Multipliers in the scale of periodic Bessel potential spaces with smoothness indices of different signs

TL;DR

This paper characterizes the multipliers between periodic Bessel potential spaces on the n-dimensional torus, especially when their smoothness indices have different signs, by analyzing a periodic analogue of a key linear operator.

Contribution

It introduces a detailed description of multipliers between such spaces with differing smoothness signs using a novel periodic analogue of the operator J_s.

Findings

Established a general description of multipliers between periodic Bessel potential spaces.

Developed a periodic analogue of the operator J_s based on Fourier coefficient asymptotics.

Demonstrated the homeomorphism between periodic distributions and Schwartz distributions.

Abstract

We prove a general type description result for the multipliers acting between two periodic Bessel potential spaces, defined on the $n$--dimensional torus, in a case when their smoothness indices are of different signs. This is done through the detailed examination of a periodic analogue of the linear operator $J_s$, which is employed in the definition of the scale of the Bessel potential space defined on the whole space $\mathbb{R}^n$. Our method of defining this periodic analogue of $J_s$ uses the results about an asymptotic behaviour of the generalized Fourier coefficients and existence of a natural homeomorphism between the spaces $\mathcal{D}'(\mathbb{T}^n)$ and $S'_{2 \cdot \pi}(\mathbb{R}^n)$, where the latter consists of all $2 \cdot \pi$--periodic distributions from the dual Schwartz space $\mathcal{S}'(\mathbb{R}^n)$.