197,209 research outputs found
Maximal Factorization of Operators Acting in Kothe-Bochner Spaces
No full text[EN] Using some representation results for Kothe-Bochner spaces of vector valued functions by means of vector measures, we analyze the maximal extension for some classes of linear operators acting in these spaces. A factorization result is provided, and a specific representation of the biggest vector valued function space to which the operator can be extended is given. Thus, we present a generalization of the optimal domain theorem for some types of operators on Banach function spaces involving domination inequalities and compactness. In particular, we show that an operator acting in Bochner spaces of p-integrable functions for any 1First author is supported by Grant MTM2011-23164 of the Ministerio de Economia y Competitividad (Spain). Second author is supported by Grant 284110 of CONACyT (Mexico). Fourth author is supported by Grant MTM2016-77054-C2-1-P of the Ministerio de Ciencia, Innovacion y Universidades, Agencia Estatal de Investigaciones (Spain) and FEDER.Calabuig, JM.; Fernández-Unzueta, M.; Galaz-Fontes, F.; Sánchez Pérez, EA. (2021). Maximal Factorization of Operators Acting in Kothe-Bochner Spaces. Journal of Geometric Analysis. 31(1):560-578. https://doi.org/10.1007/s12220-019-00290-4S560578311Abasov, N., Pliev, M.: On two definitions of a narrow operator on Köthe–Bochner spaces. Arch. Math. 111, 167–176 (2018)Bartle, R.G., Dunford, N., Schwartz, J.: Weak compactness and vector measures. Can. J. Math. 7, 289–305 (1955)Bochner, S.: Integration von Funktionen, deren Werte die Elemente eines Vectorraumes sind. Fundam. Math. 20, 262–276 (1933)Calabuig, J.M., Fernández Unzueta, M., Galaz Fontes, F., Sánchez Pérez, E.A.: Extending and factorizing bounded bilinear maps defined on order continuous Banach function spaces. RACSAM 108, 353–367 (2014)Calabuig, J.M., Jiménez-Fernández, E., Juan, M.A., Sánchez-Pérez, E.A.: Optimal extensions of compactness properties for operators on Banach function spaces. Topol. Appl. 203, 57–66 (2016)Cembranos, P., Mendoza, J.: Banach spaces of vector-valued functions, Lecture Notes in Mathematics, vol. 1676. Springer, Berlin (1997)Cerdà, J., Hudzik, H., Mastyło, M.: Geometric properties of Köthe-Bochner spaces. Math. Proc. Camb. Philos. Soc. 120(3), 521–533 (1996)Choi, C., Lee, H.H.: Operators of Fourier type p with respect to some subgroups of a locally compact abelian group. Arch. Math. 81(4), 457–466 (2003)Defant, A., Floret, K.: Tensor Norms and Operator Ideals. North-Holland, Amsterdam (1993)Defant, A., López Molina, J.A., Rivera, M.J.: On Pitt’s theorem for operators between scalar and vector-valued quasi-Banach sequence spaces. Monatshefte für Mathematik 130(1), 7–18 (2000)Diestel, J., Uhl, J.J.: Vector Measures. American Mathematical Society, Providence (1977)Duru, H., Kitover, A., Orhon, M.: Multiplication operators on vector-valued function spaces. Proc. Am. Math. Soc. 141, 3501–3513 (2013)Feledziak, K.: Absolutely continuous linear operators on Köthe–Bochner spaces. Banach Center Publ. 92, 85–89 (2011)Feledziak, K., Nowak, M.: Integral representation of linear operators on Orlicz-Bochner spaces. Collect. Math. 61, 277–290 (2010)Huerta, P.G.: Espacios de medidas vectoriales. Thesis, Universidad de Valencia, ISBN: 8437060591 (2005)Kusraev, A.G.: Dominated Operators. Springer, Dordrecht (2000)Lewis, D.R.: On integrability and summability in vector spaces. Ill. J. Math. 16, 294–307 (1972)Lin, P.-K.: Köthe-Bochner Function Spaces. Birkhauser, Boston (2004)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)Nowak, M.: Bochner representable operators on Köthe–Bochner spaces. Comment. Math. 48, 113–119 (2008)Okada, S.: Does a compact operator admit a maximal domain for its compact linear extension? In: Curbera, G., Mockenhaupt, G., Ricker, W.J. (eds.) Vector Measures, Integration and Related Topics, pp. 313–322. Basel, Birkhäuser (2009)Okada, S., Ricker, W.J., Pérez, E.A.S.: Optimal Domains and Integral Extensions of Operators acting in Function Spaces, Operator Theory Advances and Applications, vol. 180. Birkhäuser, Basel (2008)Sánchez Pérez, E.A., Szwedek, R.: Vector measures with values in and interpolation of Banach lattices. J. Convex Anal. 25, 75–92 (2018
On Foxʼs m-dimensional category and theorems of Bochner type
No full textAbstractWe show that catm(X)=cat(jm), where catm(X) is Foxʼs m-dimensional category, jm:X→X[m] is the mth Postnikov section of X and cat(X) is the Lusternik–Schnirelmann category of X. This characterization is used to give new “Bochner-type” bounds on the rank of the Gottlieb group and the first Betti number for manifolds of non-negative Ricci curvature. Finally, we apply these methods to obtain Bochner-type theorems for manifolds of almost non-negative sectional curvature
Hermite-Hadamard's Inequality and the p-HH-Norm on the Cartesian Product of Two Copies of a Normed Space
No full textThe Cartesian product of two copies of a normed space is naturally equipped with
the well-known p-norm. In this paper, another notion of norm is introduced, and will be
called the p-HH-norm. This norm is an extension of the generalised logarithmic mean and is
connected to the p-norm by the Hermite-Hadamard's inequality. The Cartesian product space
(with respect to both norms) is complete, when the (original) normed space is. A proof for the
completeness of the p-HH-norm via Ostrowski's inequality is provided. This space is embedded
as a subspace of the well-known Lebesgue-Bochner function space (as a closed subspace, when
the norm is a Banach norm). Consequently, its geometrical properties are inherited from those
of Lebesgue-Bochner space. An explicit expression of the superior (inferior) semi-inner product
associated to both norms is considered and used to provide alternative proofs for the smoothness
and reflexivity of this space
Contact Hypersurfaces of a Bochner-Kaehler Manifold
No full textWe have studied contact metric hypersurfaces of a Bochner-Kaehler manifold and obtained the following two results: (1) A contact metric constant mean curvature (C M C) hypersurface of a Bochner-Kaehler manifold is a (k, µ)-contact manifold, and (2) If M is a compact contact metric C M C hypersurface of a Bochner-Kaehler manifold with a conformal vector field V that is neither tangential nor normal anywhere, then it is totally umbilical and Sasakian, and under certain conditions on V , is isometric to a unit sphere
The Bochner Integral
No full textPrévôt C, Röckner M. The Bochner Integral. In: Prévôt C, Röckner M, eds. A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics. Vol 1905. Berlin: Springer; 2007: 105-108
Bochner technique in real Finsler manifolds
No full textUsing non-linear connection of Finsler manifold M, the existence of local coordinates which is normalized at a point x is proved, and the Laplace operator Delta on 1-form of M is defined by non-linear connection and its curvature tensor. After proving the maximum principle theorem of Hopf-Bochner on M, the Bochner type vanishing theorem of Killing vectors and harmonic 1-form axe obtained
Methods for the analysis of oscillatory integrals and Bochner-Riesz operators
No full textFor a smooth surface Γ of arbitrary codimension, one can consider the Lp mapping properties of the Bochner-Riesz multiplier
m(ζ) = dist(ζ,Γ)^α φ(ζ),
where α > 0 and φ is an appropriate smooth cutoff function. Even for the sphere, the exact Lp boundedness range remains a central open problem in Euclidean harmonic analysis. We consider the Lp integrability of the Bochner-Riesz convolution kernel for a particular class of surfaces (of any codimension). For a subclass of these surfaces the range of Lp integrability of the kernels differs substantially from the Lp boundedness range of the corresponding Bochner-Riesz multiplier operator. Extending work of Mockenhoupt, we then establish a range of operator bounds, which are sharp in the α exponent, under the assumption of an appropriate L2 restriction estimate. Hickman and Wright established sharp oscillatory integral estimates, associated with a particular class of surfaces, and derived restriction estimates. We extend this work to certain curves of standard type and corresponding surfaces of revolution. These surfaces are discussed as an explicit class for which we have Lp → Lp boundedness of the corresponding Bochner-Riesz operators.
Understanding the structure of the roots of real polynomials is important in obtaining stable bounds for oscillatory integrals with polynomial phases. For real polynomials with exponents in some fixed set,
Ψ(t)=x+y1 t^{k1} +...+yL t^{kL},
we analyse the different possible root structures that can occur as the coefficients vary. We first establish a stratification of roots into tiers containing roots of comparable sizes. We then show that at most L non-zero roots can cluster about a point. Supposing additional restrictions on the coefficients, we derive structural refinements. These structural results extend work of Kowalski and Wright and provide a characteristic picture of root structure at coarse scales. As an application, these results are used to recover the sharp oscillatory integral estimates of Hickman and Wright, using bounds for oscillatory integrals of Phong and Stein
BOCHNER TECHNIQUE ON STRONG KAHLER-FINSLER MANIFOLDS
No full textNational Natural Science Foundation of China [10571144, 10771174]; Program for New Centery Excellent Talents in Xiamen UniversityBy using the Chern-Finsler connection and complex Finsler metric, the Bochner technique on strong Kahler-Finsler manifolds is studied. For a strong Kahler-Finsler manifold M, the authors first prove that there exists a system of local coordinate which is normalized at a point v is an element of (M) over tilde = T(1,0)M\0(M), and then the horizontal Laplace operator square(H) for differential forms on PTM is defined by the horizontal part of the Chern-Finsler connection and its curvature tensor, and the horizontal Laplace operator square(H) on holomorphic vector bundle over PTM is also defined. Finally, we get a Bochner vanishing theorem for differential forms on PTM. Moreover, the Bochner vanishing theorem on a holomorphic line bundle over PTM is also obtaine
Qualitative Inquiry into Art History: A Tribute to Arthur P. Bochner
No full textThis poem is dedicated to the author\u27s mentor Arthur P. Bochner, Distinguished University Professor, University of South Florida
P-convexity of Musielak-Orlicz function spaces of Bochner type
No full textIt is proved that the Musielak-Orlicz function space LF (m ,X) of Bochner type is P-convex if and only if both spaces LF (m,R) and X are P-convex. In particular, the Lebesgue-Bochner space Lp (m ,X) is P-convex iff X is P-convex
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