1. Introduction
In this paper, we will be differentiating inequalities, so we start by summarizing the general reasoning scheme. Let us suppose that we are given a family of inequalities
\begin{equation*}
F_t \leq k_t G_t
\end{equation*} parametrized by t > 0. We choose to keep separate the families 
$F_t, G_t$ from the family kt, since in later applications 
$F_t, G_t$ will be given by some functionals computed on a fixed function f, while kt are the (optimal) constants in the inequalities. Suppose further that
• the functions
$F_t,G_t,k_t$ are actually defined and differentiable for
$t\geq0$;•
$F_0=k_0G_0$.
Then we can write
\begin{equation*}
\frac{F_t-F_0}{t} \leq \frac{k_tG_t-k_0G_0}{t}
\end{equation*} and passing to the limit for 
$t\to0^+$ we get
\begin{equation}
\partial_t F_t |_{t=0} \leq G_0 \partial_t k_t |_{t=0}+k_0 \partial_t G_t |_{t=0} .
\end{equation} These computations will be carried out, in each case, for functions in a suitable class which allows differentiation under the integral sign; in our case, it is convenient to consider the Schwartz class 
$\mathcal{S}(\mathbb{R}^d)$, which is made of regular and decaying functions and is well-behaved with respect to the Fourier transform.
For a first example, we start with the Hardy–Littlewood–Sobolev inequality
\begin{equation*}
\left| \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{f(x)g(y)}{|x-y|^\lambda} \,\mathrm{d} x \,\mathrm{d} y \right| \lesssim ||f||_p ||g||_q,
\end{equation*} which, in the case 
$f=g\geq0$ (p = q and 
$0 \lt \lambda \lt d$), becomes
\begin{equation*}
\int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{f(x)f(y)}{|x-y|^\lambda} \,\mathrm{d} x \,\mathrm{d} y \leq k_{\lambda} ||f||^2_{\frac{2d}{2d-\lambda}},
\end{equation*}where
\begin{equation*}
k_{\lambda}=\pi^{\frac{\lambda}{2}} \frac{\Gamma(\frac{d}{2}-\frac{\lambda}{2})}{\Gamma(d-\frac{\lambda}{2})}\left(\frac{\Gamma(\frac{d}{2})} {\Gamma(d)}\right)^{-1+\frac{\lambda}{d}}
\end{equation*}and the inequality becomes an equality when λ = 0. By observing that
\begin{equation*}
\frac{\mathrm{d}}{\mathrm{d} \lambda} |x-y|^{-\lambda} |_{\lambda=0} = -\log|x-y|
\end{equation*}and following the above reasoning scheme, we get the logarithmic Hardy–Littlewood–Sobolev inequality (cf. [Reference Carlen and Loss8]):
\begin{equation*}
\int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \! -\log|x-y| \ f(x) f(y) \,\mathrm{d} x \,\mathrm{d} y \leq \frac{1}{d} \int_{\mathbb{R}^d} f \log f \,\mathrm{d} x + C_0,
\end{equation*} for regular 
$f\geq0$ with 
$||f||_1=1$, where 
$C_0 = \partial_{\lambda} k_{\lambda} |_{\lambda=0}$.
Also, a quantity similar to the LHS of the previous inequality can be estimated differentiating the sharp Young inequality: as in [Reference Lieb and Loss13] (ch. 8, p. 223), we find, for any a > 0, the logarithmic Sobolev inequality in 
$\mathbb{R}^d$
\begin{equation*}
\int_{\mathbb{R}^d} |f|^2 \log |f| \,\mathrm{d} x \leq \frac{a^2}{\pi} \int_{\mathbb{R}^d} |\nabla f|^2 \,\mathrm{d} x -d(1+\log a),
\end{equation*} which holds for every 
$f\in H^1(\mathbb{R}^d)$ with 
$\left \Vert f \right \Vert_2=1$.
Concerning the Fourier transform, a classical inequality is the
Hausdorff–Young inequality for 
$1\leq p \leq 2$
\begin{equation*}
||\widehat{f}||_{p'} \lesssim ||f||_p,
\end{equation*} from which Beckner [Reference Beckner2] derived a logarithmic uncertainty principle for 
$f\in \mathcal{S}(\mathbb{R}^d)$ with 
$\left \Vert f \right \Vert_2=1$
\begin{equation*}
\int_{\mathbb{R}^d} |f|^2 \log |f| \,\mathrm{d} x + \int_{\mathbb{R}^d} |\widehat{f}|^2 \log |\widehat{f}| \,\mathrm{d} \xi
\leq \frac{d}{2} (\log(2)-1) \lt 0.
\end{equation*} So far, we have examples in which we get inequalities involving logarithmic terms, in the form of 
$\log(f)$ or the kernel 
$\log|x-y|$. Now we recall three families of inequalities which give also logarithmic weights in physical 
$(\log|x|)$ and Fourier 
$(\log|\xi|)$ domains. These are Sobolev, Hardy and Stein–Weiss inequalities, the latter being a generalization of the first two ‘endpoint’ inequalities. The results of the differentiation are summarized as follows: the Sobolev and Hardy inequalities give, respectively, (
$\left \Vert f \right \Vert_2=1$) [Reference Beckner3]
\begin{align*}
||f||_{\frac{2d}{d-2s}}&\lesssim ||f||_{\dot{H}^s} \ \longrightarrow \ \frac{2}{d}\int_{\mathbb{R}^d} |f|^2 \log |f| \,\mathrm{d} x
\leq \int_{\mathbb{R}^d} |\hat{f}|^2\log |\xi| \,\mathrm{d} \xi + c \\
\left \Vert |\cdot|^{-s} f \right \Vert_2 &\lesssim ||f||_{\dot{H}^s} \ \longrightarrow
\quad -\int_{\mathbb{R}^d} |f|^2 \log|x| \,\mathrm{d} x \leq \int_{\mathbb{R}^d} |\hat{f}|^2 \log |\xi| \,\mathrm{d} \xi + c
\end{align*}and, by interpolating, the Stein–Weiss inequality
\begin{equation*}
\left \Vert |\cdot|^{-\beta} f \right \Vert_p \lesssim ||f||_{\dot{H}^s}
\end{equation*}yields (cf. Remark 4)
\begin{equation}
\begin{split}
\frac{2(1-t)}{d}\int_{\mathbb{R}^d}& |f|^2 \log |f| \,\mathrm{d} x - t \int_{\mathbb{R}^d} |f|^2 \log|x| \,\mathrm{d} x \leq \\
&\leq \int_{\mathbb{R}^d} |\widehat{f}|^2 \log |\xi| \,\mathrm{d} \xi + c
\end{split}
\end{equation} for all 
$t\in[0,1]$. In the above inequalities, c is a constant that differs from line to line.
The main purpose of the note is to prove an improvement of the logarithmic Stein–Weiss inequality (Equation 2) when restricted to radially symmetric functions. We denote by 
$\dot{H}_\mathrm{rad}^s(\mathbb{R}^d)$ the space of radial functions which belong to 
$\dot{H}^s(\mathbb{R}^d)$. The starting point is an improvement of Stein–Weiss inequality for radial functions, which is due to Rubin.
Theorem 1 Rubin inequality, [Reference Rubin14], [Reference Bellazzini, Frank and Visciglia4], [Reference Bellazzini, Ghimenti, Mercuri, Moroz and Van Schaftingen5]
Given 
$f\in \dot{H}_\mathrm{rad}^s(\mathbb{R}^d)$ with 
$ 0\leq s \lt d/2$ and 
$p \lt +\infty,\beta$ with the relations 
$(1/0 := +\infty)$
\begin{gather*}
2 \leq p \leq \frac{2}{(1-2s)_+} \\
\beta = s+\frac{d}{p}-\frac{d}{2} ,
\end{gather*}one has the inequality
\begin{equation*}
\left \Vert \ |\cdot|^{-\beta} f \right \Vert_p \leq C(p,s) ||f||_{\dot{H}^s} .
\end{equation*}Remark 2. The latter theorem is an improvement of Stein–Weiss inequality on radial functions, meaning that the range of parameters p extends further on the right and for big values of p even positive exponents 
$-\beta$ are allowed.
We shall underline that the Stein–Weiss inequality holds in a more general form involving the space 
$\dot{W}^{s,p}(\mathbb{R}^d)$. If one restricts (as we do) just to 
$\dot H^s(\mathbb{R}^d)$ spaces, the Plancherel identity allows to interpret the above inequality, without assumptions on the radiality of the function and in the restricted range β > 0, as a special case of Pitt Inequality (see [Reference Beckner17, Appendix 1]). Moreover, due to the special form that the Fourier transform assumes on radial functions, the previous theorem, due to Rubin in the whole range of values for the parameters, follows also from the result in [Reference De Carli10] (see also [Reference Gorbachev, Liflyand and Tikhonov12, Corollary 7.1]).
We point out that in Reference RubinTheorem 1 the optimal constants 
$C(p,s)$ are not known, but via an interpolation argument (Lemma 11) we get an estimate which is sufficient to carry out the differentiation, so as to get:
Theorem 3 There exists a constant 
$c\in\mathbb{R}$ such that, given 
$f\in \mathcal{S}(\mathbb{R}^d)$ radial and with 
$\left \Vert f \right \Vert_2=1$
\begin{equation}
\begin{split}
2\int_{\mathbb{R}^d} |f|^2 &\log |f| \,\mathrm{d} x + (d-1) \int_{\mathbb{R}^d} |f|^2 \log|x| \,\mathrm{d} x \leq \\
&\leq \int_{\mathbb{R}^d} |\widehat{f}|^2 \log |\xi| \,\mathrm{d} \xi + c.
\end{split}
\end{equation}Remark 4. The argument in the proof holds, in particular, in the restricted range of parameters 
$2\leq p \leq 2d/(d-2s)$ or equivalently 
$0\leq\beta\leq s$. For this range of exponent, the Rubin inequality holds even for non-radial functions (thanks to Stein–Weiss inequality) and the differentiation gives, for any 
$0\leq t \leq 1$, again
\begin{equation*}
\frac{2(1-t)}{d}\int_{\mathbb{R}^d} |f|^2 \log |f| \,\mathrm{d} x \ - \ t \int_{\mathbb{R}^d} |f|^2 \log|x| \,\mathrm{d} x
\leq \int_{\mathbb{R}^d} |\widehat{f}|^2 \log |\xi| \,\mathrm{d} \xi + c .
\end{equation*}Notice that the relevant difference w.r.t. the radial case is only the sign of the constant in front of the second term.
Remark 5. Stein–Weiss inequality can be improved in the radial case (Rubin inequality), while Sobolev and Hardy cannot.
We underline the fact that in the previous uncertainty inequality the only term containing 
$\widehat{f}$ is 
$\int_{\mathbb{R}^d} |\widehat{f}|^2 \log |\xi| \, \mathrm{d} \xi$ while the others are in physical variables. As in the case of Gagliardo seminorms 
$[f]_s$, which are related to Sobolev norms 
$||f||_{\dot{H}^s}$ (see [Reference Di Nezza, Palatucci and Valdinoci11], [Reference Brezis, Van Schaftingen and Yung6] and [Reference Crismale, De Luca, Kubin, Ninno and Ponsiglione9]), a natural question is whether 
$\int_{\mathbb{R}^d} |\widehat{f}|^2 \log \xi \, \mathrm{d}\xi$ can also be expressed in physical variables. This is the content of our last result, that we believe can be of independent interest. For a similar result involving derivatives of logarithmic order, see [Reference Brué and Nguyen7, Theorem 1.4].
Theorem 6 For any 
$f \in \mathcal{S}(\mathbb{R}^d)$
\begin{align*}
\int_{\mathbb{R}^d} |\widehat{f}(\xi)|^2 \log|\xi| \, \mathrm{d}\xi & = C_d ||f||_2^2 + \frac{1}{2|\mathbb{S}^{d-1}|} \iint\limits_{\{|x-y|\leq 1\}} \frac{(f(x)-f(y))^2}{|x-y|^d} \, \mathrm{d}x \, \mathrm{d}y \ \\
& -\ \frac{1}{|\mathbb{S}^{d-1}|} \iint\limits_{\{|x-y| \gt 1\}} \frac{f(x)f(y)}{|x-y|^d} \, \mathrm{d}x \, \mathrm{d}y,
\end{align*}with
\begin{equation*}
C_d= - 1-\frac{1}{|\mathbb{S}^{d-1}|} \int_{B_1} \frac{z_1-\sin(z_1)}{z_1 |z|^d} \, \mathrm{d}z \ + \frac{1}{|\mathbb{S}^{d-1}|} \ \int_{B_1^c} \frac{\sin(z_1)}{z_1 |z|^d} \, \mathrm{d}z.
\end{equation*}The proof of Theorem 6 is in the appendix at the end of the paper.
1.1. Notation
$L^p(\mathbb{R}^d)$, with 
$p\in[1,\infty],$ denote the usual Lebesgue spaces, 
$\dot H^s(\mathbb{R}^d)$ stands for the usual homogeneous Sobolev space, namely the space of tempered distribution u over 
$\mathbb{R}^d$, the Fourier transform of which belongs 
$L^1_\mathrm{loc}(\mathbb{R}^d)$ and satisfies
\begin{equation*}
||u||_{\dot H^s(\mathbb{R}^d)}^2=\int_{\mathbb{R}^d} |\xi|^{2s} |\widehat{u}|^2\,\mathrm{d}\xi \lt +\infty.
\end{equation*}In this paper, we adopt the following definition for the Fourier transform:
\begin{equation*}
\mathcal{F}[f](\xi) := \widehat{f}(\xi) := (2\pi)^{-d/2} \int_{\mathbb{R}^d} f(x) \ \mathrm{e}^{-\mathrm{i} \xi \cdot x} \,\mathrm{d} x,
\end{equation*}so that, defining
\begin{equation*}
\widehat{|D|f} := |\xi|\widehat{f},
\end{equation*}one has
\begin{equation*}
\left \Vert |D| f \right \Vert_2 = \left \Vert \nabla f \right \Vert_2.
\end{equation*}For references on the properties of homogeneous and non-homogeneous Sobolev spaces, we refer to [Reference Bahouri, Chemin and Danchin1].
2. Proof of main results
In order to prove the estimate on the growth of the constant of the Rubin inequality near s = 0, we need an interpolation result due to Stein, so we start with a preliminary definition.
Definition 7. Let 
$(M, \mathcal{M},\mu), (N, \mathcal{N},\nu)$ be two σ-finite measure spaces and 
$\{T_z\}_{z\in \overline{\Sigma}}$ a family of operators defined on the space of simple functions on M and taking values in the space of measurable functions on N. The family of operators is said to be admissible if, for every couple of simple functions 
$f,g$ defined on 
$M,N$, respectively, the function
\begin{equation*}
z\mapsto \langle \overline{T_z f}, g\rangle := \int_N T_z f \ g \,\mathrm{d} \nu
\end{equation*} is holomorphic in 
$\Sigma:= (0,1)\times i\mathbb{R}\subset \mathbb{C}$ and continuous up to the boundary and for some 
$a \lt \pi$
\begin{equation*}
\sup_{z\in \overline{\Sigma}} \ \mathrm{e}^{-a |y|} \log |\langle T_z f, g \rangle | \lt +\infty.
\end{equation*}Theorem 8 Stein interpolation, [Reference Stein15], [Reference Stein and Weiss16]
Given 
$p_0, p_1, q_0, q_1 \in [1,+\infty]$ and an admissible family of operators 
$\{T_z\}_{z\in\overline{\Sigma}}$ satisfying
\begin{equation*}
||T_{iy}||_{L^{p_0}\to L^{q_0}}\leq M_0 (y) , \quad ||T_{1+iy}||_{L^{p_1}\to L^{q_1}}\leq M_1 (y)
\end{equation*}and the growth condition
\begin{equation*}
\sup_{y\in\mathbb{R}} \ \mathrm{e}^{-a|y|} \log M_j(y) \lt +\infty \quad \text{for } j=0,1
\end{equation*} with 
$a \lt \pi$, it holds
\begin{equation*}
||T_t||_{L^{p_t} \to L^{q_t}} \leq M_t ,
\end{equation*} for 
$0 \lt t \lt 1$, where
\begin{align*}
M_t := \int_{\mathbb{R}} \frac{M_0(y)}{\cosh(\pi y) -\cos(\pi t)} + \frac{M_1(y)}{\cosh(\pi y) +\cos(\pi t)} \,{\mathrm{d}} y , \\
\frac{1}{p_t} := \frac{1-t}{p_0}+\frac{t}{p_1}, \\
\frac{1}{q_t} := \frac{1-t}{q_0}+\frac{t}{q_1}.
\end{align*} We are interested in the case where the functions 
$M_0, M_1$ can be chosen to be bounded; the previous theorem becomes:
Corollary 9. Stein interpolation – weak version
Given 
$p_0, p_1, q_0, q_1 \in [1,+\infty]$ and an admissible family of operators 
$\{T_z\}_{z\in\overline{\Sigma}}$ satisfying
\begin{equation*}
||T_{iy}||_{L^{p_0}\to L^{q_0}}\leq M_0 , \quad ||T_{1+iy}||_{L^{p_1}\to L^{q_1}}\leq M_1 ,
\end{equation*}it holds
\begin{equation*}
||T_t||_{L^{p_t} \to L^{q_t}} \leq M_t ,
\end{equation*} for 
$0 \lt t \lt 1$, where
\begin{equation*}
M_t := M_0^{1-t} M_1^t , \\
\frac{1}{p_t} := \frac{1-t}{p_0}+\frac{t}{p_1}, \\
\frac{1}{q_t} := \frac{1-t}{q_0}+\frac{t}{q_1}.
\end{equation*}Remark 10. In the following, we will need to interpolate some inequalities that hold only for functions belonging to a proper subspace of an Lp space, for example, only for radial functions, so we notice that if 
$\mathcal{G}$ is a family of simple functions on M such that the hypotheses of one of the above interpolation theorems hold for 
$f\in\mathcal{G}$, for instance
\begin{equation*}
||T_{j+iy} f||_{q_j} \leq M_j(y) ||f||_{p_j} \quad (j=0,1) \quad \forall f\in \mathcal{G},
\end{equation*}then the corresponding interpolation result states that
\begin{equation*}
||T_t f||_{q_t} \leq M_t ||f||_{p_t}
\end{equation*} for every f in the closure of 
$\mathcal{G}$ as a subset of 
$L^{p_t}(\mu)$.
Lemma 11. If 
$C(p,s)$ is the optimal constant in Rubin’s inequality and 
$0\leq s \leq s_1 \lt d/2$, then we have the estimate
\begin{equation*}
C(p_\theta,s) \leq C(p_1, s_1)^{s/s_1} ,
\end{equation*} where 
$\theta=s/s_1$ and 
$1/p_\theta =(1-\theta)/2+\theta/p_1$.
Proof. Setting 
$f = |D|^{-s} \varphi$, the Rubin inequality reads
\begin{equation*}
\left \Vert |\cdot|^{-\beta} \ |D|^{-s} \varphi\right \Vert_p \leq C(p,s) ||\varphi||_2,
\end{equation*}so we are led to consider the family of operators
\begin{equation*}
\widetilde{T}_{s,\beta} : \varphi \mapsto |\cdot|^{-\beta} \ |D|^{-s}\varphi .
\end{equation*} If 
$s_1, p_1$ are fixed, to apply the interpolation result we use 
$\theta=s/s_1$ as a parameter and we set
\begin{equation*}
\frac{1}{p_\theta} = \frac{1-\theta}{2} + \frac{\theta}{p_1}
\end{equation*}and, by the relation between the parameters in Rubin inequality, we have
\begin{equation*}
\beta_\theta =\theta s_1 +\frac{d}{p_\theta} -\frac{d}{2} = \theta \left( \frac{d}{p_1} -\frac{d}{2}+s_1 \right) = \theta \beta_1.
\end{equation*}Now we have a new family of operators that depend on a single parameter
\begin{equation*}
T_\theta : \varphi \mapsto |\cdot|^{-\beta_\theta} \ |D|^{-s_\theta} \varphi = |\cdot|^{-\beta_1 \theta} \ |D|^{-s_1 \theta} \varphi
\end{equation*} that can be defined on the whole strip 
$\overline{\Sigma}$. On the left line, 
$\{x=0\}$ we have
\begin{equation*}
||T_{iy} \varphi||_2 = \left \Vert |D|^{-s_1 iy} \varphi \right \Vert_2 = \left \Vert |\cdot|^{-s_1 iy} \widehat{\varphi} \right \Vert_2 = ||\varphi||_2
\end{equation*}so that
\begin{equation*}
||T_{iy}||_{L^2 \to L^2 } \leq 1 .
\end{equation*} On the left line 
$\{x=1\}$
\begin{align*}
||T_{1+iy} \varphi||_{p_1} &= \left \Vert |\cdot|^{-\beta_1 (1+iy) } |D|^{-s_1 (1+iy)} \varphi \right \Vert _{p_1} = \\
&= \left \Vert |\cdot|^{-\beta_1} |D|^{-s_1} \left( |D|^{-s_1 iy} \varphi \right) \right \Vert_{p_1}
\end{align*}and applying Rubin’s inequality
\begin{equation*}
||T_{1+iy} \varphi||_{p_1} \leq C(p_1,s_1) \left \Vert |D|^{-s_1 iy} \varphi \right \Vert_2 = C(p_1,s_1) ||\varphi||_2.
\end{equation*} Given 
$\varphi,\psi$ simple functions,
\begin{equation*}
h(z):=\langle \overline{T_z \varphi} ,\psi \rangle = \int_{\mathbb{R}^d} |x|^{-\beta_1 z} \ |D|^{-s_1 z} \varphi(x) \ \psi(x) \,\mathrm{d} x
\end{equation*} is a continuous function on 
$\overline{\Sigma}$ and holomorphic in Σ; indeed consider the two variable function
\begin{align*}
H(z_1,z_2) &:= \int_{\mathbb{R}^d} |x|^{-\beta_1 z_1} \ |D|^{-s_1 z_2} \varphi(x) \ \psi(x) \,\mathrm{d} x = \\
&= \int_{\mathbb{R}^d} \mathcal{F}\left[ |x|^{-\beta_1 z_1} \psi(x)\right](\xi) \ |\xi|^{-s_1z_2} \widehat{\varphi}(\xi) \,\mathrm{d} \xi.
\end{align*} The first expression for H shows that, fixing z 2, 
$\frac{\partial H(z_1,z_2)}{\partial \bar z_1}=0$, while the second form gives, z 1 being fixed, 
$\frac{\partial H(z_1,z_2)}{\partial \bar z_2}=0$. Hence H is holomorphic in 
$\Sigma\times \Sigma$ (Osgood’s Lemma) and
\begin{equation*}
z\mapsto H(z,z)=h(z)
\end{equation*}is a composition of holomorphic functions, so it is holomorphic itself. Moreover, the function h is bounded on the strip, since
\begin{align*}
|h(z)| &= |\langle |D|^{-s_1 \bar{z}} \overline{\varphi} , |\cdot|^{-\beta_1 \bar{z}} \psi \rangle |
= | \langle |\cdot|^{-s_1 \bar{z}} \widehat{\overline{\varphi}} , \mathcal{F} (|\cdot|^{-\beta_1 \bar{z}} \psi)) \rangle | \leq \\
&\leq| \langle |\cdot|^{-s_1 x} \widehat{\overline{\varphi}} , \mathcal{F} (|\cdot|^{-\beta_1 \bar{z}} \psi)) \rangle |
= |\langle |D|^{-s_1 x} \overline{\varphi} , |\cdot|^{-\beta_1\bar{z}} \psi \rangle | \leq \\
&\leq |\langle |D|^{-s_1 x} \overline{\varphi} , |\cdot|^{-\beta_1 x} \psi \rangle | = |\langle \overline{T_x \varphi} ,\psi \rangle=|h(x)|,
\end{align*} where we have used repeatedly Plancherel identity. This proves that 
$|h|$ is bounded on the strip if and only if it is bounded in 
$[0,1]$; by continuity, the function 
$|h|$ is indeed bounded there. Thus we apply Stein interpolation to get
\begin{equation*}
||T_\theta \varphi||_{p_\theta} \leq \left( C(p_1,s_1)\right)^\theta ||\varphi||_2 ,
\end{equation*}which means
\begin{equation*}
\left \Vert |\cdot|^{-\beta_1 s/s_1} |D|^{-s} \varphi \right \Vert_{p_\theta} \leq \left( C(p_1,s_1)\right)^\theta ||\varphi||_2 ,
\end{equation*} for every 
$\varphi \in L_{rad}^2(\mathbb{R}^d)$ and substituting back 
$\varphi = |D|^s f$
\begin{equation*}
\left \Vert |\cdot|^{-\beta_1 s/s_1} f \right \Vert_{p_\theta} \leq \left( C(p_1,s_1)\right)^\theta ||f||_{\dot{H}^s}
\end{equation*} for every 
$f\in |D|^{-s}(L_{rad}^2(\mathbb{R}^d))= \dot{H}_{rad}^s(\mathbb{R}^d)$.
Proof. Proof of main theorem
We start by differentiating all the relevant quantities: for the weighted Lp norm
\begin{gather*}
\partial_p \left \Vert |\cdot|^{-\beta} f \right \Vert_p = \partial_p \left( \left \Vert |\cdot|^{-\beta} f \right \Vert_p^p \right)^{1/p} = \\
=\left \Vert |\cdot|^{-\beta} f \right \Vert_p \partial_p\left( \frac{1}{p} \log \left( \left \Vert |\cdot|^{-\beta} f \right \Vert_p^p \right) \right) = \\
= \left \Vert |\cdot|^{-\beta} f \right \Vert_p \left( -\frac{1}{p^2} \log \left( \left \Vert |\cdot|^{-\beta} f \right \Vert_p^p \right)
+ \frac{1}{p} \frac{\int_{\mathbb{R}^d} \partial_p \left( |f|^p |x|^{-\beta p} \right) \,\mathrm{d} x}{\left \Vert |\cdot|^{-\beta} f \right \Vert_p^p} \right)
\end{gather*}and
\begin{gather*}
\partial_{\beta} \left \Vert |\cdot|^{-\beta} f \right \Vert_p = \partial_\beta \left( \left \Vert |\cdot|^{-\beta} f \right \Vert_p^p \right)^{1/p} = \\
= \frac{1}{p} \left( \left \Vert |\cdot|^{-\beta} f \right \Vert_p^p \right)^{1/p-1} \partial_\beta \left( \left \Vert |\cdot|^{-\beta} f \right \Vert_p^p \right) = \\
= \frac{1}{p} \left( \left \Vert |\cdot|^{-\beta} f \right \Vert_p^p \right)^{1/p-1} \int_{\mathbb{R}^d} |f|^p |x|^{-\beta p} \left( -p \log |x| \right) \,\mathrm{d} x .
\end{gather*} Computing the above quantities for 
$p=2, \beta=0$
\begin{gather*}
\partial_p \left \Vert |\cdot|^{-\beta} f \right \Vert_p \Big|_{\substack{{p=2} \\ {\beta=0} }}
= ||f||_2 \left( -\frac{1}{4} \log ||f||_2^2 + \frac{1}{2} ||f||_2^{-2} \int_{\mathbb{R}^d} |f|^2 \log|f| \,\mathrm{d} x \right) \\
= \frac{1}{2} \int_{\mathbb{R}^d} |f|^2 \log |f| \,\mathrm{d} x
\end{gather*}and similarly
\begin{equation*}
\partial_\beta \left \Vert |\cdot|^{-\beta} f \right \Vert_p \Big|_{\substack{{p=2} \\ {\beta=0} }}
= - ||f||_2^{-1} \! \int_{\mathbb{R}^d} |f|^2 \log |x| \,\mathrm{d} x = - \int_{\mathbb{R}^d} |f|^2 \log |x| \,\mathrm{d} x.
\end{equation*}This means that
\begin{gather*}
\frac{\mathrm{d}}{\mathrm{d} s} \left \Vert |\cdot|^{-\beta} f \right \Vert_p |_{s=0}
= \frac{\mathrm{d} p}{\mathrm{d} s} |_{s=0} \ \partial_p \left \Vert |\cdot|^{-\beta} f \right \Vert_p \Big|_{\substack{{p=2} \\ {\beta=0} }} + \\
+\frac{\mathrm{d} \beta}{\mathrm{d} s} |_{s=0} \ \partial_\beta \left \Vert |\cdot|^{-\beta} f \right \Vert_p \Big|_{\substack{{p=2} \\ {\beta=0} }} = \\
= \frac{1}{2} \frac{\mathrm{d} p}{\mathrm{d} s} |_{s=0} \int_{\mathbb{R}^d} |f|^2 \log |f| \,\mathrm{d} x
- \frac{\mathrm{d} \beta}{\mathrm{d} s} |_{s=0} \int_{\mathbb{R}^d} |f|^2 \log |x| \,\mathrm{d} x.
\end{gather*} Now we choose 
$s_1, p_1, \beta_1$ in the permitted range of values and, referring to Lemma 11, define
\begin{equation*}
p(s) := p_{s/s_1} \quad \text{and } \quad \beta(s) := \beta_{s/s_1} .
\end{equation*}With this choice
\begin{gather*}
\frac{\mathrm{d} p}{\mathrm{d} s} |_{s=0} = \frac{4}{s_1} \left( \frac{1}{2} - \frac{1}{p_1} \right) = \frac{4}{d} \left( 1-\frac{\beta_1}{s_1} \right) \\
\frac{\mathrm{d} \beta}{\mathrm{d} s} |_{s=0} = 1-\frac{d}{s_1} \left( \frac{1}{2}-\frac{1}{p_1} \right) = \frac{\beta_1}{s_1} .
\end{gather*}Since we are considering the Rubin inequality with a constant given by Lemma 11, we need to compute
\begin{equation*}
\partial_s \left( C(p_1,s_1) \right)^{s/s_1} |_{s=0} = \frac{1}{s_1} \log C(p_1,s_1)
\end{equation*}and recalling that
\begin{equation*}
\partial_s ||f||_{\dot{H}^s} |_{s=0} = \int_{\mathbb{R}^d} |\widehat{f}|^2 \log |\xi| \,\mathrm{d} \xi
\end{equation*}we get the inequality
\begin{equation}
\begin{split}
\frac{2}{d}\left( 1-\frac{\beta_1}{s_1}\right) \int_{\mathbb{R}^d} |f|^2 \log |f| \,\mathrm{d} x
- \frac{\beta_1}{s_1} \int_{\mathbb{R}^d} |f|^2 \log |x| \,\mathrm{d} x \leq \\
\leq \frac{1}{s_1} \log C(p_1,s_1) + \int_{\mathbb{R}^d} |\widehat{f}|^2 \log |\xi| \,\mathrm{d} \xi .
\end{split}
\end{equation} In particular, we can choose the values 
$p_1=2/(1-2s_1)$ and 
$\beta_1=s_1(1-d)$ to get
\begin{equation*}
2 \int_{\mathbb{R}^d} |f|^2 \log |f| \,\mathrm{d} x + (d-1) \int_{\mathbb{R}^d} |f|^2 \log |x| \,\mathrm{d} x \leq \int_{\mathbb{R}^d} |\widehat{f}|^2 \log |\xi| \,\mathrm{d} \xi +c
\end{equation*} for some 
$c\in\mathbb{R}$.
Appendix A.
This section is devoted to the proof of Theorem 6.
Let us define the square of the Gagliardo s-seminorm as
\begin{equation*}
[f]_s^2 := \int_{\mathbb{R}^d}\int_{\mathbb{R}^d} \frac{(f(x)-f(y))^2}{|x-y|^{d+2s}} \, \mathrm{d}x \, \mathrm{d}y ,
\end{equation*} so that for 
$s\in(0,1)$ (see [Reference Di Nezza, Palatucci and Valdinoci11])
\begin{equation*}
||f||_{\dot{H}^s}^2 = \frac{C(d,s)}{2} [f]_s^2,
\end{equation*}where
\begin{equation*}
(C(d,s))^{-1} := \int_{\mathbb{R}^d} \frac{1-\cos(z_1)}{|z|^{d+2s}} \, \mathrm{d}z.
\end{equation*} We can express 
$(C(d,s))^{-1} $ as
\begin{align*}
&(C(d,s))^{-1} = \\
=& \int_{\mathbb{R}^d} \frac{1-\cos(z_1)}{|z|^{d+2s}} \, \mathrm{d}z = \int_{\mathbb{S}^{d-1}} \int_0^{+\infty} \frac{1-\cos(r \theta_1)}{r^{1+2s}} \, \mathrm{d}r \, \mathrm{d}\theta = \\
=& \int_{\mathbb{S}^{d-1}} \int_0^{+\infty} (1+2s)\frac{r\theta_1-\sin(r \theta_1)}{\theta_1 r^{2+2s}} \, \mathrm{d}r \, \mathrm{d}\theta = \\
=& \int_{\mathbb{S}^{d-1}} \int_0^1(1+2s)\frac{r\theta_1-\sin(r \theta_1)}{\theta_1 r^{2+2s}} \, \mathrm{d}r \, \mathrm{d}\theta +
\int_{\mathbb{S}^{d-1}} \int_1^{+\infty} (1+2s)\frac{r\theta_1-\sin(r \theta_1)}{\theta_1 r^{2+2s}} \, \mathrm{d}r \, \mathrm{d}\theta = \\
=& \int_{B_1} (1+2s) \frac{z_1-\sin(z_1)}{z_1 |z|^{d+2s}} \, \mathrm{d}z + |\mathbb{S}^{d-1}|\frac{1+2s}{2s} - \int_{B_1^c} (1+2s)\frac{\sin(z_1)}{z_1 |z|^{d+2s}} \, \mathrm{d}z = \\
=& \frac{|\mathbb{S}^{d-1}|}{2s} + \left( |\mathbb{S}^{d-1}| + \int_{B_1} \frac{z_1-\sin(z_1)}{z_1 |z|^d} \, \mathrm{d}z \ - \ \int_{B_1^c} \frac{\sin(z_1)}{z_1 |z|^d} \, \mathrm{d}z \right) + o(1),
\text{as } s\to 0^+.
\end{align*}In particular
\begin{align*}
&\frac{C(d,s)|\mathbb{S}^{d-1}|}{2s}= \\
=& |\mathbb{S}^{d-1}| \left( |\mathbb{S}^{d-1}| + 2 \left( |\mathbb{S}^{d-1}| + \int_{B_1} \frac{z_1-\sin(z_1)}{z_1 |z|^d} \, \mathrm{d}z \ - \ \int_{B_1^c} \frac{\sin(z_1)}{z_1 |z|^d} \, \mathrm{d}z \right) s + o(s) \right)^{-1} = \\
=& 1 - 2 \left( 1+\frac{1}{|\mathbb{S}^{d-1}|} \int_{B_1} \frac{z_1-\sin(z_1)}{z_1 |z|^d} \, \mathrm{d}z \ - \frac{1}{|\mathbb{S}^{d-1}|} \ \int_{B_1^c} \frac{\sin(z_1)}{z_1 |z|^d} \, \mathrm{d}z \right) s + o(s)
\end{align*}and
\begin{align*}
& \lim_{s\to 0} \frac{C(d,s)|\mathbb{S}^{d-1}|}{2s} = 1 \\
& \frac{d}{ds} \big|_{s=0} \frac{C(d,s)|\mathbb{S}^{d-1}|}{2s} = - 2 \left( 1+\frac{1}{|\mathbb{S}^{d-1}|} \int_{B_1} \frac{z_1-\sin(z_1)}{z_1 |z|^d} \, \mathrm{d}z \ - \frac{1}{|\mathbb{S}^{d-1}|} \ \int_{B_1^c} \frac{\sin(z_1)}{z_1 |z|^d} \, \mathrm{d}z \right) .
\end{align*}As a last ingredient, we need an easy lemma.
Lemma 12. We can write
\begin{equation*}
[f]_s^2 = \iint\limits_{\{|x-y|\leq 1\}} \frac{(f(x)-f(y))^2}{|x-y|^{d+2s}} \, \mathrm{d}x \, \mathrm{d}y \ - \ 2 \iint\limits_{\{|x-y| \gt 1\}} \frac{f(x)f(y)}{|x-y|^{d+2s}} \, \mathrm{d}x \, \mathrm{d}y \ + \ \frac{|\mathbb{S}^{d-1}|}{s} ||f||_2^2.
\end{equation*}Proof. By dividing the domain in 
$\{|x-y|\leq 1\}$ and 
$\{|x-y|\geq 1\}$, it suffices to show that
\begin{equation*}\frac{|\mathbb{S}^{d-1}|}{s} ||f||_2^2=2 \iint\limits_{\{|x-y|\geq 1\}} \frac{f(x)^2}{|x-y|^{d+2s}} \, \mathrm{d}x \, \mathrm{d}y. \end{equation*} A change of variable 
$y \rightarrow z+x$ leads to
\begin{equation*}\iint\limits_{\{|x-y|\geq 1\}} \frac{f(x)^2}{|x-y|^{d+2s}} \, \mathrm{d}x \, \mathrm{d}y=\int_{\mathbb{R}^d} \int_{|z|\geq 1}\frac{f(x)^2}{|z|^{d+2s}} \, \mathrm{d}z \, \mathrm{d}x= \frac{|\mathbb{S}^{d-1}|}{2s} ||f||_2^2.\end{equation*}Proof of Theorem 6
The previous lemma implies that
\begin{equation*}
\lim_{s\to 0} \frac{s[f]_s^2}{|\mathbb{S}^{d-1}|} = ||f||_2^2.
\end{equation*}Moreover
\begin{align*}
\frac{\frac{s[f]_s^2}{|\mathbb{S}^{d-1}|} - ||f||_2^2}{s} = \frac{[f]_s^2}{|\mathbb{S}^{d-1}|} - \frac{1}{s}||f||_2^2
=& \frac{1}{|\mathbb{S}^{d-1}|} \iint\limits_{\{|x-y|\leq 1\}} \frac{(f(x)-f(y))^2}{|x-y|^{d+2s}} \, \mathrm{d}x \, \mathrm{d}y \ - \\
-& \frac{2}{|\mathbb{S}^{d-1}|} \iint\limits_{\{|x-y| \gt 1\}} \frac{f(x)f(y)}{|x-y|^{d+2s}} \, \mathrm{d}x \, \mathrm{d}y,
\end{align*}so, thanks to the dominated convergence theorem,
\begin{equation*}
\lim_{s\to 0} \frac{\frac{s[f]_s^2}{|\mathbb{S}^{d-1}|} - ||f||_2^2}{s} = \frac{1}{|\mathbb{S}^{d-1}|} \iint\limits_{\{|x-y|\leq 1\}} \frac{(f(x)-f(y))^2}{|x-y|^d} \, \mathrm{d}x \, \mathrm{d}y \ - \ \frac{2}{|\mathbb{S}^{d-1}|} \iint\limits_{\{|x-y| \gt 1\}} \frac{f(x)f(y)}{|x-y|^d} \, \mathrm{d}x \, \mathrm{d}y .
\end{equation*}On the other hand
\begin{equation*}
\lim_{s\to0} \frac{||f||_{\dot{H}^s}^2-||f||_2^2}{s} = 2 \int_{\mathbb{R}^d} |\widehat{f}(\xi)|^2 \log|\xi| \, \mathrm{d}\xi
\end{equation*}and
\begin{equation*}
\frac{||f||_{\dot{H}^s}^2-||f||_2^2}{s} = \frac{\frac{C(d,s)}{2} [f]_s^2-||f||_2^2}{s} .
\end{equation*}Compute
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}s} ||f||_{\dot{H}^s} =& \frac{\mathrm{d}}{\mathrm{d}s} \left( \frac{C(d,s)}{2} [f]_s^2\right) = \frac{\mathrm{d}}{\mathrm{d}s} \left( \frac{C(d,s)|\mathbb{S}^{d-1}|}{2s} \frac{s}{|\mathbb{S}^{d-1}|}[f]_s^2\right) = \\
=& \frac{\mathrm{d}}{\mathrm{d}s} \left( \frac{C(d,s)|\mathbb{S}^{d-1}|}{2s} \right) \frac{s}{|\mathbb{S}^{d-1}|}[f]_s^2+ \frac{C(d,s)|\mathbb{S}^{d-1}|}{2s} \frac{\mathrm{d}}{\mathrm{d}s} \left( \frac{s}{|\mathbb{S}^{d-1}|}[f]_s^2 \right).
\end{align*}and take the limit as s → 0 to get
\begin{align*}
\lim_{s\to0} &\frac{||f||_{\dot{H}^s}^2-||f||_2^2}{s} = \\
=& - 2 \left( 1+\frac{1}{|\mathbb{S}^{d-1}|} \int_{B_1} \frac{z_1-\sin(z_1)}{z_1 |z|^d} \, \mathrm{d}z \ - \frac{1}{|\mathbb{S}^{d-1}|} \ \int_{B_1^c} \frac{\sin(z_1)}{z_1 |z|^d} \, \mathrm{d}z \right) ||f||_2^2 + \\
+& \frac{1}{|\mathbb{S}^{d-1}|} \iint\limits_{\{|x-y|\leq 1\}} \frac{(f(x)-f(y))^2}{|x-y|^d} \, \mathrm{d}x \, \mathrm{d}y \ - \ \frac{2}{|\mathbb{S}^{d-1}|} \iint\limits_{\{|x-y| \gt 1\}} \frac{f(x)f(y)}{|x-y|^d} \, \mathrm{d}x \, \mathrm{d}y.
\end{align*}
