Nirenberg's paper [*On elliptic PDEs*](http://www.numdam.org/item/?id=ASNSP_1959_3_13_2_115_0) claims that if a function $f$ on $\mathbb{R}^n$ tends to zero at infinity or is in $L^q$ for any $q < \infty$ then the "interpolation" inequality
$$
\lVert∇ f \rVert\_{L^{2p}} ≤ C \left(\lVert f\rVert\_{L^\infty} \lVert ∇^2 f \rVert\_{L^p}\right)^{1/2} 
$$
holds (theorem on p. 125).
This is proved by establishing the inequality
$$
 \lVert∇ f \rVert\_{L^{q}} ≤ C \left(\lVert f\rVert\_{L^r} \lVert ∇^2 f \rVert\_{L^p}\right)^{1/2} \label{1}\tag{1}
$$
(equation (2.5) in the proof of that theorem)
for any $1 < r,p < \infty$ and $\frac{2}{q} = \frac{1}{r} + \frac{1}{p}$ with a constant $C$ **independent of $r, p$**, and the taking the limit $r \to \infty$.
I haven't really looked at Nirenberg's proof, but as far as I can see it is "calculus-based".


On the other hand inequality \eqref{1} looks like the interpolation inequality
$$
\lVert f \rVert\_{W^{1,q}} ≤ C\_{p, r} \left(\lVert f\rVert\_{L^r} \lVert f \rVert\_{W^{p, 2}}\right)^{1/2} \label{2}\tag{2}
$$
for $1 < r, p < \infty$ and $q$ as above,
that you get by using harmonic analysis (the Mihlin multiplier theorem and the Littlewood--Paley characterization of Sobolev spaces, to be precise) to establish that $W^{1,q}$ is the $\frac{1}{2}$-interpolation space between $W^{2,p}$ and $W^{0,r} = L^r$ (e.g. Bergh--Löfström, *An introduction to interpolation spaces*, Thm 6.4.5).


Is there some way to get \eqref{1} using the same methods as for \eqref{2} (maybe only with $\lVert f \rVert\_{W^{p,2}}$ on the RHS instead of $\lVert ∇^2 f \rVert\_{L^p}$), but with the constant independent of $p, r$, so that you can conclude the case $r = \infty$?


Or even better, is there some way to modify the interpolation arguments to give the $r = \infty$ case directly?



Is it possible to obtain the inequality $\|\nabla f\|_{L^{2p}} \leq C (\|f\|_{L^\infty} \|f\|_{W^{2, p}})^{1/2}$ from interpolation/harmonic analysis?
Here is a Littlewood-Paley + interpolation style proof. If $P\_k$ denotes a Littlewood-Paley projection to frequencies $|\xi| \sim 2^k$, then the standard Littlewood-Paley characterisations of Sobolev spaces give
$$ \| \nabla f \|\_{L^{2p}} \sim \| (\sum\_k 2^{2k} |P\_k f|^2)^{1/2} \|\_{L^{2p}}$$
and
$$ \| \nabla^2 f \|\_{L^{p}} \sim \| (\sum\_k 2^{4k} |P\_k f|^2)^{1/2} \|\_{L^{p}}$$
while
$$ \| \sup\_k |P\_k f| \|\_{L^\infty} \lesssim \|f\|\_{L^\infty}$$
so by Holder's inequality it suffices to establish the pointwise bound
$$ (\sum\_k 2^{2k} |P\_k f(x)|^2)^{1/2}
\lesssim A(x)^{1/2} B(x)^{1/2}$$
where
$$ A(x) := (\sum\_k 2^{4k} |P\_k f(x)|^2)^{1/2}$$
and
$$ B(x) := \sup\_k |P\_k f(x)|.$$
In fact we have the stronger estimate
$$ 
{\sum\_k 2^{k} |P\_k f(x)| \lesssim A(x)^{1/2} B(x)^{1/2}} \label{3} \tag{$\ast$}
$$
since from the triangle inequality we have
$$ \sum\_{k \leq k\_0} 2^{k} |P\_k f(x)| \lesssim 2^{k\_0} B(x)$$
and from Cauchy-Schwarz we have
$$ \sum\_{k > k\_0} 2^{k} |P\_k f(x)| \lesssim 2^{-k\_0} A(x)$$
and the claim follows by optimising in $k\_0$. (The estimate \eqref{3} also likely follows from some interpolation lemma on weighted $\ell^p$ spaces in Bergh-Lofstrom, but I don't have the reference handy.)


This argument in fact shows that we can replace the $W^{1,p}$ norm by the slightly stronger Triebel-Lizorkin norm $F^{1,p}\_1$ if desired.


