Generalized upper and lower Legendre conjugates for weight functions
Gerhard Schindl
TL;DR
The paper develops a cohesive framework of generalized lower and upper Legendre conjugates for weight functions, linking the product and quotient of underlying weight sequences to infimal convolution and supremal anti-convolution transforms. It provides precise inequalities and equalities for how the growth indices $\gamma(\omega)$ and $\overline{\gamma}(\omega)$ transform under these conjugations and extends the theory to associated weight functions derived from sequences. A central result shows that, in the weight-sequence setting, the lower conjugate of $\omega_{\mathbf{M}}$ and $\omega_{\mathbf{N}}$ coincides with $\omega_{\mathbf{M}\cdot\mathbf{N}}$, while the upper conjugate relates to the quotient $\omega_{\mathbf{M}/\mathbf{N}}$, with careful treatment of non-standard cases. The work also develops inverse operation results demonstrating that these transforms can recover original weights under appropriate domain constraints, thereby providing robust tools for analyzing weighted spaces and their regularity properties in ultradifferentiable and ultraholomorphic contexts.
Abstract
We introduce and study new transformations between two functions satisfying some basic growth properties and generalize the known lower and upper Legendre conjugate (or envelope). We also investigate how these transformations modify recently defined growth indices for weight functions. A special but important and useful situation, to which the knowledge is then applied, is when considering associated weight functions which are expressed in terms of an underlying weight sequence. In this case these transformations precisely correspond to the point-wise product resp. point-wise division of the given sequences. Therefore, the new approach studied in this work illustrates the genuineness and importance and suggests applications for weighted spaces in different directions.