A lower bound for a Fourier transform multiplicator from \(M(L_p\to L_q)\)
type: Item
instance of: scholarly article
title: A lower bound for a Fourier transform multiplicator from \(M(L_p\to L_q)\)
zbMATH Open document ID: 0921.42011
DOI: 10.1007/BF02355472
author: E. D. Nursultanov
published in: Mathematical Notes
publication date: 1999-09-21T00:00:00Z
review text: Let \(m\) be a bounded function on \(\mathbb{R}^n\) and define the multiplier operator \(T\) by the Fourier transform equality \(\widehat {Tf}=m \widehat f\). In [Acta Math. 104, 93-140 (1960; Zbl 0093.11402)] \textit{L. Hörmander} proved that for \(1<p<2\leq q<\infty\) and \({1\over r}={1\over p}-{1\over q}\), the space weak-\(L^r\) is included in the space of multipliers from \(L^p(\mathbb{R}^n)\) to \(L^q(\mathbb{R}^n)\). Consequently, the operator norm of \(m\) as a multiplier from \(L^p\) to \(L^q\) is bounded by a constant times \[ \sup_{ | E|>0\atop E\text{ is compact}}{1\over| E|^{1/r'}} \int_E\bigl | m(x)\bigr | dx. \] The main goal of this paper is to define a suitable collection of sets of positive measure, called \textit{harmonic intervals}, that allows one to obtain a lower bound on the operator norm. In particular, the author demonstrates that for \(m\geq 0\), \[ \sup{1\over| E|^{1/r'}} \int_E m(x)dx, \] where the supremum is taken over the set of all harmonic intervals, is a lower bound on the operator norm of \(m\) as a multiplier from \(L^p\) to \(L^q\). Some related conditions are also investigated.
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 1257015
zbMATH Keywords: multipliers
zbMATH Keywords: operator norm
zbMATH Keywords: harmonic intervals
reviewed by: Douglas S. Kurtz
MaRDI profile type: Publication
cites work: Estimates for translation invariant operators in \(L^p\) spaces
full work available at URL: https://doi.org/10.1007/bf02355472
OpenAlex ID: W1986268181
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description: scientific article; zbMATH DE number 1257015