wiktionarypar boundedness bounded Boundedness or bounded may refer to Bounded set , a set that is finite in some sense Bounded function , a function or sequence whose possible values form a bounded set Bounded poset a partially ordered set which has both a greatest element and a least element Bounded set topological vector space , a set in which every neighborhood of the zero vector can be inflated to include the set Bounded operator , a linear transformation L between normed vector spaces for which the ratio of the norm of L v to that of v is bounded by the same number, over all non zero vectors v Bounded variation , a real valued functions whose total variation is bounded Boundedness axiom , the axiom schema of replacement Boundedness linguistics , whether a situation has a clearly defined beginning or end disambig Category Mathematical disambiguation ... more details
In linguistics , boundedness is an grammatical aspect aspectual feature that describes a situation as having a definite beginning or end, or both, not as continuing indefinitely. Thus the clause I ate fish describes a unitary, bounded action, as it implies both the beginning I started eating fish and the end I finished eating fish . The clause I was eating does not express a bounded action, because the verb form does not express either the beginning or the end. Similarly, I set off for home and I arrived home present the action as bounded, whereas I was going home and I was at home do not. Aspects Certain grammatical aspect s express boundedness. Boundedness is characteristic of perfective aspect s such as the Ancient Greek aorist Ancient Greek aorist and the Spanish language Spanish preterite . The simple past English simple past of English commonly expresses a bounded event I found out , but sometimes expresses, for example, a stative verb stative I knew . The perfective aspect often includes a contextual variation similar to an inchoative aspect or inchoative verb verb , and expresses the beginning of a stative verb state . See also Lexical aspect Grammatical aspect Category Grammar Category Verb types fr Aspect s cant non s cant ... more details
infinity. Topological vector spaces Local boundedness may also refer to a property of topological .... The following theorem relates local boundedness of functions with the local boundedness of topological ... bounded id 5752 DEFAULTSORT Local Boundedness Category Mathematical analysis eo Vikipedio Projekto ... more details
In mathematics , bounded function s are function mathematics functions for which there exists a lower bound and an upper bound , in other words, a constant which is larger than the absolute value of any value of this function. If we consider a family mathematics family of bounded functions, this constant can vary between functions. If it is possible to find one constant which bounds all functions, this family of functions is uniformly bounded . The uniform boundedness principle in functional analysis provides sufficient conditions for uniform boundedness of a family of operators. Definition Real line and complex plane Let math mathcal F f i X to K, i in I math be a family of functions Index set indexed by math I math , where math X math is an arbitrary set and math K math is the set of real or complex numbers. We call math mathcal F math uniformly bounded if there exists a real number math M math such that math f i x leq M qquad forall i in I quad forall x in X. math Metric space In general let math Y math be a metric space with metric math d math , then the set math mathcal F f i X to Y, i in I math is called uniformly bounded if there exists an element math a math from math Y math and a real number math M math such that math d f i x , a leq M qquad forall i in I quad forall x in X. math Examples Every uniformly convergent sequence of bounded functions is uniformly bounded. The family of functions math f n x sin nx , math defined for real number real math x math with math n math traveling through the integer s, is uniformly bounded by 1. The family of derivative s of the above family, math f n x n , cos nx, math is not uniformly bounded. Each math f n , math is bounded by math n , , math but there is no real number math M math such that math n le M math for all integers math n. math References cite book last Ma first Tsoy Wo title Banach Hilbert spaces, vector measures, group representations publisher World Scientific date 2002 isbn 9812380388, important to look up ... more details
In mathematics , the uniform boundedness principle or Banach Steinhaus theorem is one of the fundamental results in functional analysis . Together with the Hahn Banach theorem and the open mapping theorem functional analysis open mapping theorem , it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operator s and thus bounded operators whose domain is a Banach space , pointwise boundedness is equivalent to uniform boundedness in operator norm. The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus but it was also proven independently by Hans Hahn . Uniform boundedness principle The precise statement of the result is Theorem. Let X be a Banach space and Y be a normed vector space . Suppose that F is a collection of continuous linear operators from X to Y . The uniform boundedness principle states that if for all x in X we have math sup T in F T x infty, math then math sup T in F T infty. math The completeness of X enables the following short proof, using the Baire category theorem Proof. Define the closed sets X sub n sub with n 1, 2, 3, &hellip by math X n x in X sup T in F T x le n . math By hypothesis, the union of all the X sub n sub is X . Since X is a Baire space , one of the X sub n sub , say X sub m sub , has an interior point X sub n sub are closed sets , i.e. , there exists a &delta > 0 and a y in X such that all x &isin X with nowrap begin x &minus y ... Y sup &lowast sup , continuous dual of Y . By the uniform boundedness principle, the norms ... boundedness principle, the set math R x in X sup T in F T x infty math is not empty. In fact ... space of continuous functions on T , with the uniform norm . Using the uniform boundedness principle ... sub N, x sub is unbounded in C T sup &lowast sup , the dual of C T . Therefore by the uniform boundedness ... . Generalizations The least restrictive setting for the uniform boundedness principle is a barrelled ... more details
In mathematics , inverse mapping theorem may refer to the inverse function theorem on the existence of local inverse function inverses for functions with non singular derivative s the bounded inverse theorem on the bounded linear operator boundedness of the inverse for invertible bounded linear operators on Banach spaces . disambig ... more details
At least three well known results in mathematics bear the name Schur s lemma Schur s lemma from representation theory Schur s lemma from Riemannian geometry Schur test for boundedness of integral operators See also Schur s theorem mathdab ... more details
space R sup n sup is compact if and only if it is closed set closed and bounded. Boundedness ... vector spaces , then the two definitions coincide. Boundedness in order theory A set of real ... of any partially ordered set . Note that this more general concept of boundedness does not correspond ... element . Note that this concept of boundedness has nothing to do with finite size, and that a subset ... numbers. See also Bounded function Local boundedness Order theory Totally bounded Category Mathematical ... more details
In mathematics , Carath odory s theorem may refer to one of a number of results of Constantin Carath odory Carath odory s theorem conformal mapping , about the extension of conformal mappings to the boundary Carath odory s theorem convex hull , about the convex hulls of sets in Euclidean space Carath odory s theorem measure theory , about outer measures in measure theory Carath odory s existence theorem , about the existence of solutions to ordinary differential equations Carath odory s extension theorem , about the extension of a measure Borel Carath odory theorem , about the boundedness of a complex analytic function Carath odory Jacobi Lie theorem , a generalization of Darboux s theorem in symplectic topology The Carath odory kernel theorem , a geometric criterion for local uniform convergence of univalent function s DEFAULTSORT Caratheodory s theorem Category Mathematical disambiguation disambig pl Twierdzenie Carath odory ego ru ... more details
wiktionary pub Pub is a public house or bar establishment . Pub may refer to Pub Denzil album Pub Denzil album , a 1994 album by British band Denzil Pub or e Bala evi album Pub or e Bala evi album , a 1982 album by Serbian singer songwriter or e Bala evi album PUB file type , Microsoft Publisher document file format PUB Stockholm , department store in Stockholm Pub can also be Percutaneous umbilical cord blood sampling , a genetic test Politehnica University of Bucharest , Politehnica University of Bucharest Princeton University Band , the marching band and pep band of Princeton University Principle of uniform boundedness , a fundamental result of functional analysis Public Utilities Board , Singapore s national water agency Pueblo Memorial Airport , in Colorado, USA Public directories on FTP and HTTP Servers are often named pub See also Nightclub disambig fr Pub it PUB ... more details
with compactness and completeness There is a nice relationship between total boundedness ... set boundedness with total boundedness and also replace closed set closedness with completeness . There is a complementary relationship between total boundedness and the process of Cauchy completion ... is compact. This may be taken as an alternative definition of total boundedness. Alternatively, this may ... boundedness. Then it becomes a theorem that a space is totally bounded if and only if it is precompact ... section. Use of the axiom of choice The properties of total boundedness mentioned above rely in part on the axiom of choice . In the absence of the axiom of choice, total boundedness and precompactness must be distinguished. That is, we define total boundedness in elementary terms but define precompactness ... more details
In mathematics , Nemytskii operators are a class of nonlinear operator s on Lp space L sup p sup spaces with good continuous function continuity and bounded function boundedness properties. They take their name from the mathematician Viktor Vladimirovich Nemytskii . Definition Let &Omega be a domain an open set open and connected space connected set in n dimensional Euclidean space . A function f &Omega × R sup m sup &rarr R is said to satisfy the Carath odory conditions if f x , u is a continuous function of u for almost all x &isin &Omega f x , u is a measurable function of x for all u &isin R sup m sup . Given a function f satisfying the Carath odory conditions and a function u &Omega &rarr R sup m sup , define a new function F u &Omega &rarr R by math F u x f big x, u x big . math The function F is called a Nemytskii operator . Boundedness theorem Let &Omega be a domain, let 1 < p < &infin and let g &isin L sup q sup &Omega R , with math frac1 p frac1 q 1. math Suppose that f satisfies the Carath odory conditions and that, for some constant C and all x and u , math big f x, u big leq C u p 1 g x . math Then the Nemytskii operator F as defined above is a bounded and continuous map from L sup p sup &Omega R sup m sup into L sup q sup &Omega R . References cite book author Renardy, Michael and Rogers, Robert C. title An introduction to partial differential equations series Texts in Applied Mathematics 13 edition Second edition publisher Springer Verlag location New York year 2004 pages 356 id ISBN 0 387 00444 0 Section 9.3.4 Category Operator theory ... more details
seen to be 1. Equivalence of boundedness and continuity As stated in the introduction, a linear ... . math This proves that L is bounded. Linearity and boundedness Not every linear operator between normed ... . If B U , V is Banach and U is nontrivial, then V is Banach. Topological vector spaces The boundedness ... general condition of boundedness for sets in a topological vector space TVS a set is bounded if and only if it is absorbed by every neighborhood of 0. Note that the two notions of boundedness ... operator need not be continuous. Clearly, this also means that boundedness is no longer equivalent ... more details
, then the corresponding half of the boundedness theorem and the extreme value theorem hold and the values ... is a slight modification of the proofs given above. In the proof of the boundedness theorem ..., if and only if it is continuous in the usual sense. Hence these two theorems imply the boundedness ... numbers . We first prove the boundedness theorem, which is a step in the proof of the extreme value theorem. The basic steps involved in the proof of the extreme value theorem are Prove the boundedness ... continuity to show that the image of the subsequence converges to the supremum. Proof of the boundedness ... . Q.E.D. Proof of the extreme value theorem By the boundedness theorem, f is bounded from above ... value theorem at cut the knot planetmath reference id 6022 title Boundedness Theorem planetmath ... more details
mapping s between topological vector spaces preserve boundedness. A locally convex space is seminormable ... bounded space Local boundedness bounded function References cite book last Robertson first A.P. ... more details
W adys aw Roman Orlicz May 24, 1903 in Okocim , Austria Hungary now Poland August 9, 1990 in Pozna , Poland was a Poland Polish mathematician of Lw w School of Mathematics . His main interests were functional analysis and topology Orlicz space s are named after him. See also Convexity in the sense of Orlicz Drewnowski Orlicz theorem F norm of Mazur Orlicz Hardy Orlicz spaces Marcinkiewicz Orlicz space Matuszewska Orlicz indices Mazur Orlicz bounded consistency theorem Mazur Orlicz theorem on inequalities Mazur Orlicz theorem on uniform boundedness in F spaces Musielak Orlicz spaces Orlicz category theorem Orlicz interpolation theorem Orlicz norm Orlicz property Orlicz space Orlicz theorem Orlicz theorem on Weyl multipliers Orlicz Bochner space Orlicz Pettis theorem Orlicz Sobolev space External links MacTutor id Orlicz title W adys aw Orlicz date May 2001 MathGenealogy id 51907 Persondata Metadata see Wikipedia Persondata . NAME Orlicz, Wladyslaw ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH 1903 PLACE OF BIRTH DATE OF DEATH 1990 PLACE OF DEATH DEFAULTSORT Orlicz, Wladyslaw Category 1903 births Category 1990 deaths Category 20th century mathematicians Category Functional analysts Category Polish mathematicians Category Topologists Poland mathematician stub de W adys aw Orlicz fr W adys aw Orlicz la Vladislaus Orliczl nl W adys aw Orlicz pl W adys aw Orlicz pt W adys aw Orlicz sk W adys aw Orlicz ... more details
other uses2 Adversary Unreferenced date December 2009 In cryptography , an adversary rarely opponent , enemy is a malicious entity whose aim is to prevent the users of the cryptosystem from achieving their goal primarily privacy, integrity, and availability of data . An adversary s efforts might take the form of attempting to discover secret data, corrupting some of the data in the system, Spoofing attack spoof ing the identity of a message sender or receiver, or forcing system downtime. Actual adversaries, as opposed to idealized ones, are referred to as attackers . Not surprisingly, the former term predominates in the cryptographic and the latter in the computer security literature. Alice and Bob Eve, Mallory, Oscar and Trudy are all adversarial characters widely used in both types of texts. This notion of an adversary helps both intuitive and formal reasoning about cryptosystems by casting security analysis of cryptosystems as a game between the users and a centrally co ordinated enemy. The notion of security of a cryptosystem is meaningful only with respect to particular attacks usually presumed to be carried out by particular sorts of adversaries . There are several types of adversaries depending on what capabilities or intentions they are presumed to have. Adversaries may be computational boundedness computationally bounded or unbounded i.e. in terms of time and storage resources , eavesdropping or Byzantine i.e. passively listening on or actively corrupting data in the channel , static or adaptive i.e. having fixed or changing behavior , mobile or non mobile e.g. in the context of network security and so on. In actual security practice, the attacks assigned to such adversaries are often seen, so such notional analysis is not merely theoretical. How successful an adversary is at breaking a system is measured by its advantage . An adversary s advantage is the difference between the adversary s probability of breaking the system and the probability that the syst ... more details
In cryptography , concrete security or exact security is a practice oriented approach that aims to give more precise estimates of the computational complexities of Adversary cryptography adversarial tasks than Polynomial time reduction polynomial equivalence would allow. Traditionally, provable security is asymptotic it classifies the hardness of computational problems using polynomial time reducibility. Secure schemes are defined to be those in which the advantage of any computational boundedness computationally bounded adversary is negligible function cryptography negligible . While such a theoretical guarantee is important, in practice one needs to know exactly how efficient a reduction is because of the need to instantiate the security parameter it is not enough to know that sufficiently large security parameters will do. An inefficient reduction results either in the success probability for the adversary or the resource requirement of the scheme being greater than desired. Concrete security parametrizes all the resources available to the adversary, such as running time and memory, and other resources specific to the system in question, such as the number of plaintexts it can obtain or the number of queries it can make to any oracle cryptography oracles available. Then the advantage of the adversary is upper bounded as a function of these resources and of the problem size. It is often possible to give a lower bound i.e, an adversarial strategy matching the upper bound, hence the name exact security. References Mihir Bellare M. Bellare , A. Desai, E. Jokipii and Phillip Rogaway P. Rogaway . http www cse.ucsd.edu users mihir papers sym enc.html A Concrete Security Treatment of Symmetric Encryption Analysis of the DES Modes of Operation . M. Bellare and P. Rogaway. http citeseer.ist.psu.edu bellare96exact.html The Exact Security of Digital Signatures How to Sign with RSA and Rabin Category Cryptography crypto stub ... more details
Otto Marcin Nikod m 1887 &ndash 1974 was a Poland Polish mathematician. He was educated at the Universities of Lwow and Warsaw , and the Sorbonne . Nikodym taught at the Universities of Krak w and Warsaw and at the High Polytechnical School in Krak w. He came to the United States in 1948 to join the faculty of Kenyon College . He retired in 1966 and moved to Utica, N.Y., where he continued his research. Nikodym died in 1974. Nikodym worked in a wide range of areas, but his best known early work was his contribution to the development of the Lebesgue Radon Nikodym integral see Radon Nikodym theorem . His work in measure theory led him to an interest in abstract Boolean lattice s. His work after coming to the United States centered on the theory of operators in Hilbert space , based on Boolean lattices, culminating in his The Mathematical Apparatus for Quantum Theories . He was also interested in the teaching of mathematics. See also Nikodym set Radon Nikodym theorem Nikodym convergence theorem Nikodym Grothendieck boundedness theorem Frechet Nikodym metric space Radon Nikodym property of a Banach space External links http www history.mcs.st andrews.ac.uk history Biographies Nikodym.html MacTutor Entry Persondata Metadata see Wikipedia Persondata . NAME Nikodym, Otton M. ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH 1887 PLACE OF BIRTH DATE OF DEATH 1974 PLACE OF DEATH DEFAULTSORT Nikodym, Otton M. Category 1887 births Category 1974 deaths Category University of Lviv alumni Category University of Warsaw alumni Category University of Paris alumni Category Jagiellonian University faculty Category University of Warsaw faculty Category Polish emigrants to the United States Category Polish mathematicians Nikodym, Otton Martin Category Kenyon College faculty de Otton Marcin Nikod m es Otto M. Nikodym fr Otto Nikodym pl Otto M. Nikod m uk ... more details
In mathematics , particularly functional analysis , the Dunford Schwartz theorem , named after Nelson Dunford and Jacob T. Schwartz states that the averages of powers of certain norm bounded operator s on Lp space L sup 1 sup Limit of a sequence converge in a suitable sense. ref Citation last Dunford first Nelson last2 Schwartz first2 J. T. title Convergence almost everywhere of operator averages journal J. Rational Mech. Anal. volume 5 year 1956 issue pages 129 178 mr 77090 . ref Theorem. Let math T math be a linear operator from math L 1 math to math L 1 math with math T 1 leq 1 math and math T infty leq 1 math . Then math lim n rightarrow infty frac 1 n sum k 0 n 1 T kf math exists almost everywhere for all math f in L 1 math . The statement is no longer true when the boundedness condition is relaxed to even math T infty le 1 varepsilon math . ref Citation last Friedman first N. title On the Dunford Schwartz theorem journal Z. Wahrscheinlichkeitstheorie und Verw. Gebiete volume 5 issue 3 year 1966 pages 226 231 doi 10.1007 BF00533059 mr 220900 . ref See also Bartle Dunford Schwartz theorem Notes Reflist Category Functional analysis Category Theorems in functional analysis Mathanalysis stub ... more details
Orphan date September 2011 In mathematics, the T 1 theorem , first proved by harvtxt David Journ 1984 , describes when an operator T given by a Kernel integral operator kernel can be extended to a bounded linear operator on the Hilbert space L sup 2 sup R sup n sup . The name T 1 theorem refers to a condition on the Distribution mathematics distribution T 1 , given by the operator T applied to the function 1. Statement Suppose that T is a continuous operator from Schwartz functions on R sup n sup to tempered distributions, so that T is given by a kernel K which is a distribution. Assume that the kernel is standard, which means that off the diagonal it is given by a function satisfying certain conditions. Then the T 1 theorem states that T can be extended to a bounded operator on the Hilbert space L sup 2 sup R sup n sup if and only if the following conditions are satisfied T 1 is of bounded mean oscillation where T is extended to an operator on bounded smooth functions, such as 1 . T sup sup 1 is of bounded mean oscillation, where T sup sup is the adjoint of T . T is weakly bounded, a weak condition that is easy to verify in practice. References Citation last1 David first1 Guy last2 Journ first2 Jean Lin title A boundedness criterion for generalized Calder n Zygmund operators jstor 2006946 mr 763911 year 1984 journal Annals of Mathematics Annals of Mathematics. Second Series issn 0003 486X volume 120 issue 2 pages 371 397 Citation last1 Grafakos first1 Loukas title Modern Fourier analysis publisher Springer Verlag location Berlin, New York edition 2nd series Graduate Texts in Mathematics isbn 978 0 387 09433 5 doi 10.1007 978 0 387 09434 2 mr 2463316 year 2009 volume 250 Category Functional analysis ... more details
In mathematics , a paraproduct is a non commutative bilinear operator acting on function mathematics functions that in some sense is like the product mathematics product of the two functions it acts on. According to Svante Janson and Jaak Peetre, in an article from 1988 ref Svante Janson and Jaak Peetre, http www.jstor.org stable 2000875 Paracommutators Boundedness and Schatten Von Neumann Properties , Transactions of the American Mathematical Society , Vol. 305, No. 2 Feb., 1988 , pp. 467 504. ref , the name paraproduct denotes an idea rather than a unique definition several versions exist and can be used for the same purposes. This said, for a given operator math Lambda math to be defined as a paraproduct, it is normally required to satisfy the following properties It should reconstruct the product in the sense that for any pair of functions, math f, g math in its domain, math fg Lambda f, g Lambda g, f . math For any appropriate functions, math f math and math h math with math h 0 0 math , it is the case that math h f Lambda f, h f math . It should satisfy some form of the Leibnitz rule . A paraproduct may also be required to satisfy some form of H lder s inequality . Notes Reflist Further references rp d B nyi, Diego Maldonado, and Virginia Naibo, http www.ams.org notices 201007 rtx100700858p.pdf What is a Paraproduct? , Notices of the American Mathematical Society , Vol. 57, No. 7 Aug., 2010 , pp. 858 860. Category Bilinear operators algebra stub ... more details
File Max Koecher 2.jpeg thumb right Max Koechen in Munich, 1967 Max Koecher 20 January 1924 in Weimar 7 February 1990 in Lengerich Westfalen Lengerich was a German mathematician. Koecher studied mathematics and physics at the Georg August Universit t in G ttingen . In 1951 he received his doctorate under Max Deuring with his work harv Koecher 1953 on Dirichlet series with functional equation where he introduced Koecher Maass series . He qualified in 1954 at the Westf lische Wilhelms University in M nster. From 1962 to 1970 Koecher was department chair at the University of Munich. He retired in 1989. His main research area was the theory of Jordan algebras, where he introduced the Kantor Koecher Tits construction . He discovered the Koecher boundedness principle in the theory of Siegel modular form s. References Citation last1 Koecher first1 Max title ber Dirichlet Reihen mit Funktionalgleichung url http gdz.sub.uni goettingen.de dms load img ?PPN PPN243919689 0192 mr 0057907 year 1953 journal Journal f r die reine und angewandte Mathematik issn 0075 4102 volume 192 pages 1 23 Citation last1 Krieg first1 A. last2 Petersson first2 H. P. title Max Koecher zum Ged chtnis url http dml.math.uni bielefeld.de JB DMV Band95 mr 1203926 year 1993 journal Jahresbericht der Deutschen Mathematiker Vereinigung issn 0012 0456 volume 95 issue 1 pages 1 27 Citation last1 Petersson first1 Holger P. editor1 last Kaup editor1 first Wilhelm editor2 last McCrimmon editor2 first Kevin editor3 last Petersson editor3 first Holger P. title Jordan algebras Oberwolfach, 1992 url http books.google.com books?isbn 3110142511 publisher de Gruyter location Berlin isbn 978 3 11 014251 8 mr 1293319 year 1994 chapter Max Koecher s work on Jordan algebras pages 187 195 External links MathGenealogy id 21575 http commons.wikimedia.org wiki Category Max Koecher Max Koecher on Wikipedia commons DEFAULTSORT Koecher, Max Category German mathematicians sl Max Koecher ... more details
In mathematics , the Mellin inversion formula named after Hjalmar Mellin tells us conditions under which the inverse Mellin transform , or equivalently the inverse two sided Laplace transform , are defined and recover the transformed function. If math varphi s math is analytic in the strip math a Re s b math , and if it tends to zero uniformly with increasing math Im s math for any real value c between a and b , with its integral along such a line converging absolutely, then if math f x mathcal M 1 varphi frac 1 2 pi i int c i infty c i infty x s varphi s , ds math we have that math varphi s mathcal M f int 0 infty x s f x , frac dx x . math Conversely, suppose f x is piecewise continuous on the positive real numbers, taking a value halfway between the limit values at any jump discontinuities, and suppose the integral math varphi s int 0 infty x s f x , frac dx x math is absolutely convergent when math a Re s b math . Then f is recoverable via the inverse Mellin transform from its Mellin transform math varphi math . We may strengthen the boundedness condition on math varphi s math if f x is continuous. If math varphi s math is analytic in the strip math a Re s b math , and if math varphi s K s 2 math , where K is a positive constant, then f x as defined by the inversion integral exists and is continuous moreover the Mellin transform of f is math varphi math for at least math a Re s b math . On the other hand, if we are willing to accept an original f which is a generalized function , we may relax the boundedness condition on math varphi math to simply make it of polynomial growth in any closed strip contained in the open strip math a Re s b math . We may also define a Banach space version of this theorem. If we call by math L nu, p R math the weighted Lp space of complex valued functions f on the positive reals such that math f left int 0 infty x nu f x p , frac dx x right 1 p infty math where and p are fixed real numbers with p 1, then if f x is in math L nu, p R ... more details
boundedness of the Cauchy kernel to the curvature of measures. They proved that if there is some ... S. and Verdera, Joan title A geometric proof of the L sup 2 sup boundedness of the Cauchy integral ... more details
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