Monotone operators and "bigger conjugate" functions

TL;DR

This paper investigates 'bigger conjugate' functions in monotone operator theory, providing tools to answer a key open problem negatively and refuting a related conjecture, highlighting the importance of unbounded skew relations.

Contribution

It offers new analytical tools for BC-functions, answers Simons' problem negatively, and refutes a broader conjecture, advancing understanding in monotone operator theory.

Findings

Answer to Simons' problem is negative

Refutation of a generalized conjecture

Highlights role of unbounded skew relations

Abstract

We study a question posed by Stephen Simons in his 2008 monograph involving "bigger conjugate" (BC) functions and the partial infimal convolution. As Simons demonstrated in his monograph, these function have been crucial to the understanding and advancement of the state-of-the-art of harder problems in monotone operator theory, especially the sum problem. In this paper, we provide some tools for further analysis of BC--functions which allow us to answer Simons' problem in the negative. We are also able to refute a similar but much harder conjecture which would have generalized a classical result of Br\'ezis, Crandall and Pazy. Our work also reinforces the importance of understanding unbounded skew linear relations to construct monotone operators with unexpected properties.