As always, the write-up is here, while the motivation is below.
We consider the Bessel definition of the fractional Sobolev spaces, given for \(s \in \mathbb{R}\) and \(1 < p < \infty\) by $$W^{s, p}(\mathbb{R}^d) = \{f \in \mathcal{S}'(\mathbb{R}^d) : \langle \nabla \rangle^s f \in L^p(\mathbb{R}^d)\}.$$ (Here, \(\langle \nabla \rangle^s : \mathcal{S}(\mathbb{R}^d) \to \mathcal{S}(\mathbb{R}^d)\) is extended by duality, from a Schwartz space isomorphism to an isomorphism on the tempered distributions. Membership in \(L^p(\mathbb{R}^d)\) of an arbitrary tempered distribution can, of course, be defined by demanding that \(f \mapsto \langle T, f \rangle\) is an \(L^{p’}(\mathbb{R}^d)\)-bounded form on \(\mathcal{S}(\mathbb{R}^d)\).)
It turns out that these form the “right” spaces for complex interpolation, in the following sense:
Theorem: Let \(s_i, \tilde{s}_i \in \mathbb{R}\) and \(1 < p_i, \tilde{p}_i < \infty\).
Suppose \(T\) is a linear operator acting on the sum-space \(W^{s_0, p_0}(\mathbb{R}^d) + W^{s_1, p_1}(\mathbb{R}^d)\), such that \(T\) restricts to continuous mappings $$T : W^{s_0, p_0}(\mathbb{R}^d) \to W^{\tilde{s}_0, \tilde{p}_0}(\mathbb{R}^d),$$ $$T : W^{s_1, p_1}(\mathbb{R}^d) \to W^{\tilde{s}_1, \tilde{p}_1}(\mathbb{R}^d)$$ with operator norms \(C_0 \geq 0\) and \(C_1 \geq 0\), respectively.
Then for \(\theta \in (0, 1)\), and \(q_{\theta}\) defined in general for \(1 \leq q_0, q_1 \leq \infty\) by $$\frac{1}{q_{\theta}} = \frac{1 \ – \, \theta}{q_0} + \frac{\theta}{q_1},$$ and likewise \(s_{\theta} = (1 \ – \, \theta) s_0 + \theta s_1\), we have that for every \(f \in \mathcal{S}(\mathbb{R}^d)\), \(T f \in W^{s_{\theta}, p_{\theta}}\), and we have the bound $$\|T f\|_{W^{\tilde{s}_{\theta}, \tilde{p}_{\theta}}(\mathbb{R}^d)} \lesssim_{d, \vec{s}, \vec{p}} C_0^{1 \ – \, \theta} C_1^{\theta} \|f\|_{W^{s_{\theta}, p_{\theta}}(\mathbb{R}^d)}.$$
Up to the constant (which I’m not sure how to remove; it is an unavoidable artifact whenever Mikhlin is applied), I would argue that this is essentially the most general complex interpolation-type result one could have for Sobolev spaces, analogous to the sense in which Riesz-Thorin is the best all-purpose complex interpolation result in \(L^p\) spaces.
There is a somewhat-serious technical issue that prevents the straightforward implementation of the standard Stein (or Riesz-Thorin) style complex interpolation method. Ideally, we could choose analytically-varying operators like $$T_z = \langle \nabla \rangle^{a(z)} T \langle \nabla \rangle^{- b(z)},$$ with \(a(z)\) and \(b(z)\) being the usual affine functions that match with the endpoints \(s_i\), \(\tilde{s}_i\), to cancel with the differentiations. Then one would take simple functions with holomorphically-varying coefficients, \(f_z, g_z\), and use a dual pairing, like how we proceed with the standard proof of the Riesz-Thorin theorem.
The issue with this is that as \(z\) varies, the inner smoothing operator (\(\langle \nabla \rangle^{- b(z)}\)) may grow weaker, so that the output \(\langle \nabla \rangle^{- b(z_0)} f_{z_0}\) might not be of sufficient regularity to be taken-in by \(T\), and then undergo a differentiation of order \(a(z_0)\). After all, \(f_{z_0}\) is a simple function, not Schwartz. (It is very difficult to get around this, because for Riesz-Thorin, one seeks objects with \(\|f_{0 + i t}\|_{L^{p_0}(\mathbb{R}^d)} = \|f\|_{L^{p_{\theta}}(\mathbb{R}^d)}^{p_{\theta} / p}\), and other “boundary conditions” like this for \(f_z\) and \(g_z\) on \(\Re(z) = 0, 1\); it is easy to construct a holomorphically-varying family obeying such conditions if \(f\) is finitely-valued (just take the complex powers, aligning them at \(z = \theta\)), but I certainly could not manage this for Schwartz functions \(f\) and \(g\).)
After a bit of thought, it becomes clear that any quick fix like adding to the smoothing (increasing the real part of the antidifferentiation parameter \(b(z)\)) does not resolve this issue in the general case; roughly speaking, the additional gain in the smoothing to make everything well-defined would also show-up in the degree of the RHS \(\|f\|_{W^{s, p}(\mathbb{R}^d)}\), and spoil things so that \(s\) would not equal \(s_{\theta}\) any more (it would be a higher value, which would create lots of problems).
For this reason, this result has been stated in other contexts, but not fully; for instance, Tao’s 247A notes give the extra hypothesis that \(T\) maps \(\mathcal{S}(\mathbb{R}^d)\) into itself (and even then, I’m not quite sure this is sufficient to solve the issue). By contrast, our argument does not assume a priori that the operator maps into \(W^{\tilde{s}_{\theta}, \tilde{p}_{\theta}}(\mathbb{R}^d)\) at all, but instead proves this as a consequence of the bounds we obtain.
The approach I had to use was to force everything, at all stages of the interpolation, to be Schwartz: for all the distributional-pairings and operator evaluations to be unambiguously and perfectly well-defined. Specifically, I chose \(f_z\) and \(g_z\) to be analytic families of simple functions, as in Riesz-Thorin; but I also required that they vanish outside a bounded set (not just a finite-measure) one, and in the bilinear form involving \(T\) in the interpolation, I added (compactly-supported) mollifiers to act on each argument. This would mean that each argument, suitably-processed by all this machinery, would turn out to be a \(C^{\infty}_c\) function, and thus very well-behaved.
(Some careful arguing, which was tedious and involved a lot of inductive-type manipulations for the “shape” of derivatives of symbols, but not especially difficult, was first given to show that the dual pairing was a holomorphically-varying function in a neighborhood of the strip.)
From here, Young’s inequality (in the basic \((p, 1)\) averaging case) shows that these convolutions are contractions, so that on the boundary-lines of the strip, after reducing and cancelling everything out (the derivatives in the Sobolev norms, and the smoothing operators), one would get an \(L^p\) norm of a convolution, which could be estimated by the underlying function.
After obtaining the interpolation bounds, a few soft arguments then gave that we could pass from \(f, g\) simple to Schwartz functions; after this, we let the mollification parameters tend to zero, to recover \(f\) and \(g\) exactly. From the estimates on the duality pairing, we were able to deduce that \(T f \in W^{\tilde{s}_{\theta}, \tilde{p}_{\theta}}(\mathbb{R}^d)\), and obtain information on the size of its norm.
Obviously, to do this, we required uniform estimates in the mollification parameter \(\epsilon > 0\); it was very important that it not show up in the constants. While the mollifiers ensure that everything is qualitatively-well-defined, when verifying the conditions for the general three-lines lemma, it does appear in the constant in-front; differentiating a rough function convolved with an approximate identity produces negative powers of \(\epsilon\), which would be very bad if they appeared anywhere.
For this reason, we also had to obtain an interpolation principle for the strip that is independent of the constants in the a priori growth bound on the holomorphic function; for this reason, the result outlined in Tao’s notes (Exercise 8, here) was not suitable for our purposes, as the stated constant really depends on the size condition. (I discuss an alternate approach that removes this dependency in a recent comment to that page, and may outline the argument in a post later. In the write-up, I use a simpler result, based on the equality of the degrees of the polynomial terms here from Mikhlin: i.e., both \(d + 2\).)
Furthermore, to make the soft arguments work (recovering Schwartz functions from the simple functions), I had to prove an interesting and simple Sobolev-type inequality:
Corollary: Let \(s \in \mathbb{R}\) actually be a natural number, \(s = k \in \mathbb{N}\). Then for any \(f \in W^{s, p}(\mathbb{R}^d)\), we have the equivalence of norms $$\|f\|_{W^{s, p}(\mathbb{R}^d)} \simeq_{k, p, d} \|f\|_{L^p(\mathbb{R}^d)} + \sum_{i = 1}^{d} \|\partial_i^k f\|_{L^p(\mathbb{R}^d)} \simeq_{k, p, d} \sum_{|\alpha| \leq k} \|\partial^{\alpha} f\|_{L^p(\mathbb{R}^d)}.$$
This shows that the classical Sobolev spaces are indeed special cases of these Bessel potential spaces, as we would hope if we want any amount of consistency in the theory. And this showcases the general principle that the most extreme terms (here, the zeroth-order term and the highest derivatives) control the intermediate ones.
But what makes this interesting to me is that we are not considering all \(k\)th order derivatives, even, but a very special subset: just the pure directional partial derivatives, with no mixed terms (with \(\partial_i\) and \(\partial_j\) for \(i \neq j\) in \(\{1, \dots, d\}\)); just as the most extreme orders control the intermediate ones, so too does it seem that the most extreme “advances” (“movements”?) in the each direction suffice to control all other combinations of derivatives.
This was proven via Littlewood-Paley theory, invoking the dyadic characterization of Sobolev spaces. The details, of course, are in the PDF. I find this result interesting, and I wonder if there is any physical-space proof of this inequality.
This easily gives an interpolation-type inequality for the intermediate derivatives, by scaling.
Work begun: September 4, 2025. Finished: September 5, 2025.
Revised: October 7, 2025 (to fix some notational mistakes and confusion in the original).
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