I am trying to prove Q13 from Terence Tao's Fourier Analysis notes (number 4):

For every $t>0$, let $\psi_{t}:\mathbb{R}^{d}\rightarrow\mathbb{C}$ be a function obeying the estimates

$$|\nabla^{j}\psi_{t}(\xi)|\lesssim_{d} t^{-j}\min((|\xi|/t)^{-\epsilon},(|\xi|/t)^{\epsilon})$$ for all $0\leq j\leq d+2$, all $\xi\in\mathbb{R}^{d}$, and some $\epsilon>0$ (independent of $t$). Conclude that $$\|\left(\int_{0}^{\infty}|\psi_{t}(D)f|^{2}\dfrac{dt}{t}\right)^{1/2}\|_{L^{p}(\mathbb{R}^{d})}\lesssim_{p,d}\|f\|_{L^{p}(\mathbb{R}^{d})}$$ for $f\in L^{p}(\mathbb{R}^{d})$.

Presumably, $1<p<\infty$, and $(\psi_{t}(D)f)(x):=(\widehat{\psi_{t}}\widehat{f})^{\vee}(x)$.

My idea for proving the inequality is as follows. Consider the linear operator $T$ initially defined on the space of complex-valued Schwartz functions by $$T(f)(x)\in L^{2}\left(\mathbb{R}_{+},\dfrac{dt}{t}\right),\quad (Tf(x))(t)=(\psi_{t}(D)f)(x)$$ It is not hard to see from the hypotheses on $\psi_{t}$ that $T$ is bounded from $L^{2}(\mathbb{R}^{n},\mathbb{C})$ to $L^{2}(\mathbb{R}^{n},L^{2}(\mathbb{R}_{+},dt/t))$. If we can show that $T$ is associated to a kernel $K$ such that $K(x)$ is a bounded linear operator $\mathbb{C}\rightarrow L^{2}(\mathbb{R}_{+},\dfrac{dt}{t})$ which satisfies the standard size and smoothness estimates, then we can apply the vector-valued Calderon-Zygmund theorem to conclude that $T$ is $L^{p}\rightarrow L^{p}$ bounded.

My difficulty is in showing the estimates for the kernel. A priori, $\psi_{t}$ isn't compactly supported or even has sufficient decay for me to directly use Fourier inversion. So it seems like I need to use a Littlewood-Paley partition of unity applied to each $\psi_{t}$ in order to make sense of the kernel. This seems messy, and I'm wondering if there is a clearer argument.


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