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Eccentric $p-$summing Lipschitz operators and integral inequalities on metric spaces and graphs

R. Arnau, E. A. Sánchez Pérez, S. Sanjuan

Abstract

The extension of the concept of $p-$summability for linear operators to the context of Lipschitz operators on metric spaces has been extensively studied in recent years. This research primarily uses the linearization of the metric space $M$ afforded by the associated Arens-Eells space, along with the duality between $M$ and the metric dual space $M^\#$ defined by the real-valued Lipschitz functions on $M.$ However, alternative approaches to measuring distances between sequences of elements of metric spaces (essentially involved in the definition of $p-$summability) exist. One approach involves considering specific subsets of the unit ball of $M^\#$ for computing the distances between sequences, such as the real Lipschitz functions derived from evaluating the difference of the values of the metric from two points to a fixed point. We introduce new notions of summability for Lipschitz operators involving such functions, which are characterized by integral dominations for those operators. To show the applicability of our results, we use in the last part of the paper the theoretical tools obtained in the first part to the analysis of metric graphs. In particular, we show new results on the behavior of numerical indices defined on these graphs satisfying certain conditions of summability and symmetry.

Eccentric $p-$summing Lipschitz operators and integral inequalities on metric spaces and graphs

Abstract

The extension of the concept of summability for linear operators to the context of Lipschitz operators on metric spaces has been extensively studied in recent years. This research primarily uses the linearization of the metric space afforded by the associated Arens-Eells space, along with the duality between and the metric dual space defined by the real-valued Lipschitz functions on However, alternative approaches to measuring distances between sequences of elements of metric spaces (essentially involved in the definition of summability) exist. One approach involves considering specific subsets of the unit ball of for computing the distances between sequences, such as the real Lipschitz functions derived from evaluating the difference of the values of the metric from two points to a fixed point. We introduce new notions of summability for Lipschitz operators involving such functions, which are characterized by integral dominations for those operators. To show the applicability of our results, we use in the last part of the paper the theoretical tools obtained in the first part to the analysis of metric graphs. In particular, we show new results on the behavior of numerical indices defined on these graphs satisfying certain conditions of summability and symmetry.

Paper Structure

This paper contains 7 sections, 9 theorems, 58 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Absolutely close and eccentrically close sequences in metric spaces
  3. Factorization of eccentric $p-$summing and eccentrically $p-$approximating Lipschitz operators
  4. Applications: distance summing inequalities in metric graphs
  5. Weighted $p-$shortest path metrics an integral $p-$averages on graphs
  6. Eccentric $p-$summing domination for functionals on graphs and metric symmetry
  7. Conclusions

Key Result

Proposition 2.1

Let $(M,d)$ be a metric space and $p \ge 1,$ and consider two sequences $(x^1_i)_{i=1}^\infty$ and $(x^2_i)_{i=1}^\infty$ in it. Then for every $S \subseteq M,$

Figures (3)

  • Figure 1: A canonical example using the sequence $(1-1/2^i)_i$ with its limit, $1.$ The vertices $v_i$ represent the element $\{1-1/2^i\}$ of the convergent sequence, and $s$ represents the boundary vertex, that is, $s = \lim_i v_i.$ The edge values are weights between the vertices.
  • Figure 2: As in Figure \ref{['fig:grfini1']} of Example \ref{['exfin1']}, in this graph the convergent sequence $(a_k)_{k=1}^\infty$ is formed by the elements $(0, 1-1/2^k)_k$ together with its limit $s = (0, 1).$ In this figure the weights of the graph are not represented.
  • Figure 3: Graph on the circle with center $v_0$ and arc weights.

Theorems & Definitions (27)

  • Remark 1.1
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • ...and 17 more