Nirenberg's paper On elliptic PDEs 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?
Best Answer
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.