78 research outputs found
Maharam-type kernel representation for operators with a trigonometric domination
[EN] Consider a linear and continuous operator T between Banach function spaces.
We prove that under certain requirements an integral inequality for T is equivalent to a
factorization of T through a specific kernel operator: in other words, the operator T has
what we call a Maharam-type kernel representation. In the case that the inequality provides
a domination involving trigonometric functions, a special factorization through the Fourier
operator is given. We apply this result to study the problem that motivates the paper:
the approximation of functions in L2[0, 1] by means of trigonometric series whose Fourier
coefficients are given by weighted trigonometric integrals.This research has been supported by MTM2016-77054-C2-1-P (Ministerio de Economia, Industria y Competitividad, Spain).SĂĄnchez PĂ©rez, EA. (2017). Maharam-type kernel representation for operators with a trigonometric domination. Aequationes Mathematicae. 91(6):1073-1091. https://doi.org/10.1007/s00010-017-0507-6S10731091916Calabuig, J.M., Delgado, O., SĂĄnchez PĂ©rez, E.A.: Generalized perfect spaces. Indag. Math. 19(3), 359â378 (2008)Calabuig, J.M., Delgado, O., SĂĄnchez PĂ©rez, E.A.: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl. 364, 88â103 (2010)Delgado, O., SĂĄnchez PĂ©rez, E.A.: Strong factorizations between couples of operators on Banach function spaces. J. Convex Anal. 20(3), 599â616 (2013)Dodds, P.G., Huijsmans, C.B., de Pagter, B.: Characterizations of conditional expectation type operators. Pacific J. Math. 141(1), 55â77 (1990)Flores, J., HernĂĄndez, F.L., Tradacete, P.: Domination problems for strictly singular operators and other related classes. Positivity 15(4), 595â616 (2011). 2011Fremlin, D.H.: Tensor products of Banach lattices. Math. Ann. 211, 87â106 (1974)Hu, G.: Weighted norm inequalities for bilinear Fourier multiplier operators. Math. Ineq. Appl. 18(4), 1409â1425 (2015)Halmos, P., Sunder, V.: Bounded Integral Operators on L 2 Spaces. Springer, Berlin (1978)Kantorovitch, L., Vulich, B.: Sur la reprĂ©sentation des opĂ©rations linĂ©aires. Compositio Math. 5, 119â165 (1938)Kolwicz, P., LeĆnik, K., Maligranda, L.: Pointwise multipliers of CalderĂłn- Lozanovskii spaces. Math. Nachr. 286, 876â907 (2013)Kolwicz, P., LeĆnik, K., Maligranda, L.: Pointwise products of some Banach function spaces and factorization. J. Funct. Anal. 266(2), 616â659 (2014)Kuo, W.-C., Labuschagne, C.C.A., Watson, B.A.: Conditional expectations on Riesz spaces. J. Math. Anal. Appl. 303, 509â521 (2005)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)Maharam, D.: The representation of abstract integrals. Trans. Am. Math. Soc. 75, 154â184 (1953)Maharam, D.: On kernel representation of linear operators. Trans. Am. Math. Soc. 79, 229â255 (1955)Maligranda, L., Persson, L.E.: Generalized duality of some Banach function spaces. Indag. Math. 51, 323â338 (1989)Neugebauer, C.J.: Weighted norm inequalities for averaging operators of monotone functions. Publ. Mat. 35, 429â447 (1991)Okada, S., Ricker, W.J., SĂĄnchez PĂ©rez, E.A.: Optimal Domain and Integral Extension of Operators Acting in Function Spaces. Operator Theory: Adv. Appl., vol. 180. BirkhĂ€user, Basel (2008)Rota, G.C.: On the representation of averaging operators. Rend. Sem. Mat. Univ. Padova. 30, 52â64 (1960)SĂĄnchez PĂ©rez, E.A.: Factorization theorems for multiplication operators on Banach function spaces. Integr. Equ. Oper. Theory 80(1), 117â135 (2014)Schep, A.R.: Factorization of positive multilinear maps. Ill. J. Math. 28(4), 579â591 (1984)Schep, A.R.: Products and factors of Banach function spaces. Positivity 14(2), 301â319 (2010
Product factorability of integral bilinear operators on Banach function spaces
[EN] This paper deals with bilinear operators acting in pairs of Banach function spaces that factor through the pointwise product. We find similar situations in different contexts of the functional analysis, including abstract vector latticesÂżorthosymmetric maps, CÂż-algebrasÂżzero product preserving operators, and classical and harmonic analysisÂżintegral bilinear operators. Bringing together the ideas of these areas, we show new factorization theorems and characterizations by means of norm inequalities. The objective of the paper is to apply these tools to provide new descriptions of some classes of bilinear integral operators, and to obtain integral representations for abstract classes of bilinear maps satisfying certain domination properties.The first author was supported by TUBITAK-The Scientific and Technological Research Council of Turkey, Grant No. 2211/E. The second author was supported by Ministerio de Economia y Competitividad (Spain) and FEDER, Grant MTM2016-77054-C2-1-P.Erdogan, E.; SĂĄnchez PĂ©rez, EA.; Gok, O. (2019). Product factorability of integral bilinear operators on Banach function spaces. Positivity. 23(3):671-696. https://doi.org/10.1007/s11117-018-0632-zS671696233Abramovich, Y.A., Kitover, A.K.: Inverses of Disjointness Preserving Operators. American Mathematical Society, Providence (2000)Abramovich, Y.A., Wickstead, A.W.: When each continuous operator is regular II. Indag. Math. (N.S.) 8(3), 281â294 (1997)Alaminos, J., BreĆĄar, M., Extremera, J., Villena, A.R.: Maps preserving zero products. Studia Math. 193(2), 131â159 (2009)Alaminos, J., BreĆĄar, M., Extremera, J., Villena, A.R.: On bilinear maps determined by rank one idempotents. Linear Algebra Appl. 432, 738â743 (2010)Alaminos, J., Extremera, J., Villena, A.R.: Orthogonality preserving linear maps on group algebras. Math. Proc. Camb. Philos. Soc. 158, 493â504 (2015)Ben Amor, F.: On orthosymmetric bilinear maps. Positivity 14, 123â134 (2010)Astashkin, S.V., Maligranda, L.: Structure of CesĂ ro function spaces: a survey. Banach Center Publ. 102, 13â40 (2014)Beckenstein, E., Narici, L.: A non-Archimedean StoneâBanach theorem. Proc. Am. Math. Soc. 100(2), 242â246 (1987)Bu, Q., Buskes, G., Kusraev, A.G.: Bilinear maps on products of vector lattices: a survey. In: Boulabiar, K., Buskes, G., Triki, A. (eds.) Positivity: Trends in Mathematics, pp. 97â126. Springer, Birkhuser (2007)Buskes, G., van Rooij, A.: Almost f-algebras: commutativity and CauchyâSchwarz inequality. Positivity 4, 227â231 (2000)Buskes, G., van Rooij, A.: Squares of Riesz spaces. Rocky Mt. J. Math. 31(1), 45â56 (2001)Calabuig, J.M., Delgado, O., SĂĄnchez PĂ©rez, E.A.: Generalized perfect spaces. Indag. Math. (N.S.) 19(3), 359â378 (2008)CalderĂłn, A.P.: Intermediate spaces and interpolation, the complex method. Studia Math. 24, 113â190 (1964)Defant, A.: Variants of the Maurey-Rosenthal theorem for quasi Köthe function spaces. Positivity 5, 153â175 (2001)Delgado Garrido, O., SĂĄnchez PĂ©rez, E.A.: Strong factorizations between couples of operators on Banach function spaces. J. Convex Anal. 20(3), 599â616 (2013)Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators, vol. 43. Cambridge University Press, Cambridge (1995)ErdoÄan, E., Calabuig, J.M., SĂĄnchez PĂ©rez, E.A.: Convolution-continuous bilinear operators acting in Hilbert spaces of integrable functions. Ann. Funct. Anal. 9(2), 166â179 (2018)Diestel, J., Uhl, J.J.: Vector Measures. American Mathematical Society, Providence (1977)Fremlin, D.H.: Tensor products of Archimedean vector lattices. Am. J. Math. 94, 778â798 (1972)Gillespie, T.A.: Factorization in Banach function spaces. Nederl. Akad. Wetensch. Indag. Math. 43(3), 287â300 (1981)Grafakos, L., Li, X.: Uniform bounds for the bilinear Hilbert transforms I. Ann. Math. 159, 889â933 (2004)Kantorovich, K.L., Akilov, G.P.: Functional Analysis, Nauka, Moscow 1977 (Russian). English transl. Pergamon Press, Oxford, Elmsford, New York (1982)Kolwicz, P., LeĆnik, K., Maligranda, L.: Pointwise products of some Banach function spaces and factorization. J. Funct. Anal. 266(2), 616â659 (2014)Kolwicz, P., LeĆnik, K.: Topological and geometrical structure of CalderĂłnâLozanovskii construction. Math. Inequal. Appl. 13(1), 175â196 (2010)KĂŒhn, B.: BanachverbĂ€nde mit ordnungsstetiger dualnorm. Math. Z. 167(3), 271â277 (1979)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II: Function Spaces, vol. 97. Springer, Berlin (1979)Lozanovskii, G.Ya.: On some Banach lattices. Sibirsk. Mat. Zh. 10, 584-599 (1969)(Russian)English transl. in Siberian Math. J. 10(3), 419-431 (1969)Maligranda, L., Persson, L.E.: Generalized duality of some Banach function spaces. Nederl. Akad. Wetensch. Indag. Math. 51(3), 323â338 (1989)Okada, S., Ricker, W., SĂĄnchez PĂ©rez, E.A.: Optimal domain and integral extension of operators. Oper. Theory Adv. Appl. BirkhĂ€user/Springer 180 (2008)Ryan, R.: Introduction to Tensor Product of Banach Spaces. Springer, London (2002)SĂĄnchez PĂ©rez, E.A., Werner, D.: Slice continuity for operators and the Daugavet property for bilinear maps. Funct. Approx. Comment. Math. 50(2), 251â269 (2014)Schep, A.R.: Products and factors of Banach function spaces. Positivity 14(2), 301â319 (2010)Villarroya, F.: Bilinear multipliers on Lorentz spaces. Czechoslov. Math. J. 58(4), 1045â1057 (2008
FIRST CRYOMODULE TEST AT AMTF HALL FOR THE EUROPEAN X-RAY FREE ELECTRON LASER (XFEL)
Abstract The Accelerator Module Test Facility (AMTF) at DESY in Hamburg is dedicated to the tests of RF cavities and accelerating cryomodules for the European X-ray Free Electron Laser (XFEL). The AMTF hall is equipped with two vertical cryostats, which are used for RF cavities testing and three test benches that will be used for tests of the accelerating cryomodules. Recently, the first cryomodule teststand (XATB3) was commissioned and the first XFEL cryomodule (XM-2) was tested by team of physicists, engineers and technicians from The Henry
Orthogonalities and functional equations
In this survey we show how various notions of orthogonality appear in the theory of functional equations. After introducing some orthogonality relations, we give examples of functional equations postulated for orthogonal vectors only. We show their solutions as well as some applications. Then we discuss the problem of stability of some of them considering various aspects of the problem. In the sequel, we mention the orthogonality equation and the problem of preserving orthogonality. Last, but not least, in addition to presenting results, we state some open problems concerning these topics. Taking into account the big amount of results concerning functional equations postulated for orthogonal vectors which have appeared in the literature during the last decades, we restrict ourselves to the most classical equations
K\"othe-Herz Spaces: The Amalgam-Type Spaces of Infinite Direct Sums
In this paper, we introduce a class of function spaces called K\"othe-Herz
spaces . These spaces are similar to amalgam spaces and are
characterized by a local component given by a countable family
of quasi-normed
function spaces, and a global component , which is a quasi-normed sequence
space. We investigate various geometric and topological properties inherited by
from its components, such as their completeness, duality,
order continuity, ideal and Fatou properties, in an abstract setting. In
addition, we provide a Banach function space characterization for
, which allows us to understand its structure and behavior more
deeply. Furthermore, by appropriate amalgamation of Lorentz spaces (Orlicz
spaces) and Lebesgue sequence spaces, we define Lorentz-Herz spaces
(Orlicz-Herz spaces) as a particular case of , which are still
generalizations of the classical Herz spaces. In this context (especially
Lorentz-Herz spaces), we establish previously studied properties, demonstrate
interpolation results, and prove the boundedness of important sublinear
integral operators with kernels that satisfy a size condition.Comment: 32 pages, 0 figure
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