Oscillatory Integrals and the Kakeya Problem

We consider a variation of $T_{\alpha }$ using an oscillating kernel which is sometimes positive and sometimes negative. ${\tilde{T}}_{\alpha }f:=f\ast {\tilde{K}}_{\alpha },$ where ${\tilde{K}}_{\alpha }(x):=[1+|x|{]}^{-\alpha }\cos |x|.$ Let $T$ be a cylinder of length $L\gg 1$ and radius $(1/1000)L^{1/2}$. The cylinder may point in any direction. Let $v_T$ be a unit vector parallel to the axis of the cylinder. Let $f_T$ be the function $f_T(x):={\chi }_T(x)e^{i(v_T\cdot x)}.$ Let $T^{+}$ denote the cylinder we get by translating $T$ by $10L{v}_T$. We are going to study the behavior of ${\tilde{T}}_{\alpha }{f}_T$ on $T^{+}$.

Proposition

Fix a dimension $n\ge 2$. For all $L$ sufficiently large, the following holds. If $f_T$ and $T^{+}$ are defined as above, then for every $x\in T^{+}$ we have $|{\tilde{T}}_{\alpha }{f}_T(x)|\gtrsim L^{\frac{n+1}{2}-\alpha }.$

Corollary

If $\alpha <\frac{n+1}{2}$, then for every $p\in [1,\infty ]$, as $L\to \infty$, $\frac{\Vert {\tilde{T}}_{\alpha }{f}_T{\Vert }_p}{\Vert f_T{\Vert }_p}\to \infty .$ So the operator ${\tilde{T}}_{\alpha }$ does not obey any $L^p$ estimate of the form $\Vert {\tilde{T}}_{\alpha }f{\Vert }_p\lesssim \Vert f{\Vert }_p$.