Multipliers in Bessel potential spaces with smoothness indices of different sign

Abstract:

The subject of the present paper is the study of multipliers from the Bessel potential space $H^s_p(\mathbb {R}^n)$ to the Bessel potential space $H^{-t}_q(\mathbb {R}^n)$ for the case in which the smoothness indices of these spaces have different signs, i.e., $s, t \geq 0$. The space of such multipliers consists of distributions $u$ such that for all $\varphi \in H^s_p(\mathbb {R}^n)$ the product $\varphi \cdot u$ is well defined and belongs to $H^{-t}_q(\mathbb {R}^n)$. It turns out that these multiplier spaces can be described explicitly in the case where $p \leq q$ and one of the following conditions is fulfilled: \begin{equation*} s \geq t \geq 0, \ s > n/p \ \text { or } \ \ t \geq s \geq 0, \ t > n/q’ \quad (\text {where } \ 1/q +1/q’ = 1). \end{equation*} Namely, in this case we have \begin{equation*} M[H^s_p(\mathbb {R}^n) \to H^{-t}_{q}(\mathbb {R}^n)] = H^{-t}_{q, \mathrm {unif}}(\mathbb {R}^n) \cap H^{-s}_{p’, \mathrm {unif}}(\mathbb {R}^n), \end{equation*} where $H^\gamma _{r, \mathrm {unif}}(\mathbb {R}^n)$ is the space of uniformly localized Bessel potentials.

For the important case where $s = t < n/\max (p,q’)$, we prove the two-sided embeddings \begin{equation*} H^{-s}_{r_1, \mathrm {unif}}(\mathbb {R}^n) \subset M[H^s_p(\mathbb {R}^n) \to H^{-s}_q(\mathbb {R}^n)] \subset H^{-s}_{r_2, \mathrm {unif}}(\mathbb {R}^n), \end{equation*} where $r_2 = \max (p’, q)$, $r_1 =[s/n-(1/p -1/q)]^{-1}$.