I am reading the paper "P.Sjolin, Convolution with Oscillating Kernels, Indiana University Mathematics Journal Vol. 30, No. 1 (1981), pp. 47-55" where $L^p-L^p$ boundedness of the operator
$$Tf(x)=\int \frac{e^{i|x-y|^{a}}}{|x-y|^{\alpha}}
 f(y)dy$$
is studied. Here $0<\alpha<n$ and $a>0$, $a\neq 1$.


The author repeatedly utilizes the claim that:


if the inequality
$$\|Tf\|\_{p}\leq C \|f\|\_{p}\qquad (1)$$
necessitates that $p\geq p\_{0}$ for some $1< p\_{0}\leq 2$ then (1) also necessitates that that $p\leq p\_{0}^{\prime}$, the dual exponent of $p\_{0}$. That is if (1) is false for all $p<p\_{0}$ for some $1<p\_{0}\leq 2$
then (1) is also false for all $p>p\_{0}^{\prime}$.


This seems to have something to do with duality but I can't make that precise. Where does this statement come from ?



Why does failure of boundedness of this operator for $p<q$ implies its failure for $p>q^{\prime}$?
This is a standard property of self-adjoint (including convolution) operators, following from duality.


Let $Tf = k\*f$ and assume that $||Tf||\_{p} \leq C ||f||\_{p}$.


Then $$||Tf||\_{p'} = \sup\_{\substack{g \in L^p \\ ||g||\_{p}} =1} \left<Tf,g\right> =\sup\_{\substack{g \in L^p \\ ||g||\_{p}} =1} \left<f,Tg\right>.$$
From Holder we then have that this is
$$\leq \sup\_{\substack{g \in L^p \\ ||g||\_{p}} =1} ||Tg||\_{p} ||f||\_{p'} \leq C ||f||\_{p'}.$$


