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Let me consider instead the operator $$Sf(x):=\int e^{-i x y} \frac{f(y)}{|x-y|^{\alpha}}dy$$ which has the same properties as $T$ (related by conjugation). Writing $$e^{-i x y}=e^{i|x-y|^{2}/2}e^{-i|x|^{2}/2}e^{-i|y|^{2}/2}$$ the operator $S$ can be written $$Sf(x)=e^{-i|x|^{2}/2}\int \frac{e^{i|x-y|^{2}/2}}{|x-y|^{a}} e^{-i|y|^{2}/2}f(y)dy.$$ Thus $T:L^{q}\to L^{p}$ is bounded iff convolution with the oscillating kernel $|x|^{-a}e^{i|x|^{2}/2}$ is bounded. This kind of kernel has been studied extensively, for instance in: P.Sjolin, Convolution with Oscillating Kernels, Indiana University Mathematics Journal Vol. 30, No. 1 (January–February, 1981), pp. 47-55. The range of values of $p,q$ is larger than (2), e.g. Sjolin proves boundedness $L^{p}\to L^{p}$ for $p\in[p_{0},p_{0}']$ where $p_{0}=\frac{2n}{n+a}$, provided $0\le a<n$. Thus the answer to your questions seems to be: no, condition (2) is not necessary for boundendess of $T$.