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Denote $S$ as the space of Schwartz functions, for $v\in S'$, the space of tempered distributions, define an operator $T_v:f\in S \to f*v$. Then space of Hormander Multipliers $M^{p,q}$ can be defined as $\{ v\in S': \|T_v\|_{L^p\to L^q}<\infty \}$.

Do we know some results on the characterization of $M^{p,q}$, when $p < q$?

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Of course, it contains $L^r$ where $\frac1p+\frac1r=1+\frac1q$. But you must expect a more accurate answer. –  Denis Serre Apr 1 '11 at 5:28
    
@Denis Serre: And also the Lorentz space $L^{r,\infty}$ by the weak type Young's inequality. I'm wondering if Lorentz space is optimal for Young's inequality. –  Shaoming Guo Apr 1 '11 at 8:37
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I would be very surprised if there were any nice, precise characterisation for general $p,q$ (but would be happy to be proved wrong). Even if you restrict to compactly supported $v \in C^\infty$, I believe there's no nice characterisation - and furthermore, it's known that no simple characterisation in terms of growth rates is possible. –  Zen Harper Apr 1 '11 at 10:06
    
For example, I'm fairly sure there are measurable functions $v$ such that $T_v$ (extends to) a bounded operator on $L^2$, but $T_{|v|}$ does not. I expect $v$ can be made smooth. Sorry I haven't got any more precise results or references, though - that's why this is a comment, not an answer. –  Zen Harper Apr 1 '11 at 10:10
    
@ Zen Harper: Your comment is very helpful, maybe convolution is always some kind of weird, and also not that "precise", so it's not easy to get the optimal estimates. –  Shaoming Guo Apr 1 '11 at 11:12
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