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.


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.


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