Characterizations of Lorentz Type Sobolev Multiplier Spaces and Their Preduals

TL;DR

The paper generalizes Sobolev multiplier spaces to Lorentz-type scales and develops a comprehensive duality framework. It introduces Lorentz-type multiplier spaces $\mathcal{M}^{p,q}$ and $M^{p,q}$, and their preduals via block-type spaces $B^{p',q'}$, $\mathcal{B}^{p',q'}$, and $N^{p,q}$, with Köthe dual characterizations. Key results include $(M^{p,q})^{\prime\,*}\approx M^{p,q}$ and $(\mathcal{M}^{p,q})^{\prime\,*}\approx \mathcal{M}^{p,q}$, plus embedding relations and conditions under which these dualities become isometric. The work also proves boundedness of the local Hardy-Littlewood maximal operator on these Lorentz-type spaces and ties capacitary Lorentz spaces into the duality theory via $\mathcal{L}^{1,q}(\mathcal{C})^{*}\approx \mathfrak{M}$. Overall, the paper provides a robust functional-analytic toolkit for Lorentz-type Sobolev multipliers with applications to nonlinear PDEs in super-critical regimes.

Abstract

We provide several characterizations of Sobolev multiplier spaces of Lorentz type and their preduals. Block decomposition and Köthe dual of such preduals are discussed. As an application, the boundedness of local Hardy-Littlewood maximal function on these spaces will be justified.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 1.4
• Proposition 1.5
• Theorem 1.6
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• Theorem 1.8
• Theorem 1.9
• Theorem 1.10