extension by stein, both of which are presented in. A generalization of the marcinkiewicz interpolation theorem to orlicz spaces contains the conditions 1. I am trying to prove a form of the marcinkiewicz interpolation theorem. These two methods are modelled on the proofs of the marcinkiewicz theorem and the rieszthorin theorem respectively, as they are given in chapter 1.
In this section, we focus on employing the technique of atomic decomposition to establish two marcinkiewicz type interpolation theorems for weak orlicz martingale spaces. The generalized interpolation material point method 479 nite deformation analyses, and is demonstrated to very seriously degrade the accuracy of solutions obtained using mpm. 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 is obtained. On a theorem of marcinkiewicz and zygmund springerlink. Applying theorem 2 and extrapolation argument, we can obtain the following result.
The complexanalytic method used in the proof of the. Let1 p pdf available in journal of the mathematical society of japan 552003 april 2003 with 58 reads. Interpolation estimating information within a graph extrapolationextending the graph to estimate information consider the following example from the previous lesson on modelling linear relations with equations. Integration of topographic and bathymetric digital. The main aim of this section is to prove the following theorem. Strong law of large numbers slln is a central result in classical probability theory. A note on marcinkiewicz integrals along submanifolds of. Second, this result is shown to be a rather easy consequence of a celebrated inequality of hardy 2, pp. Marcinkiewicz integrals on product spaces and extrapolation. First, we approximate fby a continuous and compactly supported gi.
Suppose that one has a table listing the population of some country in 1970, 1980, 1990 and 2000, and that one wanted to estimate the population in 1994. Pdf some extensions of the marcinkiewicz interpolation. The operatorvalued marcinkiewicz multiplier theorem and maximal regularity wolfgang arendt 1, shangquan bu 1,2 1 abteilung angewandte analysis, universit at ulm, 89069 ulm, germany. Carlesons theorem states that if fis an lp periodic function for some p21. Interpolation and extrapolation interpolation and extrapolation are mathematical names given to the process of reading graphs. Combining the above result with plancherels theorem theorem 1. Chaudhury our goal is to prove the following interpolation theorem for any subadditive and, of course, linear operator acting between lp spaces. Introduction to fourier analysis and wavelets brooks cole. The generalized interpolation material point method. A generalization for fourier transforms of a theorem due to.
1437 1224 680 599 292 167 1299 1146 1454 1088 1284 1017 1145 220 1219 1455 1240 875 1067 173 1122 1401 1253 280 608 283 566 1353 1043 1286 980 443 1076 34 1194