Bilateral Continuous Embeddings for Multipliers Acting in the Scale of Bessel Potential Spaces with Nonnegative Smoothness Indices

Abstract: We investigate the problem of establishing bilateral embeddingsof the uniformly localized Bessel potential spaces into the multiplier spaces between twoBessel potential spaces with nonnegative smoothness indices. Thisproblem is considered in the most general situation when only thenatural assumptions on the indices of these Bessel potentialspaces are met yet the description theorems for the correspondingmultiplier space in terms of the spaces can not be established. The natural characterof these assumptions is demonstrated explicitly. The embedding ofa uniformly localized Bessel potential space into thecorresponding multiplier space is obtained via the criterion ofthe validity of this embedding in terms of the multiplicativefunctional estimate on the norms, while the functional estimateitself is derived from general multiplicative norm estimates inLizorkin–Triebel spaces. The uniform localization principle,which holds true not only for Bessel potential spaces but also forgeneral Lizorkin–Triebel spaces, is of the utmost importance foremploying this criterion. © 2025 Elsevier B.V., All rights reserved.