Author s : Alicandro, R. Author s : C. Author s : Figalli, A. Author s : Esposito, L. Author s : Fusco, Nicola,Pratelli, A. Author s : Cinti, Eleonora,Pratelli, Aldo. Author s : Krejcirik, David,Pratelli, Aldo. Author s : Fusco, N. Author s : Daneri, Sara,Pratelli, Aldo. Author s : Acerbi, E. Author s : Brasco, Lorenzo,Pratelli, Aldo.

Author s : Morgan, Frank,Pratelli, Aldo. Author s : Breit, Dominic,Verde, Anna. Author s : Mingione, Giuseppe. Author s : Mazzoleni, Dario,Pratelli, Aldo. Author s : G. De Philippis,F. Differential Geom. Published in: MSP Anal. PDE 8, no. Published in: European Mathematical Society Published in: Mathematical Sciences Publishers Last update: 16 July Record number: Veuillez activer JavaScript.

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### Final Report Summary - ANTEGEFI (Analytic Techniques for Geometric and Functional Inequalities)

English en. Deutsch de. No suggestions found. Sign in. Results Packs. About us. Fact Sheet. Objective Isoperimetric and Sobolev inequalities are the best known examples of geometric-functional inequalities. In recent years the PI and collaborators have obtained new and sharp quantitative versions of these and other important related inequalities. These results have been obtained by the combined use of classical symmetrization methods, new tools coming from mass transportation theory, deep geometric measure tools and ad hoc symmetrizations. Field of Science logarithm.

Activity type Higher or Secondary Education Establishments. Website Contact the organisation. Principal Investigator Nicola Fusco Prof. Administrative Contact Immacolata Diez Dr. Status Closed project. Start date 1 January End date 31 December Milovanovic Deadline: March 31, Speed 88 days to first decision for reviewed manuscripts only 53 days to first decision for all manuscripts days from submission to acceptance 14 days from acceptance to publication.

Citation Impact 1. Skip to main content Advertisement. Search all SpringerOpen articles Search. Sensitivity analysis for a new class of generalized parametric nonlinear ordered variational inequality problems in ordered Banach spaces Authors: Kottakkaran Sooppy Nisar, Mohd. An alternating iteration algorithm for solving the split equality fixed point problem with L-Lipschitz and quasi-pseudo-contractive mappings Authors: Meixia Li, Xueling Zhou and Haitao Che.

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Relations of a planar domains bounded by hyperbolas with families of holomorphic functions Authors: S. Daniell concerning the existence of certain functions of bounded variation, Littlewood proved that. Bohr , related to the convergence of Dirichlet series. Both multilinear and polynomial inequalities also hold for real scalars instead of complex scalars.

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The case of complex scalars is the original one, with strong connections with Analytic Number Theory and Dirichlet series; we mention Defant et al. The case of real scalars was just very recently explored but seems to be also very relevant as it can be seen in its applications in Quantum Information Theory, remarked by Montanaro It may sound surprising but, by now, the estimates of the constants are what really matters for the applications. Up to now, according to Bayart et al. It has been recently shown in Maia et al. For real scalars, combining information from Bayart et al.

Below we list some basic open problems in this setting:. This was conjectured to be true in Pellegrino and Teixeira but it seems to be far from being solved. One can also try to figure out a complete perspective by attacking the even more ambitious question in the multilinear setting:.

Further generalizations to the anisotropic settings were obtained in Albuquerque et al. The main goal of this paper is to survey some aspects of the Bohnenblust-Hille and Hardy-Littlewood inequalities and also present new results and perspectives. These inequalities have been exhaustively investigated in the recent years and several new results and new techniques have appeared. We are also interested in showing connections between these inequalities and some classical results of the theory of absolutely summing operators. This paper is organized as follows. In the second section we show how the investigation of the geometry of the unit ball of the spaces of multilinear forms can be potentially useful to reach the optimal constants.

In the third section we show how these kind of inequalities recover classical results of the linear theory of absolutely summing operators. In the final section we discuss the perspectives of the subject and new strategies to attack the problem. In this section we discuss how the geometry of the unit balls of Banach spaces is connected to the problem of finding optimal constants of the multilinear Bohnenblust-Hille inequality.

If the answer was positive, then a definitive answer to the optimization problem of finding the optimal constants of the Bohnenblust-Hille inequality could be eventually achieved by a kind of Lagrange Multipliers approach. Unfortunately, to estimate the norm of a general multilinear form seems to be a quite complicated task.

The following result of independent interest is a prediction:. For these values of t 0 we can see that. We may have one of the following situations:.

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## Analytic-Geometric Inequalities and Related Topics

If we consider 1 , f can be written as one of the following expressions:. So, we get. In this case, it is plain that. For other values of t we have. For these values of t 1 we have. For real scalars, for the obvious reasons, the expression of the norm is less complicated; this result can be found in Jameson :. The above expressions of the norms of bilinear forms are somewhat prohibitive for further investigations following this vein. Extreme points are important for optimization of continuous and convex functions for a simple reason: their maximum are attained in extreme points as we shall see below.

If the maximum of f is not attained in any extreme point, then. The same happens with multilinear forms in finite dimensional normed spaces as the next result which seems to be folklore asserts:. Then, denoting by B E its closed unit ball, we have.

## Analytic and Geometric Inequalities and Applications | Themistocles M. Rassias | Springer

Let us suppose that 7 is not true. We start off with four elementary lemmata ; the proofs are omitted. Case 1. Thus, T is not an extreme point. Thus, T is an extreme point.