†† 2020 Mathematical subject classification: 42B10, 42B15, 33C10

# Some inequalities for the Fourier transform and their limiting behaviour

Nicola Garofalo  Dipartimento d’Ingegneria Civile e Ambientale (DICEA)
Università di Padova
Via Marzolo, 9 - 35131 Padova, Italy<EMAIL_ADDRESS>In ricordo di
E. M. Stein

###### Abstract.

We identify a one-parameter family of inequalities for the Fourier transform
whose limiting case is the restriction conjecture for the sphere. Using
Stein’s method of complex interpolation we prove the conjectured inequalities
when the target space is $L^{2}$, and show that this recovers in the limit the
celebrated Tomas-Stein theorem.

###### Key words and phrases:

Fourier transform. Riesz Measures. Nonlocal restriction inequality.

The author has been supported in part by a Progetto SID (Investimento
Strategico di Dipartimento): “Aspects of nonlocal operators via fine
properties of heat kernels”, University of Padova, 2022. He has also been
partially supported by a Visiting Professorship at the Arizona State
University.

###### Contents

  1. 1 Introduction
  2. 2 Necessary condition for restriction
  3. 3 The Fourier transform of the kernel $A^{(s)}_{1}$
  4. 4 Proof of Theorem 1.4
  5. 5 Appendix: some classical material

##  1\. Introduction

For any $0<s<1$ and $r>0$ we consider the function

(1.1) $A^{(s)}_{r}(x)=\frac{c(n,s)r^{2s}}{(|x|^{2}-r^{2})_{+}^{s}|x|^{n}},$

where, using the standard notation
$\sigma_{n-1}=\frac{2\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}$ for the
$(n-1)$-dimensional measure of the unit sphere
$\mathbb{S}^{n-1}\subset\mathbb{R}^{n}$, we have let

(1.2) $c(n,s)=\frac{2}{\Gamma(s)\Gamma(1-s)\sigma_{n-1}}.$

Define measures on $\mathbb{R}^{n}$ by letting

(1.3) $d\mu^{(s)}_{r}(x)=A^{(s)}_{r}(x)dx.$

Since from (1.1) it immediately follows that $A^{(s)}_{r}\in
L^{1}(\mathbb{R}^{n})$, from [9, Lemma 15.3]) one has for every $s\in(0,1)$
and $r>0$

(1.4)
$\mu^{(s)}_{r}(\mathbb{R}^{n})=||A^{(s)}_{r}||_{L^{1}(\mathbb{R}^{n})}=\int_{\mathbb{R}^{n}}A^{(s)}_{r}(x)dx=\int_{\mathbb{R}^{n}}A^{(s)}_{1}(x)dx=1.$

We next define the operator

(1.5) $\mathscr{A}^{(s)}_{r}f(x)=A^{(s)}_{r}\star f(x).$

It follows from (1.4) and Young’s convolution theorem that

$\mathscr{A}^{(s)}_{r}:L^{p}(\mathbb{R}^{n})\ \longrightarrow\
L^{p}(\mathbb{R}^{n}),\ \ \ \ \ \ \ \ 1\leq p\leq\infty,$

and that for any $f\in L^{p}(\mathbb{R}^{n})$ we have

(1.6)
$||\mathscr{A}^{(s)}_{r}f||_{L^{p}(\mathbb{R}^{n})}\leq||f||_{L^{p}(\mathbb{R}^{n})},$

which shows that $\mathscr{A}^{(s)}_{r}$ is a contraction in
$L^{p}(\mathbb{R}^{n})$ for every $0<s<1$ and $r>0$.

For any $f\in\mathscr{S}(\mathbb{R}^{n})$ we denote by
$\hat{f}(\xi)=\mathscr{F}(f)(\xi)=\int_{\mathbb{R}^{n}}e^{-2\pi
i\langle\xi,x\rangle}f(x)dx$ its Fourier transform, and ask the following

Question: Let $\frac{1}{2}\leq s<1$. If $1\leq p<\frac{2n}{n+2s-1}$, is it
true that for $1\leq q\leq\frac{n+1-2s}{n+1}p^{\prime}$ and for every
$f\in\mathscr{S}(\mathbb{R}^{n})$ one has for some $C^{(s)}(n,p)>0$

(1.7)
$\left(\int_{\mathbb{R}^{n}}|\hat{f}(\xi)|^{q}A^{(s)}_{1}(\xi)d\xi\right)^{1/q}\leq
C^{(s)}(n,p)\ ||f||_{L^{p}(\mathbb{R}^{n})}.$

We note that the restriction $s\geq\frac{1}{2}$ serves to guarantee that
$\frac{2n}{n+2s-1}\leq 2$. Therefore, the hypothesis $f\in
L^{p}(\mathbb{R}^{n})$ implies that $f$ be in the Hausdorff-Young range
$[1,2]$. As a consequence, $\hat{f}$ is a function in
$L^{p^{\prime}}(\mathbb{R}^{n})$.

Our interest in the above conjecture stems from the following observations.
Assume that (1.7) does hold for any $s$ such that $\frac{1}{2}\leq s<1$. Since
$\frac{2n}{n+2s-1}\searrow\frac{2n}{n+1}$ as $s\nearrow 1$, if we take $1\leq
p<\frac{2n}{n+1}$ and $q\leq\frac{n-1}{n+1}p^{\prime}$, then it is immediate
to verify that for any $s\in[\frac{1}{2},1)$

$1\leq p<\frac{2n}{n+2s-1}\ \ \ \ \text{and}\ \ \ \ 1\leq
q\leq\frac{n+1-2s}{n+1}p^{\prime}$

(for the second of these inequalities simply note that
$q\leq\frac{n-1}{n+1}p^{\prime}<\frac{n-1+2(1-s)}{n+1}p^{\prime}=\frac{n+1-2s}{n+1}p^{\prime}$),
therefore (1.7) holds. But we now have the following fact implicitly contained
in the seminal work [14] of M. Riesz (for a proof see [9, Prop. 15.4]).

######  Proposition 1.1.

For every function $f\in\mathscr{S}(\mathbb{R}^{n})$ and $x\in\mathbb{R}^{n}$,
one has

$\underset{s\to 1}{\lim}\ \mathscr{A}^{(s)}_{r}f(x)=\mathscr{M}_{r}(f,x),$

where

$\mathscr{M}_{r}(f,x)=\frac{1}{\sigma_{n-1}r^{n-1}}\int_{S(x,r)}f(y)d\sigma(y)$

is the spherical average of $f$ over the sphere
$S(x,r)=\\{y\in\mathbb{R}^{n}\mid|y-x|=r\\}$.

Therefore, passing to the limit in (1.7) and using Proposition 1.1, if
$C^{(s)}(n,p)$ converges to a number $C(n,p)>0$ as $s\to 1$, we would infer
that for $1\leq p<\frac{2n}{n+1}$ and $q\leq\frac{n-1}{n+1}p^{\prime}$ the
following limiting inequality holds

(1.8)
$\left(\frac{1}{\sigma_{n-1}}\int_{\mathbb{S}^{n-1}}|\hat{f}(\xi)|^{q}d\sigma(\xi)\right)^{1/q}\leq
C(n,p)\ ||f||_{L^{p}(\mathbb{R}^{n})}.$

As it is well-known, this is the famous restriction conjecture of C. Fefferman
and E. Stein for the Fourier transform, see [6], [7], [8], [18], [4] and [21].

Eli Stein lecturing on the restriction problem, UChicago, 1985

One obvious advantage of (1.7) over (1.8) is that the support of the measures
(1.3) is $\mathbb{R}^{n}\setminus B(0,1)$, instead of the lower dimensional
manifold $\mathbb{S}^{n-1}\subset\mathbb{R}^{n}$. Notice that if we let
$d\sigma$ denote the surface measure concentrated on the sphere
$\mathbb{S}^{n-1}$, then Proposition 1.1 can be equivalently stated as follows

(1.9) $d\mu^{(s)}_{1}\ \underset{s\to 1}{\longrightarrow}\
\frac{1}{\sigma_{n-1}}d\sigma\ \ \ \ \ \ \text{in}\
\mathscr{S}^{\prime}(\mathbb{R}^{n}).$

Concerning the measures (1.3) we recall that in his above quoted paper M.
Riesz developed the theory of the nonlocal operators $(-\Delta)^{s}$ and their
inverses $I_{2s}$, the operators of fractional integration which play a
pervasive role in analysis, see also [17, Ch. 5]. Among other things, he
solved by inversion the Dirichlet problem

(1.10) $\begin{cases}(-\Delta)^{s}u=0\ \ \ \text{in}\ B_{r},\\\ u=f\ \ \ \ \ \
\ \ \text{in}\ \mathbb{R}^{n}\setminus B_{r},\end{cases}$

and proved that for every $x\in B_{r}$ the unique solution to (1.10) is
provided by

(1.11)
$u(x)=c(n,s)\int_{|y|>r}\left(\frac{r^{2}-|x|^{2}}{|y|^{2}-r^{2}}\right)^{s}\frac{f(y)}{|y-x|^{n}}dy,$

see formula (3) on p. 17 in [14], but also (1.6.11’) and (1.6.2) on pages 122
and 112 in [11]. It is clear from (1.11) that
$u(0)=\int_{\mathbb{R}^{n}}f(y)A^{(s)}_{r}(y)dy=\mathscr{A}^{(s)}_{r}f(0)$.
The role of the operators $\mathscr{A}^{(s)}_{r}$ is further elucidated by the
following result, see [9, Prop. 15.6].

######  Proposition 1.2 (The Blaschke-Privalov fractional Laplacian).

Let $0<s<1$ and suppose that $f\in\mathscr{L}_{s}(\mathbb{R}^{n})$ be in
$C^{2s+\varepsilon}$ in a neighborhood of $x\in\mathbb{R}^{n}$, for some
$0<\varepsilon<1$. One has

(1.12) $(-\Delta)^{s}f(x)=-\gamma(n,s)\underset{r\to
0^{+}}{\lim}\frac{\mathscr{A}^{(s)}_{r}f(x)-f(x)}{r^{2s}},$

where
$\gamma(n,s)=\frac{s2^{2s}\Gamma(\frac{n}{2}+s)}{\pi^{\frac{n}{2}}\Gamma(1-s)}$.

Here, for $0<s<1$ we have denoted by $\mathscr{L}_{s}(\mathbb{R}^{n})$ the
space of measurable functions $f:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ for
which the norm

$||u||_{\mathscr{L}_{s}(\mathbb{R}^{n})}=\int_{\mathbb{R}^{n}}\frac{|f(x)|}{1+|x|^{n+2s}}dx<\infty.$

Returning to the inequality (1.7) we mention that, similarly to the
restriction problem (1.8), it cannot possibly hold for every exponent
$p\in[1,2]$ in the Hausdorff-Young range. To understand the constraint $1\leq
p<\frac{2n}{n+2s-1}$, denote by $T:L^{p}(\mathbb{R}^{n})\to
L^{q}(\mathbb{R}^{n},d\mu^{(s)}_{1})$ the “restriction” operator in (1.7).
Then its adjoint $T^{\star}:L^{q^{\prime}}(\mathbb{R}^{n},d\mu^{(s)}_{1})\to
L^{p^{\prime}}(\mathbb{R}^{n})$ is easily seen to be given by

(1.13) $T^{\star}f(\xi)=\int_{\mathbb{R}^{n}}e^{-2\pi
i\langle\xi,x\rangle}f(x)A^{(s)}_{1}(\xi)dx,$

so that

$T^{\star}1=\widehat{A^{(s)}_{1}}.$

Assuming (1.7), we would thus have by duality

$||T^{\star}1||_{L^{p^{\prime}}(\mathbb{R}^{n})}\leq B^{(s)}(n,p)\
||1||_{L^{q^{\prime}}(\mathbb{R}^{n},d\mu^{(s)}_{1})}=B^{(s)}(n,p)<\infty,$

where in the last equality we have used (1.4). Therefore, the validity of
(1.7) for some $p$ implies that $\widehat{A^{(s)}_{1}}\in
L^{p^{\prime}}(\mathbb{R}^{n})$. Now, in Section 3 we prove the following.

######  Theorem 1.3.

Let $s\in(0,1)$ and $n\geq 2$. Then the Fourier transform of the kernel
defined by (1.1) is given by

(1.14)
$\widehat{A^{(s)}_{r}}(\xi)=\frac{2^{\frac{n}{2}-s}\Gamma(\frac{n}{2})}{\Gamma(s)}\int_{2\pi
r|\xi|}^{\infty}t^{s-\frac{n}{2}}J_{\frac{n}{2}-1+s}(t)dt.$

Using Theorem 1.3, in (3.9) of Corollary 3.2 we obtain the following important
decay at infinity

(1.15) $\widehat{A^{(s)}_{1}}(\xi)\cong\frac{1}{|\xi|^{\frac{n+1}{2}-s}},$

which shows that

(1.16) $\widehat{A^{(s)}_{1}}\in L^{p^{\prime}}(\mathbb{R}^{n})\
\Longleftrightarrow\ p^{\prime}>\frac{2n}{n+1-2s}\ \Longleftrightarrow\ 1\leq
p<\frac{2n}{n+2s-1}.$

We conclude that the inequality (1.7) cannot possibly hold for
$p\geq\frac{2n}{n+2s-1}$.

Before proceeding we pause to comment on an aspect of Theorem 1.3. As the
reader will see its proof is somewhat more involved than its well-known
counterpart in the case $s=1$. This is due to the _nonlocal_ nature of the
measure $d\mu^{(s)}_{1}$, compared to the surface measure $d\sigma$ of the
unit sphere $\mathbb{S}^{n-1}$. To explain this comment we stress that

(1.17)
$\widehat{d\sigma}(\xi)=\frac{2\pi}{|\xi|^{\frac{n}{2}-1}}J_{\frac{n}{2}-1}(2\pi|\xi|)$

is just a rescaled spherically symmetric eigenfunction of the (local)
differential operator $\Delta$ in $\mathbb{R}^{n}$. To see this, simply
observe that for every $\lambda>0$ the function

$f_{\lambda}(\xi)=\int_{\mathbb{S}^{n-1}}e^{i\sqrt{\lambda}\langle\xi,\omega\rangle}d\sigma(\omega)$

is a spherically symmetric solution of the Helmholtz equation $\Delta
f_{\lambda}=-\lambda f_{\lambda}$ in $\mathbb{R}^{n}$, which shows that (1.17)
solves such PDE with $\lambda=4\pi^{2}$. However, the equations (1.10), (1.11)
above, and Proposition 1.2 in particular, underscore that the Fourier
transform of the measure $d\mu^{(s)}_{r}$ is instead connected to the
pseudodifferential operator $(-\Delta)^{s}$, and computations with such
nonlocal operator are usually more involved. In this regard, we recall the
following quote from p. 51 in [11]:…“ _In the theory of M. Riesz kernels, the
role of the Laplace operator, which has a local character, is taken…by a non-
local integral operator…This circumstance often substantially complicates the
theory…_ ”. In Lemma 3.3 we show that for $n\geq 2$ one has for every
$\xi\in\mathbb{R}^{n}$

$\underset{s\to 1}{\lim}\
\widehat{A^{(s)}_{1}}(\xi)=\frac{1}{\sigma_{n-1}}\widehat{d\sigma}(\xi).$

This result (which can be seen as a comforting a posteriori confirmation of
the correctness of the computations leading to Theorem 1.3) is of course not
surprising in view of (1.9) above. It is worthwhile noting at this moment that
(1.1) are clearly reminiscent of the classical Bochner-Riesz kernels

(1.18) $K_{z}(x)=\frac{\left(1-|x|^{2}\right)^{z}_{+}}{\Gamma(z+1)},\ \ \ \
\Re z>-1,$

whose Fourier transform is given by

(1.19)
$\hat{K}_{z}(\xi)=\pi^{-z}|\xi|^{-\left(\frac{n}{2}+z\right)}J_{\frac{n}{2}+z}(2\pi|\xi|),\
\ \ \ \xi\in\mathbb{R}^{n}.$

If we compare this formula with (1.17) it is clear that
$\hat{K}_{z}\to\frac{1}{2}\ \widehat{d\sigma}$ as $z\to-1$. While the
computation of the Fourier transform of $A^{(s)}_{1}$ is more involved than
(1.19), there are advantages in working with (1.1) instead of (1.18). One of
them is that, as we have mentioned, $\widehat{A^{(s)}_{1}}$ is directly
connected to the eigenfunctions of the nonlocal operator $(-\Delta)^{s}$ (to
be further analysed in a future work), while this is not the case for (1.19).

We have seen that (1.7) is only possible when $p$ satisfies (1.16). But, given
a $p$ in such range what is the optimal range of $q$’s? To answer this
question we use the well-known argument of Knapp, except that because of the
presence of the measure $d\mu_{1}^{(s)}$ in (1.7), we need to work a bit more.
In Proposition 2.1 below we show that, given $p$ within the range (1.16), a
necessary condition for (1.7) to hold is that

(1.20) $1\leq q\leq\frac{n+1-2s}{n+1}p^{\prime}.$

Notice that when the target space is $L^{2}$, then in view of (1.20) the
conjecture asks whether it is true that (1.7) holds with $q=2$ for any
$f\in\mathscr{S}(\mathbb{R}^{n})$ and $2\leq\frac{n+1-2s}{n+1}p^{\prime}$.
This is equivalent to asking that
$\frac{1}{p^{\prime}}\leq\frac{n+1-2s}{2(n+1)}=\frac{1}{2}-\frac{s}{n+1}$, or
equivalently $\frac{1}{p}\geq\frac{1}{2}+\frac{s}{n+1}=\frac{n+1+2s}{2(n+1)}$,
and therefore for any

(1.21) $1\leq p\leq\frac{2(n+1)}{n+1+2s}.$

In the next result, we prove this conjecture.

######  Theorem 1.4.

For a given $s\in(0,1)$ let $p=\frac{2(n+1)}{n+1+2s}$. Then there exists a
constant $C^{(s)}(n)>0$ such that for every $f\in\mathscr{S}(\mathbb{R}^{n})$
one has

(1.22)
$\left(\int_{\mathbb{R}^{n}}|\hat{f}(\xi)|^{2}A^{(s)}_{1}(\xi)d\xi\right)^{1/2}\leq
C^{(s)}(n)\ ||f||_{L^{p}(\mathbb{R}^{n})}.$

In order to establish Theorem 1.4 we exploit Plancherel as in [22] and reduce
matters to proving that the operator $R^{(s)}f=\widehat{A^{(s)}_{1}}\star f$
maps $L^{p}(\mathbb{R}^{n})\to L^{p^{\prime}}(\mathbb{R}^{n})$. This is
ultimately achieved using Stein’s theorem of complex interpolation by
embedding $R^{(s)}$ into an analytic family of operators $\\{T_{z}\\}_{z\in
S}$ in the strip $S=\\{z\in\mathbb{C}\mid-\frac{n-1}{2}\leq\Re z\leq 1\\}$.
Specifically, we show that

$\begin{cases}T_{-\frac{n-1}{2}+iy}:L^{1}(\mathbb{R}^{n})\ \longrightarrow\
L^{\infty}(\mathbb{R}^{n}),\\\ T_{1+iy}:L^{2}(\mathbb{R}^{n})\
\longrightarrow\ L^{2}(\mathbb{R}^{n}),\end{cases}$

with appropriate bounds on the operator norms. Since the constant $C^{(s)}(n)$
in (1.22) is bounded uniformly in $s\in(0,1)$, see (4.21) below, passing to
the limit as $s\to 1$ we recover the Tomas-Stein restriction theorem, see
[22].

######  Remark 1.5.

One should observe that the threshold exponent $p=\frac{2(n+1)}{n+1+2s}$ in
(1.21) is strictly less than $2$ for any $0<s<1$. Therefore, the limitation
$\frac{1}{2}\leq s<1$ is not necessary in such case.

The plan of the paper is as follows. In Section 2 we adapt the well-known
argument of Knapp to prove that, if for a given $p$ in the range (1.16) the
restriction inequality (1.7) does hold, then we must have $1\leq
q\leq\frac{n+1-2s}{n+1}p^{\prime}$. Section 3 is devoted to proving Theorem
1.3, from which we obtain Corollary 3.2. The representation formula (3.8)
contained in it will be quite important in the remainder of the paper. Finally
in Section 4 we prove the nonlocal restriction Theorem 1.4. As a corollary of
this result we obtain the celebrated Tomas-Stein theorem. The paper closes
with an appendix in Section 5 in which we gather some well-known facts and
collect some results needed in the rest of the paper.

Acknowledgment: I thank Carlos Kenig and Agnid Banerjee for their gracious
preliminary reading of the manuscript and their feedback.

##  2\. Necessary condition for restriction

In (1.16) above we have seen that the inequality (1.7) can possibly hold only
when $1\leq p<\frac{2n}{n+2s-1}$. In this section we adapt the well-known
argument of Knapp to prove that, if for a given $p$ in such range (1.7) does
hold, then we must have $1\leq q\leq\frac{n+1-2s}{n+1}p^{\prime}$.

######  Proposition 2.1.

A necessary condition for (1.7) to hold for $1\leq p<\frac{2n}{n+2s-1}$ is
that $1\leq q\leq\frac{n+1-2s}{n+1}p^{\prime}$.

###### Proof.

For $\varepsilon>0$ small consider the parallelepiped
$K_{\varepsilon}=K^{\prime}_{\varepsilon}\times[1-\varepsilon,1]$ with sides
parallel to the coordinate axis whose projection onto
$\mathbb{R}^{n-1}\times\\{0\\}$ is the $(n-1)$-dimensional cube
$K^{\prime}_{\varepsilon}$ circumscribing the intersection of the hyperplane
$x_{n}=1-\varepsilon$ with the unit sphere $\mathbb{S}^{n-1}$. If
$\theta_{\varepsilon}$ is the angle of aperture of the right circular cone
obtained by projecting to the origin the points of the $(n-2)$-dimensional
sphere obtained intersecting $\mathbb{S}^{n-1}\cap\\{x_{n}=1-\varepsilon\\}$,
from elementary trigonometry we have $\cos\theta_{\varepsilon}=1-\varepsilon$,
$\sin\theta_{\varepsilon}=R(\varepsilon)=\sqrt{\varepsilon(2-\varepsilon)}$,
and therefore
$K^{\prime}_{\varepsilon}=[-R(\varepsilon),R(\varepsilon)]^{n-1}$.

Denoting now with
$B^{\prime}(0,r)=\\{x^{\prime}\in\mathbb{R}^{n-1}\mid|x^{\prime}|<r\\}$,
consider now the right-circular cylinders
$C_{\varepsilon}=B^{\prime}(0,R(\varepsilon))\times[1-\varepsilon,1]$ and
$C^{\star}_{\varepsilon}=B^{\prime}(0,\sqrt{n-1}\
R(\varepsilon))\times[1-\varepsilon,1]$. A moment’s thought reveals that

(2.1) $C_{\varepsilon}\cap(\mathbb{R}^{n}\setminus B(0,1))\subset
K_{\varepsilon}\cap(\mathbb{R}^{n}\setminus B(0,1))\subset
C^{\star}_{\varepsilon}\cap(\mathbb{R}^{n}\setminus B(0,1)).$

We note that

$C_{\varepsilon}\cap(\mathbb{R}^{n}\setminus
B(0,1))=\\{(x^{\prime},x_{n})\in\mathbb{R}^{n}\mid 1-\varepsilon\leq x_{n}\leq
1,\ \sqrt{1-x_{n}^{2}}\leq|x^{\prime}|\leq R(\varepsilon)\\}.$

As in Knapp’s argument, if $\mathbf{1}_{E}$ is the indicator function of a set
$E\subset\mathbb{R}^{n}$, we now consider the function
$f_{\varepsilon}=\mathscr{F}^{-1}(\mathbf{1}_{K_{\varepsilon}})$, so that
$\hat{f}_{\varepsilon}=\mathbf{1}_{K_{\varepsilon}}$. As it is well-known

$f_{\varepsilon}(\xi)=\hat{\mathbf{1}}_{K_{\varepsilon}}(\xi)=e^{i\pi(2-\varepsilon)\xi_{n}}\
\frac{\sin(\pi\varepsilon\xi_{n})}{\pi\xi_{n}}\prod_{j=1}^{n-1}\frac{\sin(2\pi
R(\varepsilon)\xi_{j})}{\pi\xi_{j}},$

and therefore $f_{\varepsilon}\in L^{p}(\mathbb{R}^{n})$ for any $p>1$ and
moreover

(2.2)
$||f||_{L^{p}(\mathbb{R}^{n})}\cong\varepsilon^{\frac{n+1}{2p^{\prime}}}.$

Next, we want to understand the asymptotic behaviour as $\varepsilon\to 0^{+}$
of the quantity

(2.3)
$\left(\int_{\mathbb{R}^{n}}|\hat{f}_{\varepsilon}(x)|^{q}A^{(s)}_{1}(x)dx\right)^{1/q}=\left(\int_{K_{\varepsilon}\cap(\mathbb{R}^{n}\setminus
B(0,1))}A^{(s)}_{1}(x)dx\right)^{1/q}.$

In view of the inclusions (2.1) it suffices to understand the asymptotic
behaviour of the right-hand side of (2.3) when the integral is performed on
the set $C_{\varepsilon}\cap(\mathbb{R}^{n}\setminus B(0,1))$. With this
objective in mind, we obtain from Cavalieri’s principle

$\displaystyle\left(\int_{C_{\varepsilon}\cap(\mathbb{R}^{n}\setminus
B(0,1))}A^{(s)}_{1}(x)dx\right)^{1/q}=\left(\int_{1-\varepsilon}^{1}\int_{\sqrt{1-t^{2}}\leq|x^{\prime}|\leq
R(\varepsilon)}A^{(s)}_{1}(x^{\prime},t)dx^{\prime}dt\right)^{1/q}$
$\displaystyle=\left(\int_{1-\varepsilon}^{1}\int_{\sqrt{1-t^{2}}\leq|x^{\prime}|\leq
R(\varepsilon)}\frac{dx^{\prime}}{(|x^{\prime}|^{2}-(1-t^{2}))^{s}(|x^{\prime}|^{2}+t^{2})^{\frac{n}{2}}}dt\right)^{1/q}$
$\displaystyle\cong\left(\int_{1-\varepsilon}^{1}\int_{\sqrt{1-t^{2}}\leq|x^{\prime}|\leq
R(\varepsilon)}\frac{dx^{\prime}}{(|x^{\prime}|^{2}-(1-t^{2}))^{s}}dt\right)^{1/q}$
$\displaystyle\cong\left(\int_{1-\varepsilon}^{1}\int_{\sqrt{1-t^{2}}}^{R(\varepsilon)}\frac{\rho^{n-2}d\rho}{(\rho^{2}-(1-t^{2}))^{s}}dt\right)^{1/q}.$

We now want to show that as $\varepsilon\to 0^{+}$ we have

(2.4)
$G(\varepsilon)=\int_{1-\varepsilon}^{1}\int_{\sqrt{1-t^{2}}}^{R(\varepsilon)}\frac{\rho^{n-2}d\rho}{(\rho^{2}-(1-t^{2}))^{s}}dt\cong\varepsilon^{\frac{n+1}{2}-s}.$

To see this we write
$G(\varepsilon)=\int_{1-\varepsilon}^{1}F(\varepsilon,t)dt$, where

$F(\varepsilon,t)=\int_{\sqrt{1-t^{2}}}^{R(\varepsilon)}\frac{\rho^{n-2}d\rho}{(\rho^{2}-(1-t^{2}))^{s}}.$

The chain rule gives

$G^{\prime}(\varepsilon)=F(\varepsilon,1-\varepsilon)+\int_{1-\varepsilon}^{1}\frac{\partial
F}{\partial\varepsilon}(\varepsilon,t)dt=\int_{1-\varepsilon}^{1}\frac{\partial
F}{\partial\varepsilon}(\varepsilon,t)dt,$

since $F(\varepsilon,1-\varepsilon)=0$. A simple computation gives

$\frac{\partial
F}{\partial\varepsilon}(\varepsilon,t)=\frac{R^{\prime}(\varepsilon)R(\varepsilon)^{n-2}}{(R(\varepsilon)^{2}-(1-t^{2}))^{s}},$

therefore

$\displaystyle G^{\prime}(\varepsilon)$
$\displaystyle=R^{\prime}(\varepsilon)R(\varepsilon)^{n-2}\int_{1-\varepsilon}^{1}\frac{dt}{(R(\varepsilon)^{2}-(1-t^{2}))^{s}}=R^{\prime}(\varepsilon)R(\varepsilon)^{n-2}\int_{1-\varepsilon}^{1}\frac{1}{(t^{2}-(1-\varepsilon)^{2})^{s}}dt$
$\displaystyle\cong
R^{\prime}(\varepsilon)R(\varepsilon)^{n-2}\int_{1-\varepsilon}^{1}\frac{dt}{(t-(1-\varepsilon))^{s}}\cong\varepsilon^{1-s}R^{\prime}(\varepsilon)R(\varepsilon)^{n-2}.$

Since
$R(\varepsilon)^{n-2}=(\varepsilon(2-\varepsilon))^{n-2}\cong\varepsilon^{\frac{n}{2}-1}$,
and $R^{\prime}(\varepsilon)\cong\varepsilon^{-\frac{1}{2}}$, we infer that
$G(\varepsilon)\cong\varepsilon^{\frac{n+1}{2}-s}$, which gives the desired
conclusion (2.4). In conclusion, we have shown that

(2.5)
$\left(\int_{\mathbb{R}^{n}}|\hat{f}_{\varepsilon}(\xi)|^{q}A^{(s)}_{1}(\xi)d\xi\right)^{1/q}\cong\varepsilon^{(\frac{n+1}{2}-s)\frac{1}{q}}.$

Combining (2.2) with (2.5) we finally infer that a necessary condition for
(1.7) to hold is

$(\frac{n+1}{2}-s)\frac{1}{q}\geq\frac{n+1}{2p^{\prime}}\ \Longleftrightarrow\
1\leq q\leq\frac{n+1-2s}{n+1}p^{\prime}.$

∎

##  3\. The Fourier transform of the kernel $A^{(s)}_{1}$

In this section we prove Theorem 1.3. Using Lemmas 5.3 and 5.4 in the
Appendix, we establish a result which provides a key step in the proof of
Theorem 1.3.

######  Lemma 3.1.

For every $0<s<1$, $r>0$ and $\xi\in\mathbb{R}^{n}\setminus\\{0\\}$ one has

$\displaystyle\int_{1}^{\infty}\rho^{-\frac{n}{2}}(\rho^{2}-1)^{-s}J_{\frac{n}{2}-1}(2\pi
r|\xi|\rho)\ d\rho$ $\displaystyle=\Gamma(1-s)(2\pi
r|\xi|)^{\frac{n}{2}-1}\left\\{\frac{\Gamma(s)}{2^{\frac{n}{2}}\Gamma(\frac{n}{2})}-\frac{1}{2^{s}\Gamma(s)}\int_{0}^{2\pi
r|\xi|}t^{s-\frac{n}{2}}J_{\frac{n}{2}-1+s}(t)dt\right\\}.$

###### Proof.

To prove Lemma 3.1 we use Lemma 5.3 in which we take

(3.1) $\nu=\frac{n}{2}-1,\ \ \ \ \ \alpha=-\frac{n}{2}+1=-\nu,\ \ \ \ \
\beta=1-s,\ \ \ \ c=2\pi r|\xi|.$

Since $s<1$, the condition $\Re\beta>0$ is guaranteed. Also,
$\alpha+2\beta<\frac{7}{2}$ is equivalent to $\frac{n}{2}+2s>-\frac{1}{2}$,
which is trivially satisfied. Furthermore, we have

$\frac{\alpha+\nu}{2}=0,\ \ \ \ \ \nu+1=\frac{n}{2},\ \ \ \ \ \
\frac{\alpha+\nu}{2}+\beta=1-s,\ \ \ \ 1-\beta-\frac{\alpha+\nu}{2}=s.$

We thus have

$\displaystyle\frac{c^{\nu}\
\Gamma(\beta)\Gamma(1-\beta-\frac{\alpha+\nu}{2})}{2^{\nu+1}\
\Gamma(\nu+1)\Gamma(1-\frac{\alpha+\nu}{2})}\
_{1}F_{2}(\frac{\alpha+\nu}{2};\nu+1,\frac{\alpha+\nu}{2}+\beta;-(\frac{c}{2})^{2})$
$\displaystyle=(2\pi
r|\xi|)^{\frac{n}{2}-1}\frac{\Gamma(1-s)\Gamma(s)}{2^{\frac{n}{2}}\Gamma(\frac{n}{2})}\
_{1}F_{2}(0;\frac{n}{2},1-s;-(\frac{2\pi|\xi|}{2})^{2})=(2\pi
r|\xi|)^{\frac{n}{2}-1}\frac{\Gamma(1-s)\Gamma(s)}{2^{\frac{n}{2}}\Gamma(\frac{n}{2})},$

since from (5.12) we have
${}_{1}F_{2}(0;\frac{n}{2},1-s;-(\frac{2\pi|\xi|}{2})^{2})\equiv 1$. We next
want to write in a more convenient form the second hypergeometric function in
the right-hand side of (5.14) below. With the above choices (3.1), we now have

$1-\beta=s,\ \ \ \ \ 2-\beta-\frac{\alpha+\nu}{2}=s+1,\ \ \ \ \ \
2-\beta+\frac{\nu-\alpha}{2}=s+1+\frac{n}{2}-1=\frac{n}{2}+s,$

and also

$\alpha+2\beta-3=-\frac{n}{2}-2s,\ \ \ \ \ 2-\alpha-2\beta=\frac{n}{2}-1+2s,\
\ \ \ \ \beta+\frac{\alpha+\nu}{2}-1=-s.$

This gives

$\displaystyle\frac{2^{\alpha+2\beta-3}c^{2-\alpha-2\beta}\Gamma(\beta+\frac{\alpha+\nu}{2}-1)}{\Gamma(2-\beta+\frac{\nu-\alpha}{2})}\
_{1}F_{2}(1-\beta;2-\beta-\frac{\alpha+\nu}{2},2-\beta+\frac{\nu-\alpha}{2};-(\frac{c}{2})^{2})$
$\displaystyle=\frac{2^{-\frac{n}{2}-2s}(2\pi
r|\xi|)^{\frac{n}{2}-1+2s}\Gamma(-s)}{\Gamma(\frac{n}{2}+s)}\
_{1}F_{2}(s;s+1,\frac{n}{2}+s;-(\frac{2\pi r|\xi|}{2})^{2})$
$\displaystyle=-\frac{2^{-\frac{n}{2}-2s}(2\pi
r|\xi|)^{\frac{n}{2}-1+2s}\Gamma(1-s)}{s\Gamma(\frac{n}{2}+s)}\
_{1}F_{2}(s;s+1,\frac{n}{2}+s;-(\frac{2\pi r|\xi|}{2})^{2})$

If we now apply Lemma 5.4 with

$a=2\pi r|\xi|,\ \ \ \ c=1,\ \ \ \ \nu=\frac{n}{2}-1+s,\ \ \ \
\frac{\alpha+\nu}{2}=s\ \Longrightarrow\ \alpha=s-\frac{n}{2}+1,$

we obtain

${}_{1}F_{2}(s;s+1,\frac{n}{2}+s;-(\frac{2\pi
r|\xi|}{2})^{2})=\frac{2s2^{\frac{n}{2}-1+s}\Gamma(\frac{n}{2}+s)}{(2\pi
r|\xi|)^{2s}}\int_{0}^{2\pi r|\xi|}t^{s-\frac{n}{2}}J_{\frac{n}{2}-1+s}(t)dt.$

Substituting in the above, and putting everything together, we find

$\displaystyle\int_{1}^{\infty}\rho^{-\frac{n}{2}}(\rho^{2}-1)^{-s}J_{\frac{n}{2}-1}(2\pi
r|\xi|\rho)\ d\rho=$ $\displaystyle\Gamma(1-s)(2\pi
r|\xi|)^{\frac{n}{2}-1}\left\\{\frac{\Gamma(s)}{2^{\frac{n}{2}}\Gamma(\frac{n}{2})}-\frac{1}{2^{s}}\int_{0}^{2\pi
r|\xi|}t^{s-\frac{n}{2}}J_{\frac{n}{2}-1+s}(t)dt\right\\},$

which finally gives the desired conclusion.

∎

###### Proof of Theorem 1.3.

We begin by observing that the left-hand side of (1.14) coincides with the
right-hand side when $\xi=0$. For this, note that on one hand (1.4) gives

$\widehat{A^{(s)}_{r}}(0)=||A^{(s)}_{r}||_{L^{1}(\mathbb{R}^{n})}=1.$

On the other, we apply Lemma 5.1 with

$a=1,\ \ \ \ \ \mu=s-\frac{n}{2},\ \ \ \ \ \nu=\frac{n}{2}+s-1.$

In such case, we have $\mu<\frac{1}{2}$ and also
$\mu+\nu=s-\frac{n}{2}+\frac{n}{2}+s-1=2s-1>-1$, since $s>0$. Since
$\frac{\nu-\mu+1}{2}=\frac{n}{2}$, we thus obtain

(3.2)
$\int_{0}^{\infty}t^{s-\frac{n}{2}}J_{\frac{n}{2}-1+s}(t)dt=2^{s-\frac{n}{2}}\frac{\Gamma(s)}{\Gamma(\frac{n}{2})}.$

This shows that when $\xi=0$ the right-hand side of (1.14) becomes

(3.3)
$\frac{2^{\frac{n}{2}-s}\Gamma(\frac{n}{2})}{\Gamma(s)}\int_{0}^{\infty}t^{s-\frac{n}{2}}J_{\frac{n}{2}-1+s}(t)dt=1,$

and therefore (1.14) does hold in $\xi=0$.

Let now $\xi\not=0$ and recall the well-known formula of Bochner for the
Fourier transform of a spherically symmetric function

(3.4)
$\hat{f}(\xi)=\frac{2\pi}{|\xi|^{\frac{n}{2}-1}}\int_{0}^{\infty}r^{\frac{n}{2}}f^{\star}(r)J(2\pi
r|\xi|)dr,$

see [3, Theor. 40 on p. 69]. Applying (3.4) to (1.1), after a simple change of
variable we find

(3.5) $\displaystyle\widehat{A^{(s)}_{r}}(\xi)$
$\displaystyle=c(n,s)\frac{2\pi}{r^{\frac{n}{2}-1}|\xi|^{\frac{n}{2}-1}}\int_{1}^{\infty}\rho^{-\frac{n}{2}}(\rho^{2}-1)^{-s}J_{\frac{n}{2}-1}(2\pi
r|\xi|\rho)\ d\rho.$

At this point we substitute Lemma 3.1 in (3.5), obtaining

(3.6) $\displaystyle\widehat{A^{(s)}_{r}}(\xi)=(2\pi)^{\frac{n}{2}}\
c(n,s)\Gamma(1-s)\left\\{\frac{\Gamma(s)}{2^{\frac{n}{2}}\Gamma(\frac{n}{2})}-\frac{1}{2^{s}}\int_{0}^{2\pi
r|\xi|}t^{s-\frac{n}{2}}J_{\frac{n}{2}-1+s}(t)dt\right\\}$
$\displaystyle=\frac{\sigma_{n-1}}{2}c(n,s)\
\Gamma(1-s)\Gamma(s)\left\\{1-\frac{2^{\frac{n}{2}-s}\Gamma(\frac{n}{2})}{\Gamma(s)}\int_{0}^{2\pi
r|\xi|}t^{s-\frac{n}{2}}J_{\frac{n}{2}-1+s}(t)dt\right\\}$
$\displaystyle=\frac{\sigma_{n-1}}{2}c(n,s)\
\Gamma(1-s)\Gamma(s)\frac{2^{\frac{n}{2}-s}\Gamma(\frac{n}{2})}{\Gamma(s)}\left\\{\int_{0}^{\infty}t^{s-\frac{n}{2}}J_{\frac{n}{2}-1+s}(t)dt-\int_{0}^{2\pi
r|\xi|}t^{s-\frac{n}{2}}J_{\frac{n}{2}-1+s}(t)dt\right\\},$

where in the last equality we have used (3.3). Using (1.2) in (3.6), we
finally obtain for every $0<s<1$ and $r>0$

(3.7)
$\widehat{A^{(s)}_{r}}(\xi)=\frac{2^{\frac{n}{2}-s}\Gamma(\frac{n}{2})}{\Gamma(s)}\int_{2\pi
r|\xi|}^{\infty}t^{s-\frac{n}{2}}J_{\frac{n}{2}-1+s}(t)dt.$

This completes the proof of Theorem 1.3.

∎

For subsequent purposes, it will be important to have the following
alternative representation of $\widehat{A^{(s)}_{r}}$.

######  Corollary 3.2.

Let $0<s<1$. For every $\xi\in\mathbb{R}^{n}$ we have

(3.8) $\displaystyle\widehat{A^{(s)}_{r}}(\xi)$
$\displaystyle=\frac{2^{\frac{n}{2}-s}\Gamma(\frac{n}{2})}{\Gamma(s)}\left\\{n\int_{2\pi
r|\xi|}^{\infty}t^{s-\frac{n}{2}-1}J_{\frac{n}{2}+s}(t)dt-\frac{1}{(2\pi
r|\xi|)^{\frac{n}{2}-s}}J_{\frac{n}{2}+s}(2\pi r|\xi|)\right\\}.$

The identity (3.8) implies, in particular, the existence of a universal
$C(n,s)>0$, such that as $|\xi|\to\infty$

(3.9)
$\left|\widehat{A^{(s)}_{1}}(\xi)\right|\leq\frac{C(n,s)}{|\xi|^{\frac{n+1}{2}-s}}.$

###### Proof.

The proof of (3.8) follows from (3.7) by applying the recursive formula (5.3)
with $\nu=\frac{n}{2}+s$ and integrating by parts. One has in fact

$\displaystyle\int_{2\pi
r|\xi|}^{\infty}t^{s-\frac{n}{2}}J_{\frac{n}{2}-1+s}(t)dt=\int_{2\pi
r|\xi|}^{\infty}t^{-n}\frac{d}{dt}\left(t^{\frac{n}{2}+s}J_{\frac{n}{2}+s}(t)\right)dt.$

Details are left to the interested reader, who should notice that, since
$s>0$, we now have $s-\frac{n}{2}-1+\frac{n}{2}+s=2s-1>-1$, and therefore the
oscillatory integral
$\int_{0}^{\infty}t^{s-\frac{n}{2}-1}J_{\frac{n}{2}+s}(t)dt$ is convergent
near $t=0$ (and can in fact be explicitly computed via Lemma 5.1). To prove
(3.9) it is enough to observe that when $|\xi|$ is sufficiently large, then by
(5.10) we have for some universal constant $C>0$ and all $t\geq 2\pi|\xi|$,

$|J_{\frac{n}{2}+s}(t)|\leq\frac{C}{t^{1/2}}.$

We thus find for the first term in the right-hand side of (3.8)

$\left|\int_{2\pi|\xi|}^{\infty}t^{s-\frac{n}{2}-1}J_{\frac{n}{2}+s}(t)dt\right|\leq
C\int_{2\pi|\xi|}^{\infty}t^{s-\frac{n}{2}-\frac{3}{2}}dt=\frac{2C}{(n+1-2s)(2\pi|\xi|)^{\frac{n+1}{2}-s}}.$

Since the second term can obviously be estimated in the same way, we are
finished.

∎

The next result provides the limiting value of $\widehat{A^{(s)}_{1}}(\xi)$ as
$s\to 1$. It represents the counterpart on the Fourier transform side of
Proposition 1.1.

######  Lemma 3.3.

Let $n\geq 2$. Then for every $\xi\in\mathbb{R}^{n}$ one has

$\underset{s\to 1}{\lim}\
\widehat{A^{(s)}_{1}}(\xi)=\frac{1}{\sigma_{n-1}}\widehat{d\sigma}(\xi)$

###### Proof.

Notice that for any $\xi\not=0$ and $t>2\pi|\xi|$ we have from (5.9)

$|t^{s-\frac{n}{2}-1}J_{\frac{n}{2}+s}(t)|\leq C(n,\xi)\ t^{-\frac{n+1}{2}}\in
L^{1}(2\pi|\xi|,\infty),$

for some constant $C(n,\xi)>0$ independent of $s\in(0,1)$. By Lebesgue
dominated convergence we thus have

$\underset{s\to
1}{\lim}\int_{2\pi|\xi|}^{\infty}t^{s-\frac{n}{2}-1}J_{\frac{n}{2}+s}(t)dt=\int_{2\pi|\xi|}^{\infty}t^{-\frac{n}{2}}J_{\frac{n}{2}+1}(t)dt=-\int_{2\pi|\xi|}^{\infty}\frac{d}{dt}\left(t^{-\frac{n}{2}}J_{\frac{n}{2}}(t)\right)dt.$

This observation and (3.8) give

$\underset{s\to 1}{\lim}\
\widehat{A^{(s)}_{1}}(\xi)=2^{\frac{n}{2}-1}\Gamma(\frac{n}{2})\frac{1}{(2\pi|\xi|)^{\frac{n}{2}-1}}J_{\frac{n}{2}-1}(2\pi|\xi|).$

In view of (1.17) above, we have reached the desired conclusion when
$\xi\not=0$. When instead $\xi=0$ for the left-hand side of (1.14) we obtain
from (1.4)

$\underset{s\to 1}{\lim}\ \widehat{A^{(s)}_{1}}(0)=\underset{s\to 1}{\lim}\
||A^{(s)}_{1}||_{L^{1}(\mathbb{R}^{n})}=1.$

Convergence to the same limit of the right-hand side follows from (3.3).

∎

##  4\. Proof of Theorem 1.4

In this section we prove Theorem 1.4. The proof will be based on two central
ideas: 1) To exploit the $L^{2}$ nature of the inequality (1.22) via the
Plancherel theorem. This reduces considerations to proving that the nonlocal
Tomas-Stein operator $R^{(s)}$ in (4.1) below maps $L^{p}$ into
$L^{p^{\prime}}$; 2) To accomplish this step, we embed $R^{(s)}$ into an
analytic family of operators $T_{z}$. For the latter we show that

$\begin{cases}T_{-\frac{n-1}{2}+iy}:L^{1}(\mathbb{R}^{n})\ \longrightarrow\
L^{\infty}(\mathbb{R}^{n})\\\ \\\ T_{1+iy}:L^{2}(\mathbb{R}^{n})\
\longrightarrow\ L^{2}(\mathbb{R}^{n}),\end{cases}$

with appropriate bounds on the operator norms.

###### Proof of Theorem 1.4.

Similarly to [22] we write for $f\in\mathscr{S}(\mathbb{R}^{n})$

$\displaystyle\int_{\mathbb{R}^{n}}|\hat{f}(\xi)|^{2}A^{(s)}_{1}(\xi)d\xi=\int_{\mathbb{R}^{n}}\overline{\hat{f}(\xi)}\hat{f}(\xi)A^{(s)}_{1}(\xi)d\xi$
$\displaystyle=\int_{\mathbb{R}^{n}}\overline{\hat{f}(\xi)}\
\hat{f}(\xi)A^{(s)}_{1}(\xi)d\xi=\int_{\mathbb{R}^{n}}\overline{\hat{f}(\xi)}\
\widehat{R^{(s)}f}(\xi)d\xi,$

where we have defined

(4.1) $R^{(s)}f(\xi)=\widehat{A^{(s)}_{1}}\star f(\xi),$

so that $\widehat{R^{(s)}f}(\xi)=\hat{f}(\xi)A^{(s)}_{1}(\xi)$. By Plancherel
we thus obtain

$\displaystyle\int_{\mathbb{R}^{n}}|\hat{f}(\xi)|^{2}A^{(s)}_{1}(\xi)d\xi=\int_{\mathbb{R}^{n}}\overline{f(x)}\
R^{(s)}f(x)dx$
$\displaystyle\leq||f||_{L^{p}(\mathbb{R}^{n})}||R^{(s)}f||_{L^{p^{\prime}}(\mathbb{R}^{n})},$

where in the last inequality we have used Hölder. The proof will be finished
if we can show that for every $0<s<1$ there exists $M_{s}=M_{s}(n)>0$ such
that for $f\in\mathscr{S}(\mathbb{R}^{n})$ one has

(4.2) $||R^{(s)}f||_{L^{p^{\prime}}(\mathbb{R}^{n})}\leq M_{s}\
||f||_{L^{p}(\mathbb{R}^{n})},$

for $p=\frac{2(n+1)}{n+1+2s}$. We want to accomplish (4.2) by interpolating
between the two endpoints $L^{1}\to L^{\infty}$ and $L^{2}\to L^{2}$. This
means we have to choose $\theta\in[0,1]$ such that

$\frac{n+1+2s}{2(n+1)}=\frac{1}{p}=\frac{1-\theta}{1}+\frac{\theta}{2}=1-\frac{\theta}{2},$

which gives

$\frac{\theta}{2}=\frac{1}{p^{\prime}}=\frac{n+1-2s}{2(n+1)},$

and therefore

(4.3) $\theta=\theta(s)=\frac{n+1-2s}{n+1}=1-\frac{2s}{n+1},\ \ \ \ \ \
1-\theta=\frac{2s}{n+1}.$

For $z\in\mathbb{C}$ such that $\Re z\leq 1$ we define a linear operator
$T_{z}:\mathscr{S}(\mathbb{R}^{n})\to\mathbb{C}$ by letting

(4.4) $\widehat{T_{z}f}(\xi)=A^{(1-z)}_{1}(\xi)\hat{f}(\xi),$

where for $\Re z\leq 1$ we have let

(4.5) $A^{(z)}_{1}(x)=\frac{c(n,z)}{(|x|^{2}-1)_{+}^{z}|x|^{n}},$

with $c(n,z)=\frac{2}{\sigma_{n-1}\Gamma(z)\Gamma(1-z)}$. Notice that
$c(n,z)=c(n,1-z)$ and that for $y\in\mathbb{R}$ we have

(4.6)
$||A^{(iy)}_{1}||_{L^{\infty}(\mathbb{R}^{n})}\leq\frac{2}{\sigma_{n-1}|\Gamma(iy)||\Gamma(1-iy)|}.$

From Plancherel theorem, (4.4), (4.6) and (5.1) we conclude for some universal
$C(n)>0$ and for every $y\in\mathbb{R}$

(4.7) $||T_{1+iy}f||_{L^{2}(\mathbb{R}^{n})}\leq M_{1}(y)\
||f||_{L^{2}(\mathbb{R}^{n})},$

with $M_{1}(y)=C(n)\ e^{\pi|y|}$.

We now introduce the kernels

(4.8)
$K_{z}(\xi)=\frac{2^{\frac{n}{2}-1+z}\Gamma(\frac{n}{2})}{\Gamma(1-z)}\bigg{\\{}n\int_{2\pi|\xi|}^{\infty}t^{-z-\frac{n}{2}}J_{\frac{n}{2}+1-z}(t)dt-\frac{1}{(2\pi|\xi|)^{\frac{n}{2}-1+z}}J_{\frac{n}{2}+1-z}(2\pi|\xi|)\bigg{\\}}.$

Notice that, according to (3.8) in Corollary 3.2, when $z=1-s$ we have
$K_{1-s}=\widehat{A^{(s)}_{1}}$, and therefore (4.4) gives

(4.9) $T_{1-s}f(\xi)=\widehat{A^{(s)}_{1}}\star f(\xi)=R^{(s)}f(\xi),$

where in the last equality we have used (4.1). Also notice that, since by
analytic continuation (3.8) continues to be valid for any $z\in\mathbb{C}$ in
the strip $0<\Re z<1$, for any such $z$ we have from (4.4)

(4.10) $T_{z}f=K_{z}\star f.$

Since by (4.8) the kernel $K_{z}$ defines an analytic function of $z$ for
$-\frac{n+1}{2}<\Re z<1$ (see Lemma 5.1), we can use (4.10) to analytically
extend the operator $T_{z}$ to the whole strip
$S=\\{z\in\mathbb{C}\mid-\frac{n-1}{2}<\Re z<1\\}$. If we let $z=x+iy$, then
we define

$S_{x+iy}=T_{-\frac{n-1}{2}+\frac{n+1}{2}x+iy}.$

Note that $S_{z}$ is now defined on the strip $\Sigma=\\{z\in\mathbb{C}\mid
0<\Re z<1\\}$, and that (4.7) now reads

(4.11) $||S_{1+iy}f||_{L^{2}(\mathbb{R}^{n})}\leq C(n)\ \sinh(\pi|y|)\
||f||_{L^{2}(\mathbb{R}^{n})}.$

Since $S_{iy}=T_{-\frac{n-1}{2}+iy}$, we next analyse the behaviour of $T_{z}$
on the line the line $L_{0}=\\{z\in\mathbb{C}\mid\Re z=-\frac{n-1}{2}\\}$.
Notice that (4.3) gives

$(1-\theta)(-\frac{n-1}{2})+\theta\cdot 1=1-s.$

Note that the equation $-\frac{n-1}{2}+\frac{n+1}{2}x=(1-s)$ gives

$x=\frac{(1-s)+\frac{n-1}{2}}{\frac{n+1}{2}}=\frac{n+1-2s}{n+1}=1-\frac{2s}{n+1}=\theta,$

see (4.3). This shows that $S_{\theta}=T_{1-s}$. In view of (4.7) we conclude
that, if we can show that $T_{-\frac{n-1}{2}+iy}:L^{1}(\mathbb{R}^{n})\to
L^{\infty}(\mathbb{R}^{n})$, with appropriate bounds on the operator norms,
then by Stein’s theorem of complex interpolation for an analytic family of
operators, see [16] or [20, Theor. 4.1, p. 205], it will follow that
$T_{1-s}:L^{p}(\mathbb{R}^{n})\to L^{p^{\prime}}(\mathbb{R}^{n})$, as desired.

In order to show that $T_{-\frac{n-1}{2}+iy}:L^{1}(\mathbb{R}^{n})\to
L^{\infty}(\mathbb{R}^{n})$ we will prove that $K_{-\frac{n-1}{2}+iy}\in
L^{\infty}(\mathbb{R}^{n})$ and that moreover for some universal constant
$C>0$ depending only on $n$, one has

(4.12) $||K_{-\frac{n-1}{2}+iy}||_{L^{\infty}(\mathbb{R}^{n})}\leq
Ce^{\frac{3\pi}{2}|y|},\ \ \ \ \ \ \ \ y\in\mathbb{R}.$

From (4.8) we obtain

(4.13)
$K_{-\frac{n-1}{2}+iy}(\xi)=\frac{2^{-\frac{1}{2}+iy}\Gamma(\frac{n}{2})}{\Gamma(\frac{n+1}{2}-iy)}\bigg{\\{}n\int_{2\pi|\xi|}^{\infty}t^{-\frac{1}{2}-iy}J_{n+\frac{1}{2}-iy}(t)dt-\frac{1}{(2\pi|\xi|)^{-\frac{1}{2}+iy}}J_{n+\frac{1}{2}-iy}(2\pi|\xi|)\bigg{\\}}.$

Note that from (5.10) there exists a universal $R=R(n)>0$ such that when
$|\xi|\geq R$ one has

(4.14)
$\left|\frac{1}{(2\pi|\xi|)^{-\frac{1}{2}+iy}}J_{n+\frac{1}{2}-iy}(2\pi|\xi|)\right|\leq
1.$

On the other hand, for $|\xi|\leq R$ we have from (5.8)

(4.15)
$\displaystyle\left|\frac{1}{(2\pi|\xi|)^{-\frac{1}{2}+iy}}J_{n+\frac{1}{2}-iy}(2\pi|\xi|)\right|\leq\frac{\Gamma(n+1)}{|\Gamma(n+1-iy)|\
\Gamma(n+\frac{1}{2}+1)}\frac{\left(2\pi|\xi|\right)^{n+1}}{2^{n+\frac{1}{2}}}$
$\displaystyle\leq\frac{\sqrt{2}\pi^{n+\frac{1}{2}}\Gamma(n)R^{n+1}}{\Gamma(n+\frac{1}{2})\
|\Gamma(n+1-iy)|}.$

We now have for $n\geq 2$

$\displaystyle|\Gamma(n+1-iy)|$
$\displaystyle=|(n-iy)||\Gamma(n-iy)|=|(n-iy)||(n-1-iy)|...|(1-iy)||\Gamma(1-iy)|$
$\displaystyle\geq(1+y^{2})^{\frac{n}{2}}|\Gamma(1-iy)|=(1+y^{2})^{\frac{n}{2}}\sqrt{\frac{\pi|y|}{\sinh\pi|y|}},$

where in the last equality we have used (5.1). This gives

(4.16)
$\frac{1}{|\Gamma(n+1-iy)|}\leq(1+y^{2})^{-\frac{n}{2}}\sqrt{\frac{\sinh\pi|y|}{\pi|y|}}\leq\sqrt{\frac{\sinh\pi|y|}{\pi|y|}}\leq\frac{3}{2}e^{\frac{\pi}{2}|y|}.$

Inserting this information in (4.15), and combining the resulting estimate
with (4.14), we conclude that there exists $C(n)>0$ such that for every
$\xi\in\mathbb{R}^{n}$ and any $y\in\mathbb{R}$ one has

(4.17)
$\left|\frac{1}{(2\pi|\xi|)^{-\frac{1}{2}+iy}}J_{n+\frac{1}{2}-iy}(2\pi|\xi|)\right|\leq
Ce^{\frac{\pi}{2}|y|}.$

Next, we show that for some $C=C(n)>0$ one has for every $\gamma\in\mathbb{R}$

(4.18) $\left|\frac{1}{\Gamma(\frac{n+1}{2}-iy)}\right|\leq
Ce^{\frac{\pi}{2}|y|}.$

To see this we apply Legendre duplication formula, see e.g. (1.2.3) in [12],
to write

$2^{n-2iy}\Gamma(\frac{n+1}{2}-iy)\Gamma(\frac{n}{2}+1-iy)=\sqrt{\pi}\Gamma(n+1-2iy).$

Using the estimate $|\Gamma(z)|\leq|\Gamma(\Re z)|$, this gives

$\displaystyle\left|\frac{1}{\Gamma(\frac{n+1}{2}-iy)}\right|\leq\frac{2^{n}|\Gamma(\frac{n}{2}+1-iy)|}{\sqrt{\pi}|\Gamma(n+1-2iy)|}\leq\frac{2^{n}\Gamma(\frac{n}{2}+1)}{\sqrt{\pi}|\Gamma(n+1-2iy)|}\leq
C(n)e^{\frac{\pi}{2}|y|},$

where in the last inequality we have used (4.16). This proves (4.18). Next, we
show that for every $\xi\in\mathbb{R}^{n}$ and every $y\in\mathbb{R}$

(4.19)
$\left|\int_{2\pi|\xi|}^{\infty}t^{-\frac{1}{2}-iy}J_{n+\frac{1}{2}-iy}(t)dt\right|\leq
C(1+|y|)e^{\frac{\pi}{2}|y|}.$

To prove (4.19) observe that, in view of (5.9) there exists $R>0$ depending on
$n$ such that for $|\xi|\geq R$ we have for $t\in[2\pi|\xi|,\infty)$

$\left|J_{n+\frac{1}{2}-iy}(t)-\sqrt{\frac{2}{\pi
t}}\cos\left(t-\frac{n+1}{2}\pi+i\frac{\pi}{2}y\right)\right|\leq t^{-3/2}.$

Since

$\displaystyle\cos\left(t-\frac{n+1}{2}\pi+i\frac{\pi}{2}y\right)=\cos(t+i\frac{\pi}{2}y)\cos(\frac{n+1}{2}\pi)+\sin(t+i\frac{\pi}{2}y)\sin(\frac{n+1}{2}\pi)$
$\displaystyle=\begin{cases}(-1)^{n+1}\sin(t+i\frac{\pi}{2}\gamma),\ \ \ \ n\
\text{even},\\\ (-1)^{n}\cos(t+i\frac{\pi}{2}\gamma),\ \ \ \ n\
\text{odd},\end{cases}$

with obvious meaning of the notation, we infer

$\displaystyle\left|\int_{2\pi|\xi|}^{\infty}t^{-\frac{1}{2}-i\gamma}J_{n+\frac{1}{2}-i\gamma}(t)dt\right|\leq\left|\sqrt{\frac{2}{\pi}}\int_{2\pi|\xi|}^{\infty}\frac{\begin{pmatrix}\sin(t+i\frac{\pi}{2}\gamma)\\\
\cos(t+i\frac{\pi}{2}\gamma)\end{pmatrix}}{t^{1+i\gamma}}dt\right|+\int_{2\pi|\xi|}^{\infty}t^{-2}dt.$

Keeping in mind that for $z=x+iy$ we have

$\cos z=\cos x\cosh y-i\sin x\sinh y,\ \ \ \ \sin z=\sin x\cosh y+i\cos x\sinh
y,$

and that integrating by parts we obtain

$\left|\int_{2\pi|\xi|}^{\infty}\frac{\begin{pmatrix}\sin t\\\ \cos
t\end{pmatrix}}{t^{1+iy}}dt\right|\leq C\frac{1+|y|}{|\xi|}\leq C(1+|y|),$

we conclude that (4.19) holds when $|\xi|\geq R$. If instead $|\xi|\leq R$,
then we write

$\displaystyle\int_{2\pi|\xi|}^{\infty}t^{-\frac{1}{2}-iy}J_{n+\frac{1}{2}-iy}(t)dt=\int_{0}^{\infty}t^{-\frac{1}{2}-iy}J_{n+\frac{1}{2}-iy}(t)dt-\int_{0}^{2\pi|\xi|}t^{-\frac{1}{2}-iy}J_{n+\frac{1}{2}-iy}(t)dt,$

and then use Lemma 5.1 and (5.8) to estimate

$\displaystyle\left|\int_{2\pi|\xi|}^{\infty}t^{-\frac{1}{2}-iy}J_{n+\frac{1}{2}-iy}(t)dt\right|\leq\frac{|\Gamma(\frac{n}{2}-iy)|}{\sqrt{2}\Gamma(\frac{n}{2}+1)}+\frac{C(n)R^{n+2}}{|\Gamma(n-iy+\frac{1}{2})|}\leq
C(n)\left(1+\frac{1}{|\Gamma(n-iy+\frac{1}{2})|}\right).$

Using again Legendre duplication formula, similarly to the proof of (4.18) we
recognise

$\frac{1}{|\Gamma(n-iy+\frac{1}{2})|}\leq C(n)e^{\frac{\pi}{2}|y|}.$

If we now use (4.17), (4.18) and (4.19) in (4.13), we conclude that (4.12)
does hold. It follows that for every $f\in\mathscr{S}(\mathbb{R}^{n})$ and any
$y\in\mathbb{R}$ one has

(4.20) $||T_{-\frac{n-1}{2}+iy}f||_{L^{\infty}(\mathbb{R}^{n})}\leq
M_{0}(y)||f||_{L^{1}(\mathbb{R}^{n})},$

with $M_{0}(y)=Ce^{\frac{3\pi}{2}|y|}$. By [20, Theor. 4.1 on p. 205] we infer
that there exists $M_{s}=M_{\theta(s)}>0$ such that (4.2) holds. This proves
Theorem 1.4.

∎

From p. 209 in [20] we see that the constant $M_{s}$ is given by

$M_{s}=\exp\left[\frac{\sin\pi\theta(s)}{2}\int_{\mathbb{R}}\left\\{\frac{\log
M_{0}(y)}{\cosh\pi y-\cos\pi\theta(s)}+\frac{\log M_{1}(y)}{\cosh\pi
y+\cos\pi\theta(s)}\right\\}dy\right],$

From (4.3) we have

$\sin\pi\theta(s)=\sin(\pi-\frac{2\pi s}{n+1})=\sin(\frac{2\pi s}{n+1}),\ \ \
\ \ \ \cos\pi\theta(s)=-\cos(\frac{2\pi s}{n+1}).$

We thus find

(4.21) $M_{s}=\exp\left[\sin(\frac{2\pi
s}{n+1})\int_{0}^{\infty}\left\\{\frac{\log M_{0}(y)}{\cosh\pi
y+\cos(\frac{2\pi s}{n+1})}+\frac{\log M_{1}(y)}{\cosh\pi y-\cos(\frac{2\pi
s}{n+1})}\right\\}dy\right].$

Since $\cos(\frac{2\pi}{n+1})\not=\pm 1$ for any $n\geq 2$, by Lebesgue
dominated convergence we find

(4.22) $\displaystyle\underset{s\to 1}{\lim}\
M_{s}=\exp\left[\sin(\frac{2\pi}{n+1})\int_{0}^{\infty}\left\\{\frac{\log
M_{0}(y)}{\cosh\pi y+\cos(\frac{2\pi}{n+1})}+\frac{\log M_{1}(y)}{\cosh\pi
y-\cos(\frac{2\pi}{n+1})}\right\\}dy\right]$ $\displaystyle=M(n)\ <\ \infty.$

Since the function $s\to\frac{2(n+1)}{n+1+2s}$ is decreasing on $(0,1)$, with
range $(\frac{2(n+1)}{n+3},2)$, if now $1\leq p\leq\frac{2(n+1)}{n+3}$, then
for any $s\in(0,1)$ we also have $1\leq p\leq\frac{2(n+1)}{n+1+2s}$. From the
proof of Theorem 1.4 we infer for every $f\in\mathscr{S}(\mathbb{R}^{n})$ one
has

(4.23)
$\left(\int_{\mathbb{R}^{n}}|\hat{f}(\xi)|^{2}A^{(s)}_{1}(\xi)d\xi\right)^{1/2}\leq\sqrt{M_{s}}\
||f||_{L^{p}(\mathbb{R}^{n})},$

with $M_{s}$ give by (4.21) above. Passing to the limit for $s\to 1$ in
(4.23), and using Proposition 1.1 and (4.22), we conclude the celebrated
Tomas-Stein theorem for the sphere

$\left(\frac{1}{\sigma_{n-1}}\int_{\mathbb{S}^{n-1}}|\hat{f}(\xi)|^{2}d\sigma(\xi)\right)^{1/2}\leq\sqrt{M(n)}\
||f||_{L^{p}(\mathbb{R}^{n})}.$

##  5\. Appendix: some classical material

In this section we collect some results needed in this paper. We recall the
gamma function

$\Gamma(z)=\int_{0}^{\infty}t^{z-1}e^{-t}dt,\ \ \ \ \ \ \ \ \ \Re z>0.$

Using the well-known formula $\Gamma(z+1)=z\Gamma(z)$ this function can be
extended as a meromorphic function to the whole $\mathbb{C}$ having simple
poles in $z=-k$, $k=0,1,...$, with residues
$\operatorname{Res}(\Gamma;-k)=\frac{(-1)^{k}}{k!}$. The reciprocal gamma
function $1/\Gamma(z)$ is an entire function with zeros at the negative
integers. The following special values of such function will be useful
subsequently, see e.g. formulas 6.1.29 and 6.1.31 on p. 256 in [1],

(5.1) $\frac{1}{|\Gamma(\pm iy)|}=|y|\sqrt{\frac{\sinh(\pi|y|)}{\pi|y|}},\ \ \
\ \ \ \ \frac{1}{|\Gamma(1\pm iy)|}=\sqrt{\frac{\sinh(\pi|y|)}{\pi|y|}}.$

Since this note is about the Fourier transform, Bessel functions (introduced
and developed in the seminal work [2]) play a pervasive role in it. We recall
the series expansion of the Bessel function $J_{\nu}(z)$ for $|\arg z|<\pi$,

(5.2)
$J_{\nu}(z)=\left(\frac{z}{2}\right)^{\nu}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{\Gamma(\nu+k+1)k!}\left(\frac{z}{2}\right)^{2k},$

see [23]. From (5.2) it is easy to obtain the following well-known formulas

(5.3) $\frac{d}{dz}\left[z^{\nu}J_{\nu}(z)\right]=z^{\nu}J_{\nu-1}(z),$ (5.4)
$-\frac{d}{dz}\left[z^{-\nu}J_{\nu}(z)\right]=z^{-\nu}J_{\nu+1}(z),$

and

(5.5) $\frac{2\nu}{z}J_{\nu}(z)-J_{\nu+1}(z)=J_{\nu-1}(z).$

For every $\nu\in\mathbb{C}$ such that $\Re\nu>-\dfrac{1}{2}$ the function
$J_{\nu}$ admits the following Poisson representation

(5.6)
$J_{\nu}(z)=\frac{1}{\Gamma(\frac{1}{2})\Gamma(\nu+\frac{1}{2})}\left(\frac{z}{2}\right)^{\nu}\int^{1}_{-1}e^{izs}(1-s^{2})^{\frac{2\nu-1}{2}}ds,$

where $z$ ranges in the complex plane cut along the negative real axis
$(-\infty,0]$. When $\nu\in\mathbb{N}_{0}$ then we can take $z\in\mathbb{C}$.
From (5.2) one easily recognises that

(5.7)
$J_{\nu}(z)\cong\frac{1}{\Gamma(\nu+1)}\left(\frac{z}{2}\right)^{\nu},\quad\text{as
}z\to 0.$

Again from (5.6) one sees that when $z\in\mathbb{R}$ and $z>0$, then for
$\Re\nu>-\dfrac{1}{2}$ one has

(5.8)
$|J_{\nu}(z)|\leq\frac{\Gamma(\Re\nu+\frac{1}{2})}{|\Gamma(\nu+\frac{1}{2})|\
\Gamma(\Re\nu+1)}\left(\frac{z}{2}\right)^{\Re\nu}.$

This estimate is not useful for large $z>0$ since $J_{\nu}(z)$ decays at
infinity with an oscillatory behaviour. When $\Re\nu>-\dfrac{1}{2}$ the
following result due to Hankel holds, see for instance [20, Lemma 3.11], or
also (5.11.6) on p. 122 in [12]. One has for $0<\delta<\pi$

(5.9) $\displaystyle J_{\nu}(z)$ $\displaystyle=\sqrt{\frac{2}{\pi
z}}\cos\left(z-\frac{\pi\nu}{2}-\frac{\pi}{4}\right)+O(z^{-\frac{3}{2}})$
$\displaystyle\quad\text{as }|z|\to\infty,\quad-\pi+\delta<\arg z<\pi-\delta.$

In particular,

(5.10) $J_{\nu}(z)=O(z^{-1/2}),\ \ \ \ \ \text{as}\ z\to+\infty.$

The following beautiful formula can be found in 6.561.14 on p. 684 of [10], or
also (19) on p. 49 in vol.2 of [5].

######  Lemma 5.1.

Let $a>0$, $-\Re\nu-1<\Re\mu<\frac{1}{2}$. Then

$\int_{0}^{\infty}t^{\mu}J_{\nu}(at)dt=2^{\mu}a^{-\mu-1}\frac{\Gamma(\frac{\nu+\mu+1}{2})}{\Gamma(\frac{\nu-\mu+1}{2})}.$

We next recall the definition of the hypergeometric functions. The Pochammer’s
symbols are defined by

(5.11) $\alpha_{0}=1,\ \ \ \
\alpha_{k}\overset{def}{=}\frac{\Gamma(\alpha+k)}{\Gamma(\alpha)}=\alpha(\alpha+1)...(\alpha+k-1),\
\ \ \ \ \ \ \ \ \ k\in\mathbb{N}.$

Notice that since the gamma function has a pole in $z=0$, we have

$0_{k}=\begin{cases}1\ \ \ \ \ \ \text{if}\ k=0\\\ 0\ \ \ \ \ \ \text{for}\
k\geq 1.\end{cases}$

######  Definition 5.2.

Let $p,q\in\mathbb{N}\cup\\{0\\}$ be such that $p\leq q+1$, and let
$\alpha_{1},...,\alpha_{p}$ and $\beta_{1},...,\beta_{q}$ be given parameters
such that $-\beta_{j}\not\in\mathbb{N}_{0}$ for $j=1,...,q$. Given a number
$z\in\mathbb{C}$, the power series

${}_{p}F_{q}(\alpha_{1},...,\alpha_{p};\beta_{1},...,\beta_{q};z)=\sum_{k=0}^{\infty}\frac{(\alpha_{1})_{k}...(\alpha_{p})_{k}}{(\beta_{1})_{k}...(\beta_{q})_{k}}\frac{z^{k}}{k!}$

is called the _generalized hypergeometric function_. When $p=2$ and $q=1$,
then the function ${}_{2}F_{1}(\alpha_{1},\alpha_{2};\beta_{1};z)$ is the
_Gauss’ hypergeometric function_ , and it is usually denoted by
$F(\alpha_{1},\alpha_{2};\beta_{1};z)$.

Using the ratio test one easily verifies that the radius of convergence of the
above hypergeometric series is $\infty$ when $p\leq q$, whereas it equals $1$
when $p=q+1$. Thus for instance it is $1$ for Gauss’ hypergeometric function
$F(\alpha_{1},\alpha_{2};\beta_{1};z)$. For later reference we record the
following facts that follow easily from Definition 5.2:

(5.12) $_{p}F_{q}(0,\alpha_{2},...,\alpha_{p};\beta_{1},...,\beta_{q};z)=1,$

and (see also p. 275 in [12])

(5.13) $F(\alpha,\beta;\beta;-z)=\ _{1}F_{0}(\alpha;-z)=(1+z)^{-\alpha}.$

The following result plays a key role in this work. The interested reader can
find it in formula 2.12.4.16 on p. 178 in [13].

######  Lemma 5.3.

Let $c>0$, $\Re\beta>0$, $\Re(\alpha+2\beta)<\frac{7}{2}$. Then

(5.14)
$\displaystyle\int_{1}^{\infty}\rho^{\alpha-1}(\rho^{2}-1)^{\beta-1}J_{\nu}(c\rho)d\rho$
$\displaystyle=\frac{c^{\nu}\
\Gamma(\beta)\Gamma(1-\beta-\frac{\alpha+\nu}{2})}{2^{\nu+1}\
\Gamma(\nu+1)\Gamma(1-\frac{\alpha+\nu}{2})}\
_{1}F_{2}(\frac{\alpha+\nu}{2};\nu+1,\frac{\alpha+\nu}{2}+\beta;-(\frac{c}{2})^{2})$
$\displaystyle+\frac{2^{\alpha+2\beta-3}c^{2-\alpha-2\beta}\Gamma(\beta+\frac{\alpha+\nu}{2}-1)}{\Gamma(2-\beta+\frac{\nu-\alpha}{2})}\
_{1}F_{2}(1-\beta;2-\beta-\frac{\alpha+\nu}{2},2-\beta+\frac{\nu-\alpha}{2};-(\frac{c}{2})^{2}).$

In order to fully exploit Lemma 5.3, we next derive a useful representation
formula of a certain hypergeometric function ${}_{1}F_{2}$ in terms of an
integral involving a Bessel function. We stress that, although for convenience
we have used the same letters, the parameters $\alpha,\nu$ in the statement of
the next result are not the same as those in Lemma 5.3.

######  Lemma 5.4.

Let $a,c>0$ and $\Re(\alpha+\nu)>0$. Then

$\
{}_{1}F_{2}\left(\frac{\alpha+\nu}{2};\frac{\alpha+\nu}{2}+1,\nu+1;-(\frac{ac}{2})^{2}\right)=\frac{(\alpha+\nu)2^{\nu}\Gamma(\nu+1)}{a^{\alpha+\nu}c^{\nu}}\int_{0}^{a}t^{\alpha-1}J_{\nu}(ct)dt.$

###### Proof.

Performing the change of variable $\tau=ct$ and using (5.2) we find

$\displaystyle\int_{0}^{a}t^{\alpha-1}J_{\nu}(ct)dt=c^{-\alpha}\int_{0}^{ac}\tau^{\alpha-1}J_{\nu}(\tau)d\tau$
$\displaystyle=2^{\alpha}c^{-\alpha}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{\Gamma(\nu+k+1)k!}\int_{0}^{ac}\left(\frac{\tau}{2}\right)^{2(k+\frac{\alpha+\nu}{2})}\frac{d\tau}{\tau}.$

The change of variable $z=\left(\frac{\tau}{2}\right)^{2}$, for which
$\frac{dz}{z}=2\frac{d\tau}{\tau}$, gives

$\displaystyle\int_{0}^{a}t^{\alpha-1}J_{\nu}(ct)dt=2^{\alpha-1}c^{-\alpha}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{\Gamma(\nu+k+1)k!}\int_{0}^{\left(\frac{ac}{2}\right)^{2}}z^{k+\frac{\alpha+\nu}{2}}\frac{dz}{z}$
$\displaystyle=2^{\alpha-1}c^{-\alpha}\left(\frac{ac}{2}\right)^{\alpha+\nu}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(\frac{\alpha+\nu}{2}+k)\Gamma(\nu+k+1)k!}\left(\frac{ac}{2}\right)^{2k}$
$\displaystyle=\frac{a^{\alpha+\nu}c^{\nu}}{2^{\nu+1}}\sum_{k=0}^{\infty}\frac{(\frac{\alpha+\nu}{2})_{k}}{(\frac{\alpha+\nu}{2}+1)_{k}(\nu+1)_{k}k!}\left(\frac{ac}{2}\right)^{2k}\frac{\Gamma(\frac{\alpha+\nu}{2})}{(\frac{\alpha+\nu}{2}+k)\Gamma(\frac{\alpha+\nu}{2}+k)}\frac{\Gamma(\frac{\alpha+\nu}{2}+k+1)}{\Gamma(\frac{\alpha+\nu}{2}+1)}\frac{\Gamma(\nu+k+1)}{\Gamma(\nu+1)\Gamma(\nu+k+1)}$
$\displaystyle=\frac{a^{\alpha+\nu}c^{\nu}}{(\alpha+\nu)2^{\nu}\Gamma(\nu+1)}\
_{1}F_{2}\left(\frac{\alpha+\nu}{2};\frac{\alpha+\nu}{2}+1,\nu+1;-(\frac{ac}{2})^{2}\right),$

where the details of the last equality are left to the reader.

∎

## References

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#  dlmontepython: A Python library for automation and analysis of Monte Carlo
molecular simulations

T. L. Underwood  J. A. Purton  J. R. H. Manning  A. V. Brukhno  K. Stratford
T. Düren  N. B. Wilding  S. C. Parker  Department of Chemistry, University of
Bath, Bath BA2 7AY, United Kingdom  Scientiﬁc Computing Department, STFC,
Daresbury Laboratory, Keckwick Lane, Warrington WA4 4AD, United Kingdom
Department of Chemical Engineering, University of Bath, Bath BA2 7AY, United
Kingdom  Department of Chemistry, University College London, 20 Gordon Street,
London WC1H 0AJ, United Kingdom  EPCC, University of Edinburgh, EH9 3FD,
Edinburgh, United Kingdom  H.H. Wills Physics Laboratory, University of
Bristol, Royal Fort, Bristol BS8 1TL, United Kingdom

###### Abstract

We present an open source Python 3 library aimed at practitioners of molecular
simulation, especially Monte Carlo simulation. The aims of the library are to
facilitate the generation of simulation data for a wide range of problems; and
to support data analysis methods which enable one to make the most of
previously generated data. The library contains a framework for automating the
task of measuring target physical properties (e.g. density) over a range of
thermodynamic parameters (e.g. temperature) calculated using a molecular
simulation program, in particular the Monte Carlo program DL_MONTE. The
library also supports analysis methods including block averaging,
equilibration detection and histogram reweighting. Here we describe the
library and provide examples to demonstrate its key functionality: we use the
library to automatically calculate isotherms to a specified precision; and to
calculate the surface tension and liquid-vapour coexistence properties of
methane.

###### keywords:

Molecular simulation , Monte Carlo , molecular dynamics , adsorption , free
energy , phase transition

Current code version |  v0.1.0
---|---
Permanent link to code/repository used for this code version |  https://gitlab.com/dl_monte/dlmontepython
Code Ocean compute capsule |  N/A
Legal Code License |  3-clause BSD
Code versioning system used |  git
Software code languages, tools, and services used |  Python 3
Compilation requirements, operating environments & dependencies |  Various Python packages: numpy, scipy, matplotlib, PyYAML
If available Link to developer documentation/manual |  See https://dl_monte.gitlab.io/dlmontepython/ for API documentation, and https://gitlab.com/dl_monte/dlmontepython for links to further documentation.
Support email for questions |<EMAIL_ADDRESS>
Table 1: Code metadata

##  1 Motivation and significance

Molecular simulation[1, 2] is used widely in many fields of the physical
sciences to study materials and fluids at prescribed thermodynamic parameters,
e.g. temperature and pressure. The technique entails setting up a virtual
representation of the system and evolving the system using one of a number of
algorithms. There are two classes of algorithm for evolving the system:
molecular dynamics (MD) and Monte Carlo (MC)[1, 2, 3]. In MD the positions of
the particles in the system are evolved through time by integrating Newton’s
equations of motion. By contrast, in MC the particle motion is stochastic: at
each timestep a Markov chain is employed to generate a new state of the system
from the current state. Of the two methods, MD is the most widely used.
However, MC is the preferred method in certain key situations[4]. For
instance, grand-canonical MC (GCMC), in which the Markov chain involves
inserting and deleting particles from the system, is widely used for studying
the liquid-vapour transition[5, 6, 3], and quantifying adsorption at surfaces
and in porous materials[3].

For both MC and MD simulations the key output is a time series 111For MC
simulations the ‘time’ in the time series is the number of elapsed MC moves,
rather than the physical quantity time. of physical quantities (e.g. the
energy or density of the system), and a common task is to extract the value
and uncertainty of a given physical quantity from the time series.
Unfortunately, this task is rendered nontrivial due to correlation in the time
series[7, 8]. For instance, data from near the start of the time series,
before the so-called _equilibration time_ , must be disregarded because it is
still correlated with the initial configuration of the system, the result
being that the data does not reflect thermodynamic equilibrium. Fortunately,
methods exist to deal with these issues[7, 9, 10].

Methods also exist which enable one to calculate the values of physical
quantities at thermodynamic parameters which are _different_ to those used in
the simulation which generated the data. Such _reweighting methods_[11] are
powerful because they make the most of the data one has already generated, and
hence reduce the computational cost associated with evaluating how a physical
quantity depends on thermodynamic parameters.

There currently exist a plethora of well-established programs for performing
MD and MC simulations (e.g. [12, 13, 14, 15, 16, 17, 18] for MD and [19, 20,
21, 22, 23, 24] for MC). This includes the general-purpose MC program
DL_MONTE[25, 19], which can apply various MC techniques to a wide range of
systems. It is desirable to have software which can interface with these
programs and facilitate the development of high-level scripts which automate
common simulation tasks, such as the calculation of an isotherm. It is also
desirable to have software which can evaluate uncertainties and figures of
merit from simulation outputs. While much such software exists[26, 27, 28],
the focus is very much on MD; less software exists which can deal with the
idiosyncrasies of MC simulation, e.g. the fact that in GCMC the number of
molecules in the simulation can vary.

With this in mind we have developed a Python library, dlmontepython[29]. The
library contains modules for applying various data analysis methods pertinent
to molecular simulation, including methods for calculating uncertainties and
equilibration times, and a number of reweighting methods. The library also
contains a framework for automating simulation tasks, such as the calculation
of an isotherm to a given precision, using DL_MONTE. Moreover this framework
is sufficiently general that it could easily be adapted to work with other
molecular simulation programs. Below we describe the library in more detail,
and provide illustrative examples demonstrating its key functionality.

##  2 Software Description

###  2.1 Overview and architecture

dlmontepython is housed on GitLab at [29], and is released under the 3-clause
BSD license. The source code is written in Python 3, and documentation is
provided for the library in the form of Jupyter notebook[30] tutorials,
example scripts, and API documentation. dlmontepython depends only on standard
packages and libraries which are widely available, in particular scipy[31] and
numpy[32]. Moreover, releases of dlmontepython can be obtained from PyPI using
pip in the standard manner[33].

dlmontepython is currently comprised of two major packages, simtask and htk,
and a number of miscellaneous stand-alone modules. Versions of the htk package
and various stand-alone modules existed before the creation of dlmontepython.
This code has been subsumed into dlmontepython and improved upon.
Functionality of old components of dlmontepython has been described in a
previous work[19]. For this reason we only provide a short description of
these components here; in Sections 2.2, 2.3 and 2.4 we describe the _new_
functionality of dlmontepython in detail.

####  2.1.1 simtask package

simtask houses a framework for automating simulation tasks using molecular
simulation programs (not only DL_MONTE), including the task of calculating a
physical property and its uncertainty over a range of thermodynamic
parameters. This package includes a module, simtask.analysis, for analysing
equilibration and correlation in time series from molecular simulation, and
applying the block averaging method to calculate uncertainties.

####  2.1.2 htk package

The htk package houses a Python interface to DL_MONTE, including modules for
constructing and manipulating DL_MONTE input files. The htk package also
includes modules for applying the single[34] and multiple[35] histogram
reweighting methods.

####  2.1.3 Stand-alone modules

The stand-alone modules also house functionality for applying reweighting
methods. The module fep contains functions for performing simple manipulations
of free energy profiles, including applying single histogram reweighting to
locate coexistence between two phases in certain special cases, in particular
liquid-vapour coexistence. Moreover, the module dlm_wham facilitates the
application of the WHAM method[36] to the output of DL_MONTE free energy
simulations.

###  2.2 simtask: Framework for automation of simulation tasks

We now describe the new functionality of dlmontepython in further detail,
beginning with simtask.

In order to calculate the value of a certain physical quantity $X$ (e.g. the
energy or density of the system) to a given precision using molecular
simulation, the following workflow is typical:

  1. 1.

Run a simulation of a prescribed length.

  2. 2.

Extract a value and uncertainty for $X$ using the time series for $X$ output
by the simulation.

  3. 3.

Check whether or not the uncertainty in $X$ is less than the desired
precision. If it is, then the task is complete. If it is not, then return to
step 1: perform another simulation, starting where the previous one left off,
extending the time series for $X$.

One could also imagine extending this workflow to repeat the above at
different thermodynamic parameters (e.g. temperature).

The simtask package facilitates the automation of such workflows using a given
molecular simulation program. The key components of the package are a module
for performing the relevant time series analysis, and a collection of Python
classes which serve as the framework for automation. We elaborate on these
below.

####  2.2.1 simtask.analysis: Time series analysis

simtask.analysis contains functions for performing time series analysis
relevant to step 2 in the above workflow. Functions are included to:

  * 1.

Determine whether a time series pertaining to a physical quantity $X$ has
equilibrated, and if it has, determine a reasonable value for the
equilibration time using a heuristic method we have developed. 222Details of
the method can be found in the source code documentation.

  * 2.

Apply the block averaging method to obtain a value and uncertainty for $X$
from the time series.

  * 3.

Analyse the correlations in the data: calculate quantities such as the
autocorrelation function, statistical inefficiency, and the autocorrelation
time, which can be used to inform the block size used in block averaging.

These functions take arrays as their input, and as such are applicable to
general time series obtained from molecular simulation.

####  2.2.2 Classes defining the automation framework

The rest of the simtask package is a collection of classes which constitute a
framework for automating the process of calculating a specified physical
quantity and its uncertainty – drawing upon the functions in simtask.analysis
just described. The framework allows the user to specify one or more physical
quantities to be evaluated, and either the precisions to which they are to be
evaluated to or a maximum wall-clock time to dedicate to the calculation. It
will then perform back-to-back simulations until enough data is gathered such
that the physical quantities are determined to the desired precisions or the
maximum wall-clock time has elapsed. The framework also allows this task to be
repeated for a range of thermodynamic parameters (e.g. temperature, pressure)
– as demonstrated below.

The key ingredients of the framework are the abstract Python classes Task and
TaskInterface. The former class represents a simulation workflow to achieve a
particular aim. Subclasses of Task included in simtask are Measurement, which
calculates one or more physical quantities and their uncertainties at a given
thermodynamic parameter; and MeasurementSweep, which applies Measurement to a
specified range of thermodynamic parameters. Each Task instance has a
TaskInterface attribute which defines how to perform common simulation tasks,
such as run a simulation, alter simulation input variables, and extract data.
The TaskInterface instance is specific to the simulation program to be
employed by the framework, e.g. DL_MONTE; for each program, a TaskInterface
interface must be written which tells Task instances how to perform the common
tasks. A TaskInterface class corresponding to DL_MONTE is included in simtask:
DLMonteInterface. Calling the run function of a Task instance instigates the
workflow corresponding to the Task.

####  2.2.3 Example: Automated calculation of an isotherm

Figure 1: Isotherms for the Lennard-Jones fluid automatically generated using
the simtask framework.

The most widely used model for fluids in molecular simulation is the Lennard-
Jones fluid, a system of particles interacting via the Lennard-Jones potential

$\phi(r)=4\epsilon\biggl{[}\Bigl{(}\frac{\sigma}{r}\Bigr{)}^{12}-\Bigl{(}\frac{\sigma}{r}\Bigr{)}^{6}\biggr{]},$
(1)

where $\epsilon$ and $\sigma$ are the free parameters in the potential. Figure
1 shows the output of the simtask framework when tasked to evaluate the
$k_{B}T/\epsilon=$2, 3 and 5 isotherms of the Lennard-Jones fluid, using GCMC
in DL_MONTE as the underlying simulation method. 333$k_{B}$ denotes the
Boltzmann constant, and $T$ denotes temperature. 444The simulations employed a
cubic box of dimension $8\sigma$, a cut-off radius for the potential of
$2.5\sigma$, and standard long-range corrections[1] were applied. The figure
shows $\rho\sigma^{3}$ versus $\ln z$, where $z$ denotes the thermodynamic
activity 555$z$ is defined as $z\equiv\exp(\mu/(k_{B}T))$, where $\mu$ is the
chemical potential. and $\rho$ denotes the density, i.e. the number of
particles per unit volume. The framework was used to calculate
$\rho\sigma^{3}$ at each set of thermodynamic parameters to a precision of
0.02. The Python code given in Figure 2 demonstrates how the framework was
used to calculate each isotherms in Figure 1. This code was invoked in a
directory containing DL_MONTE input files corresponding to a GCMC simulation
at the target temperature.

⬇

import dlmontepython.simtask.dlmonteinterface as interface

1001import dlmontepython.simtask.task as task

import dlmontepython.simtask.measurement as measurement

1003import numpy as np

1005# Array containing thermodynamic activities to examine

activities = np.logspace(-3.0, 1.3333333333, 26)

1007# Threshold precision for density at each activity

density_prec = 0.02

1009# Volume used in the simulations (used to convert ’density_prec’

# to corresponding precision for number of molecules)

1011volume = 8.0**3

1013# Create interface to DL_MONTE executable

interface = interface.DLMonteInterface(”DLMONTE-SRL.X”)

1015

# Create class corresponding to property to measure

1017# (observable): the number of molecules, N, in the system.

# The density can be obtained from N by dividing it by

1019# ’volume’.

nmol_obs = task.Observable( (”nmol”,0) )

1021observables = [ nmol_obs ]

1023# Dictionary storing threshold precisions for observables,

# i.e. threshold precision for N

1025prec= { nmol_obs : density_prec*volume }

1027# Object defining nature of calculation at each activity

M = measurement.Measurement(

1029 interface, observables, precisions=prec)

1031# Object defining nature of calculations over all

# activities

1033MS = measurement.MeasurementSweep(

param=”molchempot”, paramvalues=activities,

1035 measurement_template=M, outputdir=”isotherm”)

1037# Invoke framework, generating output of N vs. activity

MS.run()

Figure 2: Python code used to generate an isotherm in Figure 1.

###  2.3 htk.multihistogram: Multiple histogram reweighting

simtask can be used to automatically generate data for specified thermodynamic
parameters. We now turn to functionality of dlmontepython which enables one to
make the most of such data.

Histogram reweighting[11] is a powerful method which allows simulation data
obtained at certain thermodynamic parameters to be used to make predictions at
nearby thermodynamic parameters not explored by the simulation. Single
histogram reweighting[34] is the simplest incarnation of the method, and
involves using data obtained from a single simulation. Multiple histogram
reweighting (MHR)[35] is its generalisation in which data from _multiple_
simulations at different thermodynamic parameters can be pooled and used to
make predictions at thermodynamic parameters not explored by any of the
simulations. We now describe the MHR functionality, which is housed in the
htk.multihistogram module.

####  2.3.1 Methodology

The abstraction in this module is designed so that MHR can be applied to a
wide range of thermodynamic ensembles – following the approach taken
previously with the single histogram functionality[19]. To elaborate, a MHR
function is provided for applying to thermodynamic ensembles with the general
form

$p_{i}\propto\exp(\mathbf{b}\cdot\mathbf{X}_{i}),$ (2)

where $p_{i}$ is the probability of a configuration $i$, where $\mathbf{b}$ is
a vector of thermodynamic parameters, $\mathbf{X}_{i}$ is a vector of physical
quantities for $i$ ‘conjugate’ to those in $\mathbf{b}$, and ‘$\cdot$’ denotes
the conventional dot product between a pair of vectors. This form covers the
most common thermodynamic ensembles used in molecular simulation, including
the canonical (constant number of particles $N$, volume $V$ and temperature
$T$), isothermal-isobaric ($N$, pressure $P$, $T$) and grand-canonical
(chemical potential $\mu$, $V$, $T$) ensembles. For example, in the grand-
canonical ensemble

$p_{i}\propto\exp\bigl{[}-(\beta E_{i}+\beta\mu N_{i})\bigr{]},$ (3)

where $\beta\equiv 1/(k_{B}T)$, $E_{i}$ is the energy of configuration $i$,
and $N_{i}$ is the number of particles in $i$. This can be recovered from the
generalised ensemble [Eqn. (2)] by using the following definitions for
$\mathbf{b}$ and $\mathbf{X}_{i}$:

$\mathbf{b}=\bigl{(}-\beta,\beta\mu\bigr{)},\quad\mathbf{X}_{i}=(E_{i},N_{i}).$
(4)

In other words, the above definitions for $\mathbf{b}$ and $\mathbf{X}_{i}$
retrieve the grand-canonical ensemble from the generalised ensemble.

MHR in the generalised ensemble described above entails using time series over
$\mathbf{X}$ and some physical property of interest $O$, obtained from
$N_{\text{sims}}$ simulations performed using thermodynamic parameters
$\mathbf{b}_{1},\mathbf{b}_{2},\dotsc,\mathbf{b}_{N_{\text{sims}}}$, to
determine the expected value of $O$ at another set of thermodynamic parameters
$\mathbf{b}$. The workhorse function reweight_observable in htk.multihistogram
performs such reweighting. 666Details of the equations and algorithm used to
do this can be found in the source code documentation. However, for the
convenience of users MHR functions are also provided for the canonical, grand-
canonical, and isothermal-isobaric ensembles which take familiar quantities
such as $E_{i}$ and $N_{i}$ as arguments, to save users from having to
’translate’ into the generalised ensemble. An example in the grand-canonical
ensemble is given below.

####  2.3.2 Example: Interpolation of an isotherm

Figure 3: $k_{B}T/\epsilon=1.5$ isotherm for the Lennard-Jones fluid obtained
by applying multiple histogram reweighting (MHR) to data from simulations
performed at $\ln z=-3.40$, -3.33 and -3.27. The blue squares are the
densities obtained directly from the simulations.

Figure 3 shows the results of applying MHR using htk.multihistogram to trace
the $k_{B}T/\epsilon=1.5$ isotherm of the Lennard-Jones fluid _between_ values
of the thermodynamic activity, $z$, explored explicitly by GCMC simulations,
namely $\ln z=-3.40$, $-3.33$ and $-3.27$. Details of the GCMC simulations are
the same as given in Section 2.2.3. The Python code snippet given in Figure 4
demonstrates how the MHR isotherm in Figure 3 was generated.

⬇

import dlmontepython.htk.sources.dlmonte as dlmonte

1001import dlmontepython.htk.multihistogram as multihistogram

import numpy as np

1003

# List of directories containing DL_MONTE simulations at

1005# different chemical potentials or activities

simdirs = [ ”simdir_1”, ”simdir_2”, ”simdir_3” ]

1007

# List of chemical potentials to calculate density at

1009# using reweighting

mu_target = np.linspace(-5.2,-4.8,20)

1011

# Data and thermodynamic parameters for all simulations.

1013# kT[n] will be the temperature of simulation ’n’, and

# similarly for mu. E[n][i] will be the energy of

1015# simulation ’n’ at timestep ’i’, and similarly for

# N and rho.

1017kT, mu, E, N, rho = ([] for i in range(0,5))

1019# Import simulation data from directories, using classes

# in htk package to make importing easier.

1021for simdir in simdirs:

1023 dlminput = dlmonte.DLMonteInput.from_directory(simdir)

dlmoutput = dlmonte.DLMonteOutput.load(simdir)

1025

kT_sim = dlminput.control.temperature

1027 V_sim = dlminput.config.volume()

z_sim = dlminput.control.get_molecule_gcmc_potential(”lj”)

1029 mu_sim = np.log(z_sim)*kT_sim

1031 kT.append(kT_sim)

mu.append(mu_sim)

1033

E_sim = dlmoutput.yamldata.time_series(”energy”)

1035 N_sim = dlmoutput.yamldata.time_series(”nmol”)

N_sim = np.reshape(N_sim,len(N_sim))

1037 rho_sim = N_sim / V_sim

1039 E.append(E_sim)

N.append(N_sim)

1041 rho.append(rho_sim)

1043# Perform the reweighting and output

for mu_new in mu_target:

1045

rho_new = multihistogram.reweight_observable_muvt(

1047 kT, mu, E, N, rho, kT[0], mu_new)

1049 print(mu_new, rho_new)

Figure 4: Python code used to generate the multiple histogram reweighting
isotherm in Figure 3.

###  2.4 fep: Reweighting and analysis of free energy profiles

####  2.4.1 Motivation

Free energy methods are a broad class of simulation methods which calculate
the free energy profile $F(M)$ of a system over some order parameter or
reaction coordinate $M$[4, 37]. An important application of free energy
methods is to pinpoint the location of phase transitions[4], in particular the
liquid-vapour transition in fluids and soft matter[38, 5, 6, 3, 39, 40]. This
in turn enables related quantities such as the latent heat to be determined.
Such free energy calculations involve first defining an order parameter $M$
such that, for some value $M_{\text{thresh}}$, $M<M_{\text{thresh}}$
corresponds to phase 1 and $M>M_{\text{thresh}}$ corresponds to phase 2.
Locating the phase transition, i.e. where the two phases coexist, then entails
calculating $F(M)$ at many thermodynamic parameters; one searches for the
parameter at which the two phases have equal free energies – something which
can be deduced from $F(M)$.

####  2.4.2 Functionality

The fep module was developed to assist with the types of calculations just
described. In particular, for specific situations detailed in a moment, fep
can exploit histogram reweighting to massively reduce the computational cost
associated with locating coexistence. The underlying idea is that, given a
$F(M)$ for a thermodynamic parameter close to coexistence, reweighting can be
used to evaluate $F(M)$ at nearby parameters without the need for further
costly ‘explicit’ calculations of $F(M)$.

The type of reweighting which is possible in the fep module is as follows.
Consider the generalised thermodynamic ensemble described by Eqn. (2), in the
special case where $M$ is one of the physical quantities in the vector
$\mathbf{X}$. Let the thermodynamic parameter in $\mathbf{b}$ _conjugate_ to
$M$ (i.e. the $j$th element in $\mathbf{b}$ if $M$ is the $j$th element in
$\mathbf{X}$) be denoted as $b$. Now, in this special case, if $b$ is the only
parameter we wish to tune to obtain a new $F(M)$ via reweighting, then
reweighting $F(M)$ becomes especially simple. The relevant equation is

$F(M;b_{\text{new}})=-k_{B}T\ln\Biggl{\\{}\int_{-\infty}^{\infty}dM\exp\Bigl{[}-F(M;b_{\text{old}})/(k_{B}T)+(b_{\text{new}}-b_{\text{old}})M\Bigr{]}\Biggr{\\}},$
(5)

where the reweighting is used to deduce the free energy profile at
$b_{\text{new}}$ from the free energy profile $F(M;b_{\text{old}})$ at
$b_{\text{old}}$. The function reweight can be used to apply the above
equation. Moreover the function reweight_to_coexistence can be used to apply
reweighting in a search for a value of $b$ which corresponds to coexistence,
given $b_{\text{old}}$, $F(M;b_{\text{old}})$, a lower and upper bound of $b$
to use in the search, and a value for $M_{\text{thresh}}$ to define the order
parameter ranges which correspond to phase 1 and phase 2.

The fep module also contains functions for extracting certain physical
quantities from a free energy profile. As well as generic quantities such as
the probabilities and expected values of $M$ for each phase, it also contains
functions specific to the liquid-vapour problem. For instance, the functions
vapour_pressure and surface_tension can be used to calculate, respectively,
the pressure and surface tension from a free energy profile vs. number of
molecules in the system[41, 42], as demonstrated in the following example.

####  2.4.3 Example: liquid-vapour coexistence properties of methane

Figure 5: Free energy vs. density for methane modelled by the TraPPE-EH force
field[43] at 175K and various $\ln z$. All curves were obtained by applying
reweighting to the $\ln z=-6.950$ curve, which was obtained from simulation.
-6.919 corresponds to liquid-vapour coexistence.

We have used fep to efficiently pinpoint liquid-vapour coexistence at 175K for
methane modelled by the TraPPE-EH[43] force field. Moreover we also used fep
to calculate the saturation pressure and liquid-vapour surface tension at this
temperature. First, a free energy simulation (transition-matrix GCMC,[42]
using DL_MONTE) was first performed at $\ln z=-6.950$. Figure 5 figure shows
the free energy profile over the density – the relevant order parameter for
the liquid-vapour problem – obtained from this simulation, as well as at
nearby $\ln z$ obtained via reweighting using the fep module. The fep module
was used to locate the $\ln z$ corresponding to coexistence. This was found to
be $\ln z=-6.919$ and the free energy profile corresponding to this is also
shown in the figure. Finally, using the free energy profile at coexistence,
the saturation pressure and surface tension were calculated using the fep
functions vapour_pressure and surface_tension, yielding 32.6 bar and 0.9
mNm-1, respectively. The Python code used to generate Figure 5 is given in
Figure 6.

⬇

import dlmontepython.fep.fep as fep

1001

# k_B*T used in simulation, units of kJ/mol

1003kt = 175 * 0.0083144621

# Value of ln(z) the simulation was performed at

1005lnz = -6.950

# Target ln(z) values for reweighting

1007lnz_target = [ -7.000, -6.900 ]

# Range of ln(z) to consider in search for coexistence

1009lnz_lbound = -7.0

lnz_ubound = -6.9

1011# Volume used in simulation (angstrom^3)

V = 30**3

1013# Molar mass of molecule (g/mol)

molarmass = 16.04

1015# Threshold between vapour and liquid densities (g/ml):

# systems with densities < rho_thresh are considered a vapour

1017rho_thresh = 0.17

1019# Name of file containing free energy profile vs. number

# of molecules

1021filename = ”FEDDAT.000_001”

1023# Import free energy profile over number of molecules

N, fe = fep.from_file(filename)

1025

# Output the free energy profile over the density

1027rho = N * molarmass / (V * 0.60221409)

fep.to_file(rho, fe, ”fep_vs_rho_”+str(lnz)+”_sim.dat”)

1029

# Reweight to target ln(z) and output

1031for lnz_new in lnz_target:

fe_new = fep.reweight(N, fe, lnz, lnz_new)

1033 fep.to_file(rho, fe_new, ”fep_vs_rho_”+str(lnz_new)+”.dat”)

1035# Locate coexistence ln(z) and output

N_thresh = rho_thresh * V * 0.60221409 / molarmass

1037lnz_co, p_co, fe_co = fep.reweight_to_coexistence(

N, fe, lnz, lnz_lbound, lnz_ubound,

1039 N_thresh)

fep.to_file(rho, fe_co, ”fep_vs_rho_”+str(lnz_co)+”_co.dat”)

1041

# Calculate cooexistence pressure ((kJ/mol)/A^3)

1043P = fep.vapour_pressure(N, fe_co, N_thresh, kt, V)

print(”coexistence pressure (bar) = ”, P * 1.0E5 / 6.02214076)

1045

# Calculate surface tension ((kJ/mol)/A^2)

1047A = V**(2.0/3.0)

tension = fep.surface_tension(N, fe_co, N_thresh, kt, A)

1049print(”surface tension (N/m) = ”, tension / 6.02214076)

Figure 6: Python code used to generate the free energy profiles in Figure 3.

##  3 Conclusions and Impact

We have described a Python library, dlmontepython, which has two overarching
aims:

  1. 1.

to facilitate the development of Python scripts which automate common
simulation tasks, e.g. the calculation of an isotherm to a given precision;

  2. 2.

to facilitate the application of data analysis methods pertinent to molecular
simulation, especially Monte Carlo.

As well as providing an overview of key current functionality of the library,
we also provided three examples to demonstrate its utility. Specifically, we
used the library to automatically calculate isotherms to a specified
precision; used histogram reweighting to interpolate isotherms between points
explored by simulation; and used reweighting to pinpoint liquid-vapour
coexistence and calculate the saturation pressure and surface tension for
methane. These examples reflect common applications of Monte Carlo simulation.
For example, such calculations are routinely employed in the development of
accurate models for fluids, and in the design of new materials for energy
applications. We therefore believe that dlmontepython will prove useful to
practitioners of molecular simulation, especially those who use Monte Carlo
simulation and DL_MONTE.

## Acknowledgements

TLU acknowledges support from the embedded CSE programme of the ARCHER UK
National Supercomputing Service (http://www.archer.ac.uk) [project eCSE11-3],
and the Engineering and Physical Sciences Research Council [grant number
EP/P007821/1]. TD and JRHM acknowledge support from the European Research
Council (ERC) (grant agreement No. 648283 “GROWMOF”). We thank Megan Stalker
for assistance with early testing of dlmontepython.

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