1. A lower estimate for weak-type Fourier multipliers.
- Author
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Karlovich, Alexei and Shargorodsky, Eugene
- Subjects
- *
OPERATOR theory , *FUNCTION spaces , *BANACH spaces , *COMMERCIAL space ventures , *LORENTZ spaces , *MULTIPLIERS (Mathematical analysis) - Abstract
Asmar et al. [Note on norm convergence in the space of weak type multipliers. J Operator Theory. 1998;39(1):139–149] proved that the space of weak-type Fourier multipliers acting from L p into L p , ∞ is continuously embedded into L ∞ . We obtain a sharper result in the setting of abstract Lorentz spaces Λ q (X) with 0 < q ≤ ∞ built upon a Banach function space X on R n . We consider a source space S and a target space T in the class of admissible spaces A := { X , Λ q (X) : 0 < q ≤ ∞ }. Let M S , T 0 denote the space of Fourier multipliers acting from S to T . We show that if the space X satisfies the weak doubling property, then the space M Λ q (X) , Λ ∞ (X) 0 is continuously embedded into L ∞ for every 0 < q ≤ ∞. This implies that M S , T 0 is a quasi-Banach space for all choices of source and target spaces S , T ∈ A . [ABSTRACT FROM AUTHOR]
- Published
- 2022
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