Application of Factorization Theory to Oscillatory Integral Estimates

Question

In the article "Some New Estimates on Oscillatory Integrals" by Bourgain in the book Essays in Honor of Elias M. Stein, Bourgain considers operators of the form $$S_{N}g(x):=\int_{\mathbb{R}^{n}}g(y)e^{iN|x-y|}b(x-y)dy, \quad x\in\mathbb{R}^{n}$$ where $g\in\mathcal{S}(\mathbb{R}^{n})$, $N>0$, and $b$ is a standard bump function adapted to a neighborhood of the origin in $\mathbb{R}^{n}$. The goal is to prove an estimate of the form $$\|S_{N}g\|_{L^{p}}\lesssim_{\epsilon}N^{-\frac{n}{p}+\epsilon}\|g\|_{L^{p}}$$ for all $0<\epsilon\ll 1$, where $\frac{n}{p'}<\lambda+\frac{n+1}{2}$, where $\lambda>0$ is fixed. The context for this operator is proving $L^{p}$ bounds for the Bochner-Riesz multiplier.

Assume $p>2$. Bourgain claims that "by general factorization theory and the rotational invariance, it suffices to get the $L^{\infty}\rightarrow L^{p}$ estimate $$\|S_{N}f\|_{L^{p}}\lesssim_{n,p} N^{-n/p}\|f\|_{L^{\infty}}$$

Since he refers earlier to the article, in the context of Fourier restriction/extension estimates, to Nikishin-Maurey theory, I am assuming he means something along the lines of the following theorem of Nikishin, which I have taken by Garcia-Cuerva, Rubio De Francia's Weighted Inequalities

Theorem (Nikishin) Let $(Y,\mu)$ be an arbitrary measure space, $(X,m)$ be a $\sigma$-finite measure space, and let $T: L^{p}(\mu)\rightarrow L^{0}(m)$ be a continuous sublinear operator, with $0<p<\infty$. Then there exists a weight $w(x)>0$ a.e. such that $T: L^{p}(\mu)\rightarrow L^{q,\infty}(wdm)$, where $q=p\wedge 2$; i.e. $$\int_{\{|Tf|>\lambda\}}w(x)dm(x)\leq (\|f\|_{L^{p}}/\lambda)^{q}, \quad f\in L^{p}(\mu); \lambda >0$$

One can take the weight $w$ to be bounded above.

If we apply Nikishin's theorem to the adjoint operator $S_{N}^{*}$, we obtain that there is such a weight $w$ such that $S_{N}^{*}: L^{p'}(\mathbb{R}^{n})\rightarrow L^{p',\infty}(\mathbb{R}^{n}, w(x)dx)$, for $p>2$. By translation invariance, we can also take $w$ to be continuous and everywhere positive. By rotation invariance and averaging over the orthogonal group, we can take $w$ to be radial.

How does one eliminate the weight completely to obtain that $S_{N}^{*}: L^{p'}(\mathbb{R}^{n})\rightarrow L^{p',\infty}(\mathbb{R}^{n})$, which I presume is the goal (it's quite possible I am completely offbase here)? If my understanding is correct, in setting where the operator maps to functions on a homogeneous space, such as the sphere, one can use the associated group action to average out the weight; however, this doesn't seem to work in the current setting. I was unable to find a reference where this argument was sketched. Other papers seem to refer to the aforementioned Bourgain paper.

Answer 1

So here's a stab at how to obtain the desired estimate; I still don't know if it's what Bourgain intended. I would appreciate if someone could comment if my understanding of a result from the theory of absolutely summing operators is correct.

Observe that the operator $S_{N}$ is local in the sense that if $\text{supp}(f)\subset B_{r}(c)$, then $\text{supp}(S_{N}f)\subset B_{Cr}(c)$, where $C>0$ is some constant which only depends on the dimension $n$ and the function $b$. So since $S_{N}$ is translation invariant, it suffices to show that $$\|S_{N}f\|_{L^{p}}\lesssim_{n,p,\epsilon}N^{-\frac{n}{p}+\epsilon}\|f\|_{L^{p}}, \quad \text{supp}(f)\subset Q:=[0,1]^{n}$$

Fix $q>p>2$ (in particular, $q$ can be outside the admissible range for Bochner-Riesz). Since $L^{p'}(Q)$ has type $p'$ and $\|S_{N}\|_{(\infty,p)}\lesssim N^{-n/p}$, the theory of absolutely summing operators (See [Pisier 86]) tells us that there exists a probability measure $\mu$ on $Q$, such that $$\|S_{N}f\|_{L^{p}(dx)}\lesssim_{n,p,q}N^{-n/p}\|f\|_{L^{q}(\mu)}$$ By translation invariance, we have that for any $h\in\mathbb{R}^{n}$, $$\|S_{N}f\|_{L^{p}(dx)}\lesssim_{n,p,q}N^{-n/p}\|f(\cdot-h)\|_{L^{q}(\mu)}$$ Now integrating both sides of the inequality over the cube $2\sqrt{n}Q$ with respect to $dh$ and using Fubini together with the fact that $\mu$ is a probability measure, we obtain $$C_{n}\|S_{N}f\|_{L^{p}(dx)}^{q}\lesssim_{n,p,q} N^{-n/p}\|f\|_{L^{q}(dx)}^{q}$$ To obtain the desired $(p,p)$ estimate, we use complex interpolation with the trivial $(2,2)$ estimate. Specifically, writing $\frac{1}{p}=\frac{\theta}{q}+\frac{1-\theta}{2}$, we obtain $$\|S_{N}f\|_{L^{p}}\lesssim_{n,p,q}N^{-\theta\frac{n}{p}}\|f\|_{L^{p}}$$ from which the desired conclusion follows.