A lower bound for a Fourier transform multiplicator from \(M(L_p\to L_q)\) (Q1277509)

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.