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On the conjugate weight function and ultradifferentiable classes of entire functions

Gerhard Schindl

TL;DR

The work develops a comprehensive bridge between weight functions and weight sequences in ultradifferentiable theory by introducing the conjugate weight function $\omega^{\ast}$ and analyzing its interaction with associated weight matrices, Legendre envelopes, and growth indices. It systematically transfers non-standard regularity notions from weight sequences to the Braun-Meise-Taylor framework, showing how ultradifferentiable classes can be described as weighted spaces of entire functions via conjugate objects. The results illuminate when a single weight function suffices to capture regularity (matrix constancy) and when matrix flexibility is essential, with applications to operator theory on Hilbert spaces, particularly in detecting boundedness of normal operators through non-standard Gevrey-like classes. The analysis clarifies intricate links between two parallel formalisms (weight functions and weight sequences) and extends Markin-type regularity results to the BMT setting, highlighting both opportunities and restrictive regimes depending on growth indices and Legendre–envelope behavior. Overall, the paper provides a unified, technically rich framework for ultradifferentiable spaces, their weighted-entire-function representations, and operator-regularity criteria in non-standard settings.

Abstract

We introduce the new notion of a conjugate weight function and provide a detailed study of this operation and its properties. Then we apply this knowledge to study classes of ultradifferentiable functions defined in terms of fast growing weight functions in the sense of Braun-Meise-Taylor and hence violating standard regularity requirements. Therefore, we transfer recent results shown by the author and D.N. Nenning from the weight sequence to the weight function framework. In order to proceed and to complete the picture we also define the conjugate associated weight matrix and investigate the relation to conjugate weight sequences via the corresponding associate weight functions. Finally, as it has already been done in the weight sequence case, we generalize results by M. Markin from the small Gevrey-setting and show how the corresponding non-standard ultradifferentiable function classes can be used to detect boundedness of normal linear operators on Hilbert spaces (associated with an evolution equation problem). On the one hand, when involving the weight matrix here the crucial information concerning regularity of the weak solutions can be expressed in terms of only one weight, namely of the given weight function. But, on the other hand, for the connection to the weighted entire setting the required conditions on the weight function are too restrictive in the general case.

On the conjugate weight function and ultradifferentiable classes of entire functions

TL;DR

The work develops a comprehensive bridge between weight functions and weight sequences in ultradifferentiable theory by introducing the conjugate weight function and analyzing its interaction with associated weight matrices, Legendre envelopes, and growth indices. It systematically transfers non-standard regularity notions from weight sequences to the Braun-Meise-Taylor framework, showing how ultradifferentiable classes can be described as weighted spaces of entire functions via conjugate objects. The results illuminate when a single weight function suffices to capture regularity (matrix constancy) and when matrix flexibility is essential, with applications to operator theory on Hilbert spaces, particularly in detecting boundedness of normal operators through non-standard Gevrey-like classes. The analysis clarifies intricate links between two parallel formalisms (weight functions and weight sequences) and extends Markin-type regularity results to the BMT setting, highlighting both opportunities and restrictive regimes depending on growth indices and Legendre–envelope behavior. Overall, the paper provides a unified, technically rich framework for ultradifferentiable spaces, their weighted-entire-function representations, and operator-regularity criteria in non-standard settings.

Abstract

We introduce the new notion of a conjugate weight function and provide a detailed study of this operation and its properties. Then we apply this knowledge to study classes of ultradifferentiable functions defined in terms of fast growing weight functions in the sense of Braun-Meise-Taylor and hence violating standard regularity requirements. Therefore, we transfer recent results shown by the author and D.N. Nenning from the weight sequence to the weight function framework. In order to proceed and to complete the picture we also define the conjugate associated weight matrix and investigate the relation to conjugate weight sequences via the corresponding associate weight functions. Finally, as it has already been done in the weight sequence case, we generalize results by M. Markin from the small Gevrey-setting and show how the corresponding non-standard ultradifferentiable function classes can be used to detect boundedness of normal linear operators on Hilbert spaces (associated with an evolution equation problem). On the one hand, when involving the weight matrix here the crucial information concerning regularity of the weak solutions can be expressed in terms of only one weight, namely of the given weight function. But, on the other hand, for the connection to the weighted entire setting the required conditions on the weight function are too restrictive in the general case.
Paper Structure (25 sections, 33 theorems, 119 equations)

This paper contains 25 sections, 33 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Weight functions and growth properties
  3. General notation
  4. Basic definitions and growth relations
  5. Growth indices $\gamma(\omega)$ and $\overline{\gamma}(\omega)$
  6. Generalized lower and upper Legendre conjugates
  7. The conjugate weight function
  8. Definition and basic properties
  9. On growth relations
  10. On the generalized Legendre envelopes
  11. The conjugate operation and growth indices
  12. The weight sequence setting
  13. Weight sequences
  14. Associated weight functions
  15. The conjugate sequence $\mathbf{M}^{\ast}$
  16. ...and 10 more sections

Key Result

Lemma 3.1

Let $\omega$ be a weight function, then the following are equivalent:

Theorems & Definitions (52)

  • Definition 2.1
  • Remark 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Example 3.3
  • Remark 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Corollary 3.7
  • Lemma 3.8
  • ...and 42 more