# Dimension-free inequalities for low and high degree functions on the Hamming
cube

Komla Domelevo  Faculty of Mathematics and Computer Science
Institute of Mathematics
Emil-Fischer-Str. 41
97074 Würzburg
Germany<EMAIL_ADDRESS>,  Polona Durcik  Schmid
College of Science and Technology
Chapman University
One University Drive
Orange, CA 92866, USA<EMAIL_ADDRESS>,  Valentia Fragkiadaki  Department
of Mathematics
Texas A&M University
College station
TX 77843, USA<EMAIL_ADDRESS>,  Ohad Klein  School of Computer Science
and Engineering
Hebrew University of Jerusalem
Jerusalem
Israel<EMAIL_ADDRESS>,  Diogo Oliveira e Silva  Center for Mathematical
Analysis, Geometry and Dynamical Systems & Departamento de Matemática
Instituto Superior Técnico
Av. Rovisco Pais
1049-001 Lisboa, Portugal<EMAIL_ADDRESS>,  Lenka
Slavíková  Department of Mathematical Analysis
Faculty of Mathematics and Physics
Charles University
Sokolovská 83, 186 75 Praha 8, Czech Republic<EMAIL_ADDRESS>and  Błażej Wróbel  Institute of Mathematics
of the Polish Academy of Sciences
Śniadeckich 8
00-656 Warsaw
Poland & Institute of Mathematics
University of Wrocław
Plac Grunwaldzki 2
50-384 Wrocław
Poland<EMAIL_ADDRESS>
###### Abstract.

We prove two types of optimal dimension-free $L^{p}$ inequalities,
$1<p<\infty$, for low and high degree scalar-valued functions on the Hamming
cube. More precisely, any function $f$ with spectrum bounded from above by $d$
satisfies the Bernstein–Markov type inequality

$\|\Delta^{k}f\|_{L^{p}(\\{-1,1\\}^{n})}\leq
Q(f)^{\varepsilon}C(p,\varepsilon)^{k}\,d^{k}\,\|f\|_{L^{p}(\\{-1,1\\}^{n})},$

for any $\varepsilon>0$ and $k\in\mathbb{N}$, and any function $f$ with
spectrum bounded from below by $d$ satisfies the upper bound on the decay of
the heat semigroup

$\|e^{-t\Delta}f\|_{L^{p}(\\{-1,1\\}^{n})}\leq
Q(f)^{\varepsilon}\exp(-c(p,\varepsilon)td)\|f\|_{L^{p}(\\{-1,1\\}^{n})},$

for any $t,\varepsilon>0$. Here, $Q(f)$ denotes the quotient between the
largest and smallest non-zero values of $f$, and the constants
$c(p,\varepsilon)$ and $C(p,\varepsilon)$ depend only on $p$ and
$\varepsilon$; crucially, they are independent of the dimension $n$. In
particular, if $f$ is Boolean-valued or
$f\colon\\{-1,1\\}^{n}\to\\{-1,0,1\\},$ then the bounds we obtain are
independent of $Q(f)$. We also provide counterexamples at the endpoint
$p=\infty$, for which a linear growth in $d$ does not suffice when $k=1$.

###### Key words and phrases:

Hamming cube, Bernstein–Markov inequality, heat semigroup, dimension-free
estimates.

###### 2020 Mathematics Subject Classification:

Primary 42C10; Secondary 41A17, 41A63, 47A60.

##  1\. Introduction

Consider the Hamming cube $\\{-1,1\\}^{n}$ equipped with the uniform
probability measure. Let $X$ be a Banach space. Given $p\in[1,\infty)$, the
$L^{p}$ norm of a function $f\colon\\{-1,1\\}^{n}\to X$ is given by

$\|f\|_{L^{p}(\\{-1,1\\}^{n};X)}:=\left(\frac{1}{2^{n}}\sum_{\delta\in\\{-1,1\\}^{n}}\|f(\delta)\|_{X}^{p}\right)^{1/p}.$

We also set
$\|f\|_{L^{\infty}(\\{-1,1\\}^{n};X)}:=\max_{\delta\in\\{-1,1\\}^{n}}\|f(\delta)\|_{X}.$
When $X=\mathbb{C}$ we abbreviate
$L^{p}(\\{-1,1\\}^{n}):=L^{p}(\\{-1,1\\}^{n};X).$ The symbol $\langle
f,g\rangle$ denotes the (Hermitian, complex) inner product on
$L^{2}(\\{-1,1\\}^{n}).$

Any $f\colon\\{-1,1\\}^{n}\to X$ can be represented as its Hamming cube
Fourier–Walsh expansion via

$f=\sum_{S\subseteq[n]}\widehat{f}(S)w_{S},$ (1.1)

with $w_{S}$ being the Walsh functions $w_{S}(\delta)=\prod_{j\in
S}\delta_{j},$ $\delta=(\delta_{1},\ldots,\delta_{n})\in\\{-1,1\\}^{n}$,
$S\subseteq[n]:=\\{1,\ldots,n\\}.$ The coefficients $\widehat{f}(S)$ in (1.1)
are the Fourier coefficients of $f$ given by

$\widehat{f}(S):=\frac{1}{2^{n}}\sum_{\delta\in\\{-1,1\\}^{n}}f(\delta)w_{S}(\delta).$

Given $d\in[n]$, we let

$\displaystyle\operatorname{Deg}_{\leq d}$
$\displaystyle=\\{f\colon\\{-1,1\\}^{n}\to X\colon f=\sum_{|S|\leq
d}\widehat{f}(S)\,w_{S}\\},$ $\displaystyle\operatorname{Deg}_{\geq d}$
$\displaystyle=\\{f\colon\\{-1,1\\}^{n}\to X\colon f=\sum_{|S|\geq
d}\widehat{f}(S)\,w_{S}\\}$

denote the spaces of functions of degree at most $d$ and of degree at least
$d$, respectively. In other words, $\operatorname{Deg}_{\leq d}$ contains
those functions $f\colon\\{-1,1\\}^{n}\to X$ for which $\widehat{f}(S)=0$ if
$|S|>d$, and $f\in\operatorname{Deg}_{\geq d}$ means that $\widehat{f}(S)=0$
if $|S|<d.$

Define the Hamming cube Laplacian by

$\Delta f=\sum_{j=1}^{n}\partial_{j}f,$ (1.2)

with

$\partial_{j}f(\delta)=\frac{f(\delta_{1},\ldots,\delta_{j},\ldots,\delta_{n})-f(\delta_{1},\ldots,-\delta_{j},\ldots,\delta_{n})}{2}.$
(1.3)

In this way, $w_{S}$ is an eigenfunction of $\Delta$ with eigenvalue $|S|.$
Thus, if $f$ is decomposed as in (1.1), then

$\Delta f=\sum_{S\subseteq[n]}|S|\,\widehat{f}(S)\,w_{S}.$

This leads to the following definition of the heat semigroup for $f$ given by
(1.1):

$e^{-t\Delta}f=\sum_{S\subseteq[n]}e^{-t|S|}\,\widehat{f}(S)\,w_{S}.$

It is straightforward to verify that

$\|\Delta f\|_{L^{2}(\\{-1,1\\}^{n};X)}\leq
d\,\|f\|_{L^{2}(\\{-1,1\\}^{n};X)}$ (1.4)

whenever $f\in\operatorname{Deg}_{\leq d}.$

The main purpose of this note is to establish a generalization of (1.4) to
$L^{p}$ spaces when the target Banach space $X$ is the field of scalars
${\mathbb{C}}$. This will be achieved at the expense of reducing the class of
functions to which the inequality applies. In such cases, we will give a
partial answer to the following conjectures formulated by Eskenazis–Ivanisvili
in [2, Sections 1.2.2–1.2.3]. The definition of a $K$-convex Banach space can
be recalled from [2, Section 1.2].

######  Conjecture 1.1.

Let $(X,\|\cdot\|_{X})$ be a $K$-convex Banach space. For every
$p\in(1,\infty),$ there exists $C(p,X)$ such that, for every $d\in[n]$ and
every $f\in\operatorname{Deg}_{\leq d}$,

$\|\Delta f\|_{L^{p}(\\{-1,1\\}^{n};X)}\leq C(p,X)\cdot
d\,\|f\|_{L^{p}(\\{-1,1\\}^{n};X)}.$ (1.5)

######  Conjecture 1.2.

Let $(X,\|\cdot\|_{X})$ be a $K$-convex Banach space. For every
$p\in(1,\infty),$ there exist constants $c(p,X)$ and $C(p,X)$ such that, for
every $d\in[n]$ and every $f\in\operatorname{Deg}_{\leq d}$,

$\|e^{-t\Delta}f\|_{L^{p}(\\{-1,1\\}^{n};X)}\geq c(p,X)\cdot\exp(-C(p,X)\cdot
td)\|f\|_{L^{p}(\\{-1,1\\}^{n};X)}.$ (1.6)

In what follows, we denote by $\operatorname{supp}f$ the support of a function
$f\colon\\{-1,1\\}^{n}\to X$. If $f\neq 0$, then we set

$M(f)=\max_{x\in\operatorname{supp}f}|f(x)|,\qquad
m(f)=\min_{x\in\operatorname{supp}f}|f(x)|,$

and

$Q(f)=\frac{M(f)}{m(f)}.$

We also set $Q(f)=1$ if $f=0.$ Note that $Q(f)\geq 1$ for any $f.$ The
following result is the main contribution of our note.

######  Theorem 1.3.

Let $p\in(1,\infty)$ and take $\varepsilon>0.$ Take $d\in[n]$ and let
$f\in\operatorname{Deg}_{\leq d}$. Then, for any $k\in{\mathbb{N}}$,

$\|\Delta^{k}f\|_{L^{p}(\\{-1,1\\}^{n})}\leq Q(f)^{\varepsilon}\cdot
C(p,\varepsilon)^{k}\cdot d^{k}\,\|f\|_{L^{p}(\\{-1,1\\}^{n})},$ (1.7)

for a constant $C(p,\varepsilon)$ depending on $p,\varepsilon$ but not on
$d,k$ nor on the dimension $n.$

######  Remark 1.

By setting $\varepsilon=1$, we obtain the inequality

$\|\Delta^{k}f\|_{L^{p}(\\{-1,1\\}^{n})}\leq Q(f)\cdot C_{p}^{k}\cdot
d^{k}\,\|f\|_{L^{p}(\\{-1,1\\}^{n})},$ (1.8)

which is valid for any function $f\in\operatorname{Deg}_{\leq d}$.

The proof of Theorem 1.3 is based on two ingredients: Hadamard’s three-lines
theorem and dimension-free $L^{p}$ bounds for the imaginary powers
$\Delta^{iu}$. We are not aware of any previous application of these results
to the analysis of functions on the Hamming cube. Our proof is similar to the
proofs of both Riesz–Thorin interpolation theorem and Stein’s interpolation
theorem for analytic families of operators (see e.g. [5, Sections 1.3.2 and
1.3.3] and [11]), with a twist. Indeed, we apply Hadamard’s theorem in a
slightly different way, by removing an instance of one single letter of the
alphabet: instead of the holomorphic function
$z\mapsto\langle\Delta^{z}f^{z},g^{z}\rangle$, we consider a similar function
in which $f^{z}$ does not depend on $z,$ namely
$\langle\Delta^{z}f,g^{z}\rangle.$ This comes at the cost of introducing
$Q(f).$ It is vital that $\Delta$ generates a symmetric contraction semigroup,
and thus its imaginary powers $\Delta^{iu},$ $u\in{\mathbb{R}},$ satisfy
$L^{p}$ bounds, $1<p<\infty$, with explicit constants which depend only on $p$
and $u$; see e.g. [1, 12]. At the $L^{2}$ endpoint we merely use spectral
properties of $\Delta^{z},$ $\operatorname{Re}z\geq 0,$ on $L^{2},$ which
imply

$\|\Delta^{z}f\|_{L^{2}(\\{-1,1\\}^{n})}\leq
d^{\operatorname{Re}z}\,\|f\|_{L^{2}(\\{-1,1\\}^{n})},\,\text{ for all
}f\in\operatorname{Deg}_{\leq d}.$

Theorem 1.3 is proved in detail in Section 2. We proceed to discuss its
consequences. The first corollary is a Bernstein–Markov inequality on
$L^{p}(\\{-1,1\\}^{n})$ for scalar valued functions, i.e., when
$X=\mathbb{C}.$

######  Corollary 1.4.

Let $p\in(1,\infty)$ and $d\in[n].$ Then, for $f\in\operatorname{Deg}_{\leq
d}$,

$\|\Delta f\|_{L^{p}(\\{-1,1\\}^{n})}\leq C_{p}\cdot
Q(f)\cdot\,d\,\|f\|_{L^{p}(\\{-1,1\\}^{n})},$ (1.9)

for some constant $C_{p}$ depending on $p$ but not on the dimension $n.$

######  Remark 2.

The inequality (1.9) is (1.8) in the case $k=1$. Corollary 1.4 improves [2,
Theorem 8] for scalar valued functions which satisfy $Q(f)\leq R$ with a
universal constant $R$. There the authors proved that for a general $K$-convex
Banach space $X$ one has

$\|\Delta f\|_{L^{p}(\\{-1,1\\}^{n};X)}\leq C(p,X)\cdot
d^{\alpha(p,X)}\,\|f\|_{L^{p}(\\{-1,1\\}^{n};X)},$

for all functions of degree at most $d$, where the constant $C(p,X)$ and the
exponent $\alpha(p,X)\in[1,2)$ depend both on $p$ and on the underlying Banach
space $X$. In particular, in the scalar case when $X=\mathbb{C},$ one may take
$\alpha(p,\mathbb{C})=2-\pi^{-1}\arcsin(2(p-1)^{1/2}/p)$; see [2, Theorem 14].

The second corollary is a lower bound on the decay of the heat semigroup
acting on low degree functions. Again, we restrict to the scalar case
$X=\mathbb{C}$. Corollary 1.5 below is equivalent to the inequality

$\|e^{t\Delta}f\|_{L^{p}(\\{-1,1\\}^{n};X)}\leq\exp(C_{p}\cdot Q(f)\cdot
td)\|f\|_{L^{p}(\\{-1,1\\}^{n};X)},\,\text{ for all
}f\in\operatorname{Deg}_{\leq d}.$

This inequality is a consequence of Theorem 1.3 via a Taylor expansion
argument.

######  Corollary 1.5.

Let $p\in(1,\infty)$ and $d\in[n].$ Then, for $f\in\operatorname{Deg}_{\leq
d}$ and all $t>0$,

$\|e^{-t\Delta}f\|_{L^{p}(\\{-1,1\\}^{n})}\geq\exp(-C_{p}\cdot Q(f)\cdot
td)\|f\|_{L^{p}(\\{-1,1\\}^{n})},$ (1.10)

where $C_{p}$ is the constant from (1.9).

######  Remark 3.

Corollary 1.5 improves [2, Theorem 7] for scalar valued functions which
satisfy $Q(f)\leq R$ with a universal constant $R$. In [2], the authors proved
the following variant of (1.10) for functions of degree at most $d$ taking
values in a $K$-convex Banach space $X$:

$\|e^{-t\Delta}f\|_{L^{p}(\\{-1,1\\}^{n};X)}\geq
c(p,X)\cdot\exp(-C(p,X)\cdot\max\\{t,t^{\eta(p,X)}\\}\cdot
d)\|f\|_{L^{p}(\\{-1,1\\}^{n};X)}.$

The price of this generality is that $t$ is locally replaced by a power
function $t^{\eta(p,X)}$ with $\eta(p,X)\in(1/2,1).$

######  Remark 4.

It was observed in [2, Remark 33] that Conjecture 1.1 implies Conjecture 1.2.
Indeed, since the Hamming cube Laplacian $\Delta$ preserves the space
$\operatorname{Deg}_{\leq d}$, a repeated application of (1.5) yields

$\|\Delta^{k}f\|_{L^{p}(\\{-1,1\\}^{n};X)}\leq C(p,X)^{k}\cdot
d^{k}\,\|f\|_{L^{p}(\\{-1,1\\}^{n};X)},\qquad k\in{\mathbb{N}}.$

From this inequality, we easily obtain via Taylor expansion that

$\|e^{t\Delta}f\|_{L^{p}(\\{-1,1\\}^{n};X)}\leq\exp(C(p,X)\cdot
td)\|f\|_{L^{p}(\\{-1,1\\}^{n};X)},\,\text{ for all
}f\in\operatorname{Deg}_{\leq d},$

which is equivalent to Conjecture 1.2 with $c(p,X)=1.$

However, the above argument can not be applied for concluding that Corollary
1.4 implies Corollary 1.5. Indeed, $\Delta$ preserves
$\operatorname{Deg}_{\leq d}$ but it may fail to preserve $Q(f)$. In fact,
there may be an arbitrarily large gap between $Q(f)$ and $Q(\Delta f)$ once
$d\geq 2$. To see this, consider
$f=x_{1}x_{2}+x_{3}x_{4}+(1-\varepsilon)x_{5}x_{6}$ with any small
$\varepsilon>0$. We have $Q(f)=3+O(\varepsilon)$, while $\Delta
f=\sum_{i=1}^{6}x_{i}-\varepsilon(x_{5}+x_{6})$ has $Q(\Delta
f)>1/\varepsilon$. In particular, applying Corollary 1.4 twice in a row to the
functions of the above form would result in

$\|\Delta^{2}f\|_{L^{p}(\\{-1,1\\}^{n})}\leq
C_{p}^{2}\cdot\frac{d}{\varepsilon}\,\|f\|_{L^{p}(\\{-1,1\\}^{n})},$

which is not a satisfactory estimate to derive 1.5, since $\varepsilon>0$ is
arbitrary. In summary, a repeated application of Corollary 1.4 is not
permitted.

In view of Remark 4, Corollary 1.5 does not follow from Corollary 1.4, but
both of these results are a consequence of Theorem 1.3. As a further
consequence of Corollaries 1.4 and 1.5 we obtain the following results, which
are straightforward once we observe that $Q(f)=1$ for all functions that take
values in $\\{-1,0,1\\}.$

######  Corollary 1.6.

Let $p\in(1,\infty)$ and $d\in[n].$ Take $f\in\operatorname{Deg}_{\leq d}$
such that $f\colon\\{-1,1\\}^{n}\to\\{-1,0,1\\}.$ Then

$\|\Delta f\|_{L^{p}(\\{-1,1\\}^{n})}\leq C_{p}\cdot
d\,\|f\|_{L^{p}(\\{-1,1\\}^{n})}$ (1.11)

for some constant $C_{p}$ depending on $p$ but not on the dimension $d.$

######  Corollary 1.7.

Let $p\in(1,\infty)$ and $d\in[n].$ Take $f\in\operatorname{Deg}_{\leq d}$
such that $f\colon\\{-1,1\\}^{n}\to\\{-1,0,1\\}.$ Then

$\|e^{-t\Delta}f\|_{L^{p}(\\{-1,1\\}^{n})}\geq\exp(-C_{p}\cdot
dt)\|f\|_{L^{p}(\\{-1,1\\}^{n})},$ (1.12)

where $C_{p}$ is the constant from (1.11).

It is worth noting that Theorem 1.3 and Corollary 1.4 cannot be deduced from
interpolation between the trivial $L^{2}$ bound and an $L^{\infty}$ bound.
Indeed, Corollary 1.4 does not hold at the endpoint $p=\infty.$

######  Proposition 1.8.

Given $n\in{\mathbb{N}}$ and $d\in[n]$, let ${\bf C}(n,d)$ denote the best
constant in the inequality

$\|\Delta f\|_{L^{\infty}(\\{-1,1\\}^{n})}\leq{\bf C}(n,d)\cdot
Q(f)\cdot\|f\|_{L^{\infty}(\\{-1,1\\}^{n})},\qquad
f\in\operatorname{Deg}_{\leq d}.$

Then it holds that $\liminf_{n\to\infty}{\bf C}(n,d)\geq\frac{1}{8}d^{2}.$

Proposition 1.8 is justified in Section 3. Its proof is based on properties of
Chebyshev polynomials. The argument is almost the same as in [2, proof of
Theorem 5, p. 254], but we repeat it here for the convenience of the reader.
Note that the observation from [2, Remark 24] already implies that the
conjectured estimate (1.5) cannot hold at $L^{\infty}$ for general functions
$f\colon\\{-1,1\\}^{n}\to\mathbb{C}$.

Corollary 1.6 also does not hold at the endpoint $p=\infty$. A counterexample
is given by the Kushilevitz function; see [6, Section 6.3] and [13]. This was
brought to our attention by Paata Ivanisvili.

######  Proposition 1.9.

Let $k\in\mathbb{N}$ and take $n=6^{k}$ and $d=3^{k}.$ Then there exists a
function $f\colon\\{-1,1\\}^{n}\to\\{-1,1\\}$ such that
$f\in\operatorname{Deg}_{\leq d}$ but

$\|\Delta f\|_{L^{\infty}(\\{-1,1\\}^{n})}\geq d^{\log 6/\log
3}\|f\|_{L^{\infty}(\\{-1,1\\}^{n})}.$

Proposition 1.9 is proved in Section 3. Note that $\log 6/\log 3\approx 1.63$,
and it would be interesting to see if one can improve the growth to a constant
times $d^{2}$ as in Proposition 1.8.

The methods from our note can be adapted to yield $L^{p}$ estimates on tail
spaces for $\Delta f$ or $e^{-t\Delta}f$ when $f\in\operatorname{Deg}_{\geq
d}.$ In this way, one may for instance prove a slightly more general variant
of [7, Theorem 1.3] by Heilman–Mossel–Oleszkiewicz. In any case, Theorem 1.11
below can be also proved by using the methods from [7].

######  Theorem 1.10.

Let $p\in(1,\infty)$ and take $\varepsilon>0.$ Take $d\in[n]$ and let
$f\in\operatorname{Deg}_{\geq d}$. Then, for any $t>0$,

$\|e^{-t\Delta}f\|_{L^{p}(\\{-1,1\\}^{n})}\leq
Q(f)^{\varepsilon}\cdot\exp(-2c(p,\varepsilon)\cdot
td)\|f\|_{L^{p}(\\{-1,1\\}^{n})},$ (1.13)

where $c(p,\varepsilon)$ is a constant that depends only on $p$ and
$\varepsilon.$

The proof of Theorem 1.10 can be found at the end of Section 2. It is easily
adapted to yield the following version with $\varepsilon=1$ and
$c(p,1)=\min\\{1/p,1/p^{\prime}\\}$, where $p^{\prime}:=p/(p-1)$ denotes the
dual exponent.

######  Theorem 1.11.

Let $p\in(1,\infty)$, $d\in[n],$ and take $f\in\operatorname{Deg}_{\geq d}$.
Then, for all $t>0$,

$\|e^{-t\Delta}f\|_{L^{p}(\\{-1,1\\}^{n})}\leq
Q(f)\exp(-2\min\\{1/p,1/p^{\prime}\\}\cdot td)\|f\|_{L^{p}(\\{-1,1\\}^{n})}.$
(1.14)

Theorems 1.10 and 1.11 establish a restricted case of the heat smoothing
conjecture of Mendel–Naor [8, Remark 5.5] for functions satisfying $Q(f)\leq
R$ for a universal constant $R$. We remark that proving [7, Theorem 1.3] with
our methods is easier than proving Theorem 1.3. We merely apply Hadamard’s
three-lines theorem to the holomorphic function $\langle
e^{t\Delta}f,g^{z}\rangle.$ In this way, one works with a fixed operator
$e^{t\Delta}$ and there is no need to treat families of analytic operators,
such as complex powers $\Delta^{z}$.

For related work on Gaussian spaces, we refer to [3, 4, 9].

### Notation

In the remainder of the paper, for $p\in[1,\infty]$ we abbreviate
$L^{p}:=L^{p}(\\{-1,1\\}^{n})$ and $\|\cdot\|_{L^{p}}=\|\cdot\|_{p}.$

### Acknowledgments

This project grew out of the workshop Analysis on the hypercube with
applications to quantum computing held at the American Institute of
Mathematics (AIM) in June 2022. We thank the organizers of the workshop and
the staff of AIM for creating a superb environment for collaboration; in
particular, we are grateful to Irina Holmes Fay who was a member of our
research group. We also thank Paata Ivanisvili for helpful remarks and
suggestions.

PD is supported by the grant NSF DMS-2154356. OK is supported in part by a
grant from the Israel Science Foundation (ISF Grant No. 1774/20), and by a
grant from the US-Israel Binational Science Foundation and the US National
Science Foundation (BSF-NSF Grant No. 2020643). DOS is partially supported by
FCT/Portugal through CAMGSD, IST-ID, projects UIDB/04459/2020 and
UIDP/04459/2020 and by IST Santander Start Up Funds. LS is supported by the
Primus research programme PRIMUS/21/SCI/002 of Charles University and by
Charles University Research Centre program No. UNCE/24/SCI/005. BW is
supported by the National Science Centre, Poland, grant Sonata Bis
2022/46/E/ST1/ 00036.

##  2\. Proof of the positive results: Theorems 1.3 and 1.10

Theorem 1.3 will follow if we justify that

$\|(\Delta+\gamma I)^{k}f\|_{p}\leq
Q(f)^{\varepsilon}\,C(p,\varepsilon)^{k}\,(d+\gamma)^{k}\,\|f\|_{p},$ (2.1)

uniformly in $\gamma>0$. In the remainder of the proof we focus on (2.1) and
abbreviate

$L=\Delta+\gamma I.$ (2.2)

Note that the complex powers $L^{z}$ are well defined for
$\operatorname{Re}z\geq 0$ by the spectral theorem; in our case, they are
given explicitly by

$L^{z}f=\sum_{S\subseteq[n]}(|S|+\gamma)^{z}\,\widehat{f}(S)\,w_{S}.$ (2.3)

The operator $L$ in (2.2) is clearly self-adjoint on $L^{2}.$ Moreover, it is
not hard to see that $e^{-tL}$ is a contraction semigroup on all $L^{p}$
spaces, $1<p<\infty.$ Indeed, since $e^{-tL}=e^{-\gamma t}e^{-t\Delta}$ we
have the pointwise domination $|e^{-tL}f(x)|\leq e^{-t\Delta}|f|(x).$
Furthermore, recalling the definitions (1.2)–(1.3) and denoting

$Bf(\delta)=\frac{1}{2}\sum_{j=1}^{n}f(\delta_{1},\ldots,-\delta_{j},\ldots,\delta_{n}),$

we may rewrite $\Delta f=\frac{n}{2}f-Bf.$ Since
$\|Bf\|_{p}\leq\frac{n}{2}\|f\|_{p}$ for $p\in[1,\infty]$, we obtain

$\|e^{-tL}f\|_{p}\leq\|e^{-t\Delta}f\|_{p}=e^{-nt/2}\|e^{tB}f\|_{p}\leq
e^{-nt/2}e^{nt/2}\|f\|_{p}=\|f\|_{p},$

where the last inequality above comes from expanding $e^{tB}$ into a power
series and estimating it term by term. In summary, we verified that $L$
generates a symmetric contraction semigroup in the sense of [1]. Hence, we may
use [1, Corollary 1] in order to obtain an estimate for the imaginary powers
$L^{iu},$ $u\in{\mathbb{R}}.$

######  Proposition 2.1.

Let $u\in{\mathbb{R}}.$ For each $1<p<\infty$ there exists a constant
$C_{p}>1,$ depending only on $p$, such that

$\|L^{iu}f\|_{p}\leq C_{p}\cdot\exp((\pi|1/p-1/2|+1)|u|)\|f\|_{p},$ (2.4)

for all $f\in L^{p}$.

The above proposition is slightly weaker than [1, Corollary 1] but will
suffice for our purposes.

We are now ready to prove Theorem 1.3.

###### Proof of Theorem 1.3.

Given $p\in(1,\infty)$, we continue to denote by $p^{\prime}$ its dual
exponent. By duality, it is enough to justify that, for all $\varepsilon>0$,

$|\langle L^{k}f,g\rangle|\leq
Q(f)^{\varepsilon}C(p,\varepsilon)^{k}(d+\gamma)^{k},$ (2.5)

uniformly for all real-valued functions $f\in L^{p}$ and $g\in L^{p^{\prime}}$
such that $\|f\|_{p}=1$ and $\|g\|_{p^{\prime}}=1.$ In what follows, we fix
such functions $f$ and $g$, and focus on the case $p\in(2,\infty)$. The proof
for $p\in(1,2)$ is analogous.

Take $q\in(p,\infty)$ (to be chosen later in the proof) and let
$\theta\in(0,1)$ be such that

$\frac{1}{p}=\frac{1-\theta}{2}+\frac{\theta}{q},\qquad\text{ i.e.
}\theta=\frac{\frac{1}{2}-\frac{1}{p}}{\frac{1}{2}-\frac{1}{q}}.$ (2.6)

Note that $q$ (and therefore $\theta$) will be chosen to depend on $p$ and
$\varepsilon$. Further let

$N=\frac{k}{1-\theta}.$ (2.7)

Let $A=\\{\delta\in\\{-1,1\\}^{n}\colon g(\delta)\neq 0\\}$ be the support of
$g.$ Decompose $g=\sum_{\delta\in A}g(\delta){\mathds{1}_{{\delta}}}$ and,
recalling that $g$ is real-valued, let

$g_{z}=\sum_{\delta\in
A}\operatorname{sgn}(g(\delta))|g(\delta)|^{p^{\prime}(1-z)/2+p^{\prime}z/q^{\prime}}{\mathds{1}_{{\delta}}},\qquad
z\in\mathbb{C},$ (2.8)

where $q^{\prime}$ is the dual exponent of $q.$ The exact formula for $g_{z}$
is immaterial and we will only need to use some of its properties. Firstly,
note that for fixed $s$ the function $g_{z}(s)$ is holomorphic in $z$.
Moreover, by (2.6) we have $1/p^{\prime}=(1-\theta)/2+\theta/q^{\prime},$ so

$g_{\theta}=g,\qquad|g_{z}|=|g|^{p^{\prime}/2}\textrm{ if
}\operatorname{Re}z=0,\qquad|g_{z}|=|g|^{p^{\prime}/q^{\prime}}\textrm{ if
}\operatorname{Re}z=1.$ (2.9)

Let $\Sigma=\\{z\in\mathbb{C}\colon 0\leq\operatorname{Re}z\leq 1\\}.$ Given
$z\in\Sigma$, we define the function

$\varphi(z)=\exp(z^{2}N\pi|1/q-1/2|)\langle
L^{N(1-z)}f,\overline{g_{z}}\rangle.$ (2.10)

Since the sum in (2.8) is finite, the function $\varphi$ is holomorphic in the
interior of $\Sigma.$ It also follows that $|\varphi(z)|$ is bounded in
$\Sigma,$ i.e. $|\varphi(z)|\leq C(N,\gamma,q,p,f,g).$ Thus we may apply
Hadamard’s three-lines theorem to the function $\varphi$, provided we
appropriately estimate its boundary values at $\operatorname{Re}z=0$ and
$\operatorname{Re}z=1.$ We claim that

$|\varphi(z)|\leq(d+\gamma)^{N}\|f\|_{2},\qquad\operatorname{Re}z=0,\text{ and
}$ (2.11) $|\varphi(z)|\leq(C_{p,q})^{k}\|f\|_{q},\qquad\operatorname{Re}z=1,$
(2.12)

for a constant $C_{p,q}>1$ depending only on $p$ and $q.$

Assuming the claim for the moment, we apply Hadamard’s three-lines theorem [5,
Lemma 1.3.5] with

$B_{0}=(d+\gamma)^{N}\|f\|_{2},\qquad B_{1}=(C_{p,q})^{k}\|f\|_{q},$

and obtain (recall (2.7))

$|\varphi(\theta)|\leq(C_{p,q})^{k\theta}(d+\gamma)^{k}\|f\|_{2}^{1-\theta}\|f\|_{q}^{\theta}.$

Furthermore, since $m(f)\leq|f(\delta)|\leq M(f),$
$\delta\in\operatorname{supp}f,$ and $\|f\|_{p}=1,$ we have

$\|f\|_{2}^{2}=\|{\mathds{1}_{{\operatorname{supp}f}}}|f|^{2}\|_{1}=\|{\mathds{1}_{{\operatorname{supp}f}}}|f|^{2-p}|f|^{p}\|_{1}\leq
m(f)^{2-p}$ (2.13)

and

$\|f\|_{q}^{q}=\|{\mathds{1}_{{\operatorname{supp}f}}}|f|^{q}\|_{1}=\|{\mathds{1}_{{\operatorname{supp}f}}}|f|^{q-p}|f|^{p}\|_{1}\leq
M(f)^{q-p},$ (2.14)

so that

$|\varphi(\theta)|\leq(C_{p,q})^{k\theta}(d+\gamma)^{k}m(f)^{(2-p)(1-\theta)/2}M(f)^{(q-p)\theta/q}.$

Now, a short calculation based on (2.6) shows that

$(2-p)(1-\theta)/2=(p-q)\theta/q,$ (2.15)

so that we may continue with

$|\varphi(\theta)|\leq(M(f)/m(f))^{(q-p)\theta/q}(C_{p,q})^{k\theta}(d+\gamma)^{k}\leq
Q(f)^{\varepsilon}C(p,\varepsilon)^{k}(d+\gamma)^{k}.$

provided the exponent $q$ (and therefore $\theta$, according to (2.6)) is
chosen in order to satisfy $\frac{(q-p)\theta}{q}\leq\varepsilon$ and
$C(p,\varepsilon)=(C_{p,q})^{\theta}$. At this point, to complete the proof of
(2.5) we note that (2.7) and (2.9) imply

$\langle L^{k}f,g\rangle=\exp(-\theta^{2}N\pi|1/q-1/2|)\varphi(\theta).$

We are left with proving the claimed inequalities (2.11) and (2.12). We start
with a proof of (2.11). Take $z=iu,$ $u\in{\mathbb{R}}$, and note that
$\|L^{N(1-iu)}f\|_{2}\leq(d+\gamma)^{N}\|f\|_{2}.$ Since
$f\in\operatorname{Deg}_{\leq d},$ this inequality follows from (2.3) and the
orthogonality of the Walsh functions $w_{S}$. We remark that this is the only
place where the assumption $f\in\operatorname{Deg}_{\leq d}$ is used. Now, by
the Cauchy–Schwarz inequality and the second identity in (2.9), we obtain

$|\varphi(z)|\leq\exp(-u^{2}N\pi|1/q-1/2|)(d+\gamma)^{N}\|f\|_{2},$

which is even better than the claimed (2.11).

Finally, we prove (2.12). Take $z=1+iu,$ $u\in{\mathbb{R}}$, and apply
Hölder’s inequality with exponents $q$ and $q^{\prime}$ to the formula (2.10)
defining $\varphi.$ This yields

$|\varphi(z)|\leq\exp((1-u^{2})N\pi|1/q-1/2|)\|L^{-Niu}f\|_{q},$

where we used the third identity in (2.9). Applying Proposition 2.1 with $q$
in place of $p$ and using (2.7), we obtain

$\displaystyle|\varphi(z)|$ $\displaystyle\leq
C_{q}\exp\left[(1-u^{2})N\pi|1/q-1/2|+N(\pi|1/q-1/2|+1)|u|\right]\|f\|_{q}$
$\displaystyle=C_{q}(B_{p,q}(u))^{k}\|f\|_{q},$

where $C_{q}>1$ is a constant that depends only on $q$, and

$B_{p,q}(u):=\exp\left[\frac{(1-u^{2})\pi|1/q-1/2|+(\pi|1/q-1/2|+1)|u|}{1-\theta}\right].$

Denoting by $B^{\max}_{p,q}$ the maximal value of $B_{p,q}(u)$, we reach

$|\varphi(z)|\leq(C_{q}B^{\max}_{p,q})^{k}\|f\|_{q}.$ (2.16)

This proves that (2.12) holds with $C_{p,q}=C_{q}B^{\max}_{p,q}$ and completes
the proof of Theorem 1.3. ∎

###### Proof of Theorem 1.10.

We treat only the case when $p\in(2,\infty)$, as the proof for $p\in(1,2)$ is
analogous. Let $f\in L^{p}$ and $g\in L^{p^{\prime}}$ be real-valued and
normalized, in the sense that $\|f\|_{p}=\|g\|_{p^{\prime}}=1$. Let $q>p$ be
as in (2.6),

$\frac{1}{p}=\frac{1-\theta}{2}+\frac{\theta}{q},\qquad\text{ i.e.
}\theta=\frac{\frac{1}{2}-\frac{1}{p}}{\frac{1}{2}-\frac{1}{q}}.$

At the end of the proof, we will choose $q$ close to $p$, which amounts to
choosing $\theta$ close to $1$. Given $z\in\mathbb{C}$, define the function
$g_{z}$ via (2.8). As in the proof of Theorem 1.3, we have that
$\|g_{z}\|_{2}=1$ if $\operatorname{Re}(z)=0$, $\|g_{z}\|_{q^{\prime}}=1$ if
$\operatorname{Re}(z)=1$, and $g_{\theta}=g$. Given
$z\in\Sigma=\\{z\in\mathbb{C}\colon 0\leq\operatorname{Re}z\leq 1\\}$, define

$\Psi(z):=\langle e^{-t\Delta}f,g_{z}\rangle.$

This is a bounded function which is holomorphic in the interior of $\Sigma.$
We proceed to estimate $\Psi(z)$ on the boundary lines of the strip $\Sigma$.
If $\operatorname{Re}(z)=0$, then the Cauchy–Schwarz inequality and the
assumption $f\in\operatorname{Deg}_{\geq d}$ together imply

$|\Psi(z)|\leq\|e^{-t\Delta}f\|_{2}\|g\|_{2}\leq e^{-td}\|f\|_{2}\leq
e^{-td}m(f)^{\frac{2-p}{2}},$

where the last inequality follows for all $t>0$ from (2.13). If
$\operatorname{Re}(z)=1$, then Hölder’s inequality and the fact that
$e^{-t\Delta}$ is a contraction on $L^{q}$ together imply

$|\Psi(z)|\leq\|e^{-t\Delta}f\|_{q}\|g_{z}\|_{q^{\prime}}\leq\|f\|_{q}\leq
M(f)^{\frac{q-p}{q}},$

where the last inequality follows from (2.14). Hadamard’s three-lines theorem
and (2.15) then imply

$|\Psi(\theta)|=|\langle
e^{-t\Delta}f,g\rangle|\leq(e^{-td}m(f)^{\frac{2-p}{2}})^{1-\theta}(M(f)^{\frac{q-p}{q}})^{\theta}\\\
=e^{-td(1-\theta)}\left(\frac{M(f)}{m(f)}\right)^{\frac{q-p}{q}\theta}=e^{-td\frac{\frac{1}{p}-\frac{1}{q}}{\frac{1}{2}-\frac{1}{q}}}Q(f)^{\frac{q-p}{q}\frac{\frac{1}{2}-\frac{1}{p}}{\frac{1}{2}-\frac{1}{q}}}.$

To conclude, given $\varepsilon>0$ choose $q=q(p,\varepsilon)>p$ such that
$\varepsilon\geq\frac{q-p}{q}\frac{\frac{1}{2}-\frac{1}{p}}{\frac{1}{2}-\frac{1}{q}}$.
This finishes the proof and yields
$c(p,\varepsilon)=\frac{1}{2}(\frac{1}{p}-\frac{1}{q(p,\varepsilon)})(\frac{1}{2}-\frac{1}{q(p,\varepsilon)})^{-1}.$
∎

##  3\. Proofs of counterexamples: Propositions 1.8 and 1.9

We start with the proof of Proposition 1.8, by essentially repeating the proof
of Theorem 5 from [2] with one twist in the definition (3.1) below.

###### Proof of Proposition 1.8.

Denote by $T_{d}$ the $d$-th Chebyshev polynomial of the first kind and let
$f$ be the function

$f(\delta)=T_{d}\left(\frac{\delta_{1}+\cdots+\delta_{n}}{n}\right)+3,\qquad\delta\in\\{-1,1\\}^{n}.$
(3.1)

Adding the constant $3$ results in the estimates $\|f\|_{\infty}\leq 4$ and
$Q(f)\leq 2.$ Indeed, since Chebyshev polynomials take values in $[-1,1],$ the
function $f$ takes values in the interval $[2,4].$ This implies that
$\|f\|_{\infty}\leq 4$ and $2\leq m(f)\leq M(f)\leq 4,$ so that $Q(f)\leq 2.$
Since $T_{d}$ is a polynomial of degree $d$ we see that
$f\in\operatorname{Deg}_{\leq d}.$

Since $\Delta$ anihilates the constant function 3, using identities [2, eqs.
(89)–(91)] with $\varepsilon=(1,\ldots,1)$, we see that

$(\Delta
f)(1,\ldots,1)=\frac{n}{2}\left(T_{d}(1)-T_{d}(1-\frac{2}{n})\right),$

and consequently

$\|\Delta
f\|_{\infty}\geq\frac{n}{2}\left(T_{d}(1)-T_{d}(1-\frac{2}{n})\right)\geq\frac{n}{16}\left(T_{d}(1)-T_{d}(1-\frac{2}{n})\right)\cdot
Q(f)\cdot\|f\|_{\infty}.$

Since $T_{d}^{\prime}(1)=d^{2},$ letting $n\to\infty$ we have
$\lim_{n\to\infty}{\bf C}(n,d)\geq\frac{1}{8}d^{2}$, and the proof is
complete. ∎

###### Proof of Proposition 1.9.

The proof is a translation of the Kushilevitz function from [6, Section 6.3]
into our setting. We need to change the Hamming cube labelled by bits $0$ and
$1$ to the one we use and to realize that the translated function has the
desired properties.

Given two boolean functions $\tilde{f}\colon\\{0,1\\}^{m}\to\\{0,1\\}$ and
$\tilde{g}\colon\\{0,1\\}^{n}\to\\{0,1\\}$, we define their composition (which
is a function of $mn$ variables) via

$(\tilde{f}\diamond\tilde{g})(x_{11},\ldots,x_{mn})=\tilde{f}(\tilde{g}(x_{11},\ldots,x_{1n}),\ldots,\tilde{g}(x_{m1},\ldots,x_{mn})).$

Let $n=6^{k}$ and $d=3^{k}.$ The Kushilevitz function on the set
$\\{0,1\\}^{n}$ is defined via the $k$-fold composition
$\tilde{f}_{k}=\tilde{h}\diamond\tilde{h}\diamond\cdots\diamond\tilde{h}.$ The
function $\tilde{h}$ being composed is

$\tilde{h}(x_{1},\ldots,x_{6})=\sum_{i=1}^{6}x_{i}-\sum_{{i,j}\in{[6]\choose
2}}x_{i}x_{j}+\sum_{\\{i,j,k\\}\in K}x_{i}x_{j}x_{k},$

where $[6]\choose 2$ denotes the set of all $2$ element subsets of
$\\{1,\ldots,6\\}$ while

$\displaystyle K=\\{$
$\displaystyle\\{1,2,5\\},\\{1,2,6\\},\\{1,3,4\\},\\{1,3,6\\},\\{1,4,5\\},$
$\displaystyle\\{2,3,4\\},\\{2,3,5\\},\\{2,4,6\\},\\{3,5,6\\},\\{4,5,6\\}\\}.$

A straightforward but somewhat cumbersome case distinction reveals that the
function $\tilde{h}$ is {0,1}-valued. We refer to [6, Section 6.3] or [13,
Section 3.1] for further details.

Let $\tilde{f}\colon\\{0,1\\}^{n}\to\\{0,1\\}$ be a function. We say that a
coordinate (bit) $i$ is sensitive for $x\in\\{0,1\\}^{n}$ if flipping the
$i$-th bit results in flipping the output of $\tilde{f}$. The sensitivity of
$\tilde{f}$ on the input $x,$ denoted by $s(\tilde{f},x),$ is the number of
bits that are sensitive for $\tilde{f}$ on the input $x.$ The sensitivity of
$\tilde{f}$ is defined as the maximum
$s(\tilde{f})=\max_{x\in\\{0,1\\}^{n}}s(\tilde{f},x).$ It turns out that
$s(\tilde{f}_{k})=6^{k}=n$; see [13, Claim 3.2.1]. We also have
$\deg(\tilde{f}_{k})=3^{k}$; see [13, Claim 3.2.2]. Here,
$\deg(\tilde{f}_{k})$ denotes the degree of $\tilde{f}_{k}$ as a multilinear
polynomial.

Now, for each $k\in\mathbb{N}$ define the function
$f_{k}\colon\\{-1,1\\}^{n}\to\\{-1,1\\}$ via

$f_{k}(\delta)=2\tilde{f}_{k}\left(\frac{\delta_{1}+1}{2},\ldots,\frac{\delta_{n}+1}{2}\right)-1,\qquad\delta\in\\{-1,1\\}^{n}.$

Defining sensitivity for $f_{k}$ exactly as for $\tilde{f}_{k}$, it is clear
that $s(f_{k})=s(\tilde{f}_{k})=6^{k}.$ The second formula in [10, Proposition
2.37] reveals that $\|\Delta f_{k}\|_{\infty}=s(f_{k})=6^{k}.$ Since
$\deg(f_{k})=3^{k}$, we have $f_{k}\in\operatorname{Deg}_{\leq 3^{k}}$, and
thus

$\|\Delta f_{k}\|_{\infty}=6^{k}=3^{k\log 6/\log 3}=d^{\log 6/\log
3}\|f_{k}\|_{\infty}.$

This completes the proof of Proposition 1.9. ∎

## References

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# Superconducting Gap Structure of Filled Skutterudite LaOs4As12 Compound
through $\mu$SR Investigations

A. Bhattacharyya<EMAIL_ADDRESS>Department of Physics,
Ramakrishna Mission Vivekananda Educational and Research Institute, Belur
Math, Howrah 711202, West Bengal, India  D. T. Adroja
<EMAIL_ADDRESS>ISIS Facility, Rutherford Appleton Laboratory,
Chilton, Didcot Oxon, OX11 0QX, United Kingdom  Highly Correlated Matter
Research Group, Physics Department, University of Johannesburg, PO Box 524,
Auckland Park 2006, South Africa  A. D. Hillier  ISIS Facility, Rutherford
Appleton Laboratory, Chilton, Didcot Oxon, OX11 0QX, United Kingdom  P. K.
Biswas  Deceased ISIS Facility, Rutherford Appleton Laboratory, Chilton,
Didcot Oxon, OX11 0QX, United Kingdom

(August 28, 2024)

###### Abstract

Filled skutterudite compounds have gained attention recently as an innovative
platforms for studying intriguing low-temperature superconducting properties.
Regarding the symmetry of the superconducting gap, contradicting findings from
several experiments have been made for LaRu4As12 and its isoelectronic
counterpart, LaOs4As12. In this vein, we report comprehensive bulk and
microscopic results on LaOs4As12 utilizing specific heat analysis and muon-
spin rotation/relaxation ($\mu$SR) measurements. Bulk superconductivity with
$T_{C}$ = 3.2 K was confirmed by heat capacity. The superconducting ground
state of the filled-skutterudite LaOs4As12 compound is found to have two key
characteristics: superfluid density exhibits saturation type behavior at low
temperature, which points to a fully gapped superconductivity with gap value
of $2\Delta/k_{B}T_{C}$ = 3.26; additionally, the superconducting state does
not show any sign of spontaneous magnetic field, supporting the preservation
of time-reversal symmetry. These results open the door for the development of
La-based skutterudites as special probes for examining the interplay of
single- and multiband superconductivity in classical electron–phonon systems.

##  I Introduction

Due to their potential as thermoelectric materials for either refrigeration or
power generation applications, many filled skutterudite compounds with RT4X12
stoichiometry (R = alkali metals, alkaline earth metals, lanthanides, or light
actinides; T = Fe, Os, Ru; X = P, As, Sb) have lately been the focus of
several investigations [1, 2, 3]. With two formula units RT4X12 per unit cell,
these compounds form a body-centered cubic structure (space group $Im\bar{3}$,
No: 204). The structures consist of rigid covalently bonded cage-forming
frameworks T4X12 that encapsulate various bonded guest atoms R. This leads to
local anharmonic thermal vibrations (rattling modes), which would reduce
phononic heat conduction and open the door to their potential as promising
thermoelectric materials. Because of the significant hybridization between the
$4f$ band manifold and electronic conduction states, as well as the degree of
freedom provided by the R-$f$-derived multipole momenta of the cubically
symmetric X12 cages, those compounds may include a variety of distinct
electronic and magnetic ground states. For examples, consider unconventional
superconductivity [4, 5, 6, 7, 8], Kondo effect [9, 10, 11, 12, 13], heavy
fermios [14], non-Fermi liquid behavior [9], etc.

The majority of the Pr- and Ce-based filled skutterudite compounds are
hybridized gap semiconductors or show magnetic transitions, however PrOs4Sb12
[4, 5], PrRu4Sb12 [6] and PrRu4As12 [15] show superconducting transitions at
1.8 K, 0.97 K and 2.4 K, respectively. PrOs4Sb12 is highly intriguing for a
variety of reasons [16], including: (i) it is the first known example of a
heavy-fermion superconductor containing Pr; (ii) it shows unconventional
strong-coupling superconductivity that breaks time-reversal symmetry; and
(iii) instead of magnetic fluctuations, electric quadrupole fluctuations may
be involved in the superconducting pairing process. The unique band structure
of these compounds and the hybridization effects between localized $f$
electrons and conduction electrons appear to play a crucial role, in addition
to the fact that the origin of the majority of those unconventional
phenomenologies is unknown. It was recently revealed that the Fermi level of
La compounds is placed at a prominent peak arising from the T-$d$ band
manifold, which might contribute to electronic instability [1, 17]. Several
La-based compounds LaT4X12 are especially remarkable within the filled
skutterudite class due to their remarkable superconducting properties. For
examples, LaFe4P12 ($T_{C}$ = 4.1 K) [18], LaOs4P12 ($T_{C}$ = 1.8 K) [18,
19], and LaRu4Sb12 ($T_{C}$ = 3.6 K) [20, 9], with a special attention to the
LaRu4As12 ($T_{C}$ = 10.3 K, $H_{c2}$ = 10.2 T)- with the highest
superconducting transition temperature. [19, 15, 21].

The ratio of the heat capacity jump $\Delta C$ to $\gamma$TC is
$\Delta$C/($\gamma$TC)=1.75 for LaRu4As12 comparison to the BCS value of 1.43
[15]. While the majority of La-based filled skutterudites are completely
gapped superconductors, past research has shown numerous unique aspects of
LaRu4As12, such as a positive curvature of $H_{c2}$, nonexponential behavior
of the electronic heat capacity, and square root field dependency of the
Sommerfeld coefficient ($\gamma$) [22]. We recently reported unambiguous
evidence of multiband $s+s$-wave superconductivity in LaRu4As12 using muon-
spin rotation measurements, with $2\Delta_{1}/k_{B}T_{C}$ = 3.73 for the
larger gap and $2\Delta_{2}/k_{B}T_{C}$ = 0.144 for the smaller gap [23].
Furthermore, inelastic X-ray scattering experiments indicated essentially
temperature-independent phonon modes between 300 K and 20 K, with the
exception of 2 K, where a weak softening of the specific phonon modes is
detected [23]. All of these results demonstrate the relevance of the
electron–phonon interaction in the superconductivity of LaRu4As12, and they
accord well with the DFT-based phonon simulations [24].

Another isostructural La-based filled skutterudite compound, LaOs4As12, has
been reported by Shirotani et al. to exhibit superconductivity with $T_{C}$. =
3.2 K [21]. LaOs4As12 has also shown some signs of multiband
superconductivity, such as the upward curving of the upper critical field
around the transition temperature and unusual behavior in the electronic
specific heat data [25]. A single-gap, s-wave superconducting ground state,
however, is suggested by a recent study of the temperature dependency of lower
critical field [26]. Another study found that the high-amplitude lanthanum
phonons dominate the vibrational eigenmodes at low energies based on the
phonon dispersion relation determined from inelastic neutron scattering
experiments [27].

We have thus performed systematic muon-spin rotation and relaxation ($\mu$SR)
measurements to examine the superconducting pairing process in the LaOs4As12
compound. Contrary to prior experimental work asserting two-band
superconductivity [25], we demonstrate that the low-temperature behavior of
the superfluid density points to a fully gapped superconducting Fermi surface.
Furthermore, the preservation of time-reversal symmetry is confirmed by the
lack of spontaneous magnetic fields in the superconducting state, ruling out
unusual pairing processes. The transition from two-band to single-band
superconductivity in LaRu4As12 to LaOs4As12 is caused by differences in
interband coupling strength in the Fermi surface, as evidenced by the
different degrees of hybridization and electronic properties observed in the
Fermi surfaces of both compounds [28]. These results underline the
significance of LaRu4As12 and LaOs4As12 compounds as an important platform for
investigating filled skutterudites for the competition between single-band and
multiband superconductivity in electron–phonon driven systems.

Figure 1: A unit cell of the body-centered cubic LaOs4As12 structure with the
space group $Im\bar{3}$ that crystallizes within a CoAs3-type skutterudite
structure packed with La atoms. Green: As, Orange: Os, and Blue: La. Figure 2:
(a) Low-temperature specific heat of the filled skutterudite compound
LaOs4As12, expressed as $C_{e}$ vs $T$ in a zero magnetic field. The specific
heat data is shown in the inset. (b) The normalized electronic specific heat
($C_{e}/\gamma T_{C}$) versus inverse reduced temperature ($T_{C}/T$). The
heat capacity data are from [25].

##  II Experimental Details

The high-temperature molten-metal-flux technique, described in [29], was used
to grow single crystals of LaOs4As12. In a quartz ampule, elements with
purities higher than 99.9% and a molar ratio of La:Os:Cd:As $\rightarrow$
1:4:12:48 were combined. The details on the single crystal growth can be found
in [29]. The relaxation approach was used to measure the heat capacity in a
Quantum Design physical properties measurement (PPMS) system. Temperatures as
low as 0.38 K were attained utilizing a He-3 attachment to the PPMS [25].

The $\mu$SR measurements were carried out using small size unaligned single
crystals (0.1 mm $\times$ 0.1 mm $\times$ 0.1 mm, total mass 1 $\rm{g}$),
which gave powder average muon signal, of LaOs4As12. The MuSR spectrometer at
the Rutherford Appleton Laboratory, ISIS Neutron and Muon Source in the UK was
used to perform the $\mu$SR measurements [30]. In a $\mu$SR experiment, the
sample is injected with 100% spin-polarized muons. Each implanted muon
thermalizes, at which point it decays (lifetime $\tau_{\mu}$ = 2.2 $\mu$s)
into a positron (and two neutrinos) which is preferentially released in the
direction of the muon spin at the moment of decay. Utilizing detectors
carefully placed around the sample, the decay positrons are detected and time-
stamped. It is possible to calculate the asymmetry in the positron emission as
a function of time, $A(t)$, using the collected histograms from the forward
(F) and backward (B) detectors, $A(t)=\frac{N_{\mathrm{F}}(t)-\alpha
N_{\mathrm{B}}(t)}{N_{\mathrm{F}}(t)+\alpha N_{\mathrm{B}}(t)}$, where
$\alpha$ is a calibration factor for the instrument and $N_{\mathrm{F}}(t)$
and $N_{\mathrm{B}}(t)$ are the number of positrons counted in the forward and
backward detectors, respectively. Detectors are placed longitudinally during
ZF-$\mu$SR, and a correction coil is used to cancel out any stray magnetic
fields up to 10-4 mT. To investigate the time reversal symmetry ZF-$\mu$SR
measurements were carried out [31]. In the vortex state, TF-$\mu$SR
measurements were performed with applied fields of 20, 30, 40, 50, and 60 mT,
which is greater than the lower critical field $H_{\mathrm{c1}}$ ($\sim$5 mT)
and lower than the upper critical field $H_{\mathrm{c2}}$ ($\sim$1 T) [21].
The sample was covered using a thin silver foil after being mounted onto a
high purity (99.995%) silver sample holder using diluted GE-varnish. The
sample was cool down to 300 mK using a dilution refrigerator. To generate the
vertex lattice by trapping the applied TF, we applied field above
$T_{\mathrm{C}}$ and then sample was cooled in the field to the base
temperature of 300 mK. We used WiMDA [32] software to analyze the $\mu$SR
data.

##  III Results and discussion

###  III.1 Crystal Structure & Physical Properties

LaOs4As12 crystallizes in a CoAs3-type skutterudite structure packed with La
atoms and has a body-centered cubic structure with the space group $Im\bar{3}$
(No. 204) as shown in Figure 1. The large icosahedron cage made of As atoms is
located around the electropositive La sites, which lack four-fold rotational
symmetry. Between the cages, a transition metal ion called Os forms a cubic
sublattice. The low temperature specific heat measurements $C_{P}$ as a
function of temperature at zero magnetic field are shown in the inset of
Figure 2a. Using the equations $C_{P}=\gamma T+\beta T^{3}$, the normal state
heat capacity is fitted. We calculated the lattice contribution to the
specific heat $\beta$ = 0.613 mJ/mol K4 and the electronic parameter
(Sommerfeld’s coefficient) $\gamma$ = 90.47 mJ/mol K2 from this. The Debye
temperature is determined using the Debye model as
$\Theta_{D}=\left(\frac{12\pi^{4}nR}{5\beta}\right)^{1/3}$, where $R$ is the
universal gas constant, which is 8.314 J/mol-K, and $n$ denotes the number of
atoms in the compound (n = 17). The value of $\Theta_{D}$ is thus calculated
to be approximately 377 K, which agrees with the previous measurement [25,
22]. Figure 2a displays the low-$T$ electronic specific heat $C_{e}$ that was
produced after the phonon contribution was taken into account. The heat
capacity jump at $T_{C}$ ($\Delta C_{e}/\gamma T_{C}$) is calculated to be
1.2, which is less than 1.43 the value expected for a weak-coupling BCS
superconductivity. The fit to the exponential temperature dependency of
$C_{e}(T)$ yields $\Delta(0)=0.40$ meV, which is close to the 0.45 meV value
obtained from the TF-$\mu$SR data analysis (discussed in section-B). Thus, the
value of $2\Delta(0)/k_{B}T_{C}$ = 2.9, which is less than the 3.53
anticipated for weak-coupling BCS superconductors. However, the linear fitting
shown in Figure 2b shows that this material exhibits BCS behavior with a
single isotropic gap.

Figure 3: Left panel: Asymmetry spectra of the TF-$\mu$SR in the low time
region obtained in 20 mT and 50 mT applied magnetic fields at (a–c) $T$ = 0.3
K (i.e., below $T_{\mathrm{C}}$) and (b–d) $T$ = 3.5 K (i.e., above
$T_{\mathrm{C}}$). Right panel: (e) The superconducting depolarization rate
$\sigma_{sc}$ as a function of temperature in the presence of an applied field
of 20 $\leq\mu_{0}$ H $\leq$ 60 mT. (f) The magnetic field dependence of the
muon spin depolarization rate is shown for a range of different temperatures.
The solid lines are the results of fitting the data using Brandt’s equation as
discussed in Equation (2).

###  III.2 Superconducting Gap Structure: TF-$\mu$SR

The pairing mechanism and superconducting gap structure of the LaOs4As12 were
investigated by TF-$\mu$SR experiments down to 0.3 K. The TF-$\mu$SR asymmetry
time spectra in the presence of 20 mT and 50 mT applied magnetic fields at
above and below $T_{\mathrm{C}}$ are shown in Figures 3a–d. Because of the
extra inhomogeneous field distribution of the vortex lattice generated inside
the superconducting mixed state of LaOs4As12, the spectrum in Figure 3a,c in
the superconducting state at 0.3 K demonstrate a greater relaxation. Using the
Gaussian damped decay function, the asymmetry spectra were fitted [33, 34, 35]
using the following equation,

$\begin{split}A_{\mathrm{TF}}(t)=A_{\mathrm{sc}}\exp\left(-\frac{\sigma_{TF}^{2}t^{2}}{2}\right)\cos(\gamma_{\mu}B_{sc}t+\phi)+\\\
A_{\mathrm{bg}}\cos(\gamma_{\mu}B_{bg}t+\phi).\end{split}$ (1)

Figure 4: Left panel : (a) The inverse magnetic penetration depth squared as a
function of temperature is shown here. The lines show the fits using $s$-wave
(red), $s+s$-wave (light green) and $d$-wave (blue) gap functions. Right panel
: (b) Shows the ZF-$\mu$SR spectra for LaOs4As12 at 0.3 K (blue) and 4 K
(light green). The solid line fits to the experimental data points, as stated
in the text.

The gyromagnetic muon ratio is $\gamma_{\mu}/2\pi$ = 135.53 MHz/T, and the
initial asymmetries of muons stopping on the sample and on the silver holder
are $A_{sc}$ and $A_{bg}$, respectively (constant across the entire
temperature range). The local fields Bsc and Bbg represent muons stopping on
the sample and on the sample holder, respectively, whereas $\phi$ represents
initial phase value and $\sigma_{TF}$ represents the Gaussian depolarization
rate. We calculated the values of $A_{sc}$ = 76% and $A_{bg}$ = 24% of the
total asymmetries by fitting 0.3 K data. When additional temperature data were
analyzed, $A_{bg}$ was kept constant and $A_{sc}$ was found nearly temperature
independent. The emergence of bulk superconductivity is indicated by an
increase in the $\sigma_{TF}$ rate as the system approaches the
superconducting state. With the use of the following formula, the
superconducting contribution to the relaxation $\sigma_{sc}$ was determined,
$\sigma_{\mathrm{sc}}=\sqrt{\sigma_{\mathrm{TF}}^{2}-\sigma_{\mathrm{nm}}^{2}}$,
where the nuclear magnetic dipolar contribution, is denoted by the symbol
$\sigma_{nm}$, which is derived from high-temperature fits and is temperature
independent. Figure 3e depicts the temperature dependence of $\sigma_{sc}$ in
several applied TF fields. Due to low $H_{\mathrm{c2}}$ value, as seen in
Figure 3f, $\sigma_{\mathrm{sc}}$ depends on the applied field. Brandt
demonstrated that the London penetration depth $\lambda_{L}(T)$ is linked to
$\sigma_{\mathrm{sc}}$ for a superconductor with
$H_{\mathrm{ext}}/H_{\mathrm{c2}}$ $\leq$ 0.25 [36, 37].

$\begin{split}\sigma_{\mathrm{sc}}[\mu s^{-1}]=4.83\times
10^{4}(1-H_{\mathrm{ext}}/H_{\mathrm{c2}})\\\
\times\\{1+1.21\left[1-\sqrt{{(H_{\mathrm{ext}}/H_{\mathrm{c2}}})}\right]^{3}\\}\lambda_{L}^{-2}[nm].\end{split}$
(2)

This relationship has been used to compute the temperature dependency of
$\lambda_{L}(T)$. As demonstrated in Figure 3f, isothermal cuts perpendicular
to the temperature axis of $\sigma_{sc}$ data sets were utilized to estimate
the $H$-dependence of the depolarization rate $\sigma_{sc}(H)$. The normalized
$\lambda_{L}^{-2}(T)/\lambda_{L}^{-2}(0)$ temperature variation, which is
directly proportional to superfluid density, is shown in Figure 4a. The data
were fitted using the following equation [38, 39]:

$\displaystyle\frac{\sigma_{sc}(T)}{\sigma_{sc}(0)}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{L}^{-2}(T)}{\lambda_{L}^{-2}(0)}$
$\displaystyle=$ $\displaystyle
1+\frac{1}{\pi}\int_{0}^{2\pi}\int_{\Delta(T)}^{\infty}\left(\frac{\delta
f}{\delta E}\right)\times\frac{EdEd\phi}{\sqrt{E^{2}-\Delta(T,\phi})^{2}},$

where $f=[1+\exp(\frac{E}{k_{B}T})]^{-1}$ is the Fermi function. We take
$\Delta_{k}(T,\phi)=\Delta(T)\mathrm{g}_{k}(\phi)$, where we assume a
temperature dependence that is universal
$\Delta(T)=\Delta_{0}\tanh[1.82\big{\\{}1.018(T_{\mathrm{C}}/T-1)\big{\\}}^{0.51}]$.
The magnitude of the gap at 0 K is $\Delta_{0}$, and the function
$\mathrm{g}_{k}$ denotes the gap’s angular dependence, which is equal to 1 for
one isotropic energy gap $s$, 1 for two isotropic $s+s$ wave energy gap and
$\cos(2\phi)$ for d-wave gap, where $\phi$ is the azimuthal angle along the
Fermi surface.

Figure 4a illustrates our comparison of three distinct gap models: employing a
single isotropic $s$-gap wave, a multigap $s+s$-wave gap, and a nodal $d$-wave
gap. As seen in the figure, the superfluid density saturates at low
temperatures, which is a characteristic of the $s$-wave model with a single
gap. An isotropic single-band $s-$wave model with a gap value of 0.45 meV
provides the best representation of the data, with a gap to $T_{\mathrm{C}}$
ratio $2\Delta(0)/k_{\mathrm{B}}T_{\mathrm{C}}$ = 3.26, which is less than the
BCS weak-coupling limit (=3.53). On the other hand, the substantial rise in
the $\chi^{2}$ value puts the $d$-wave model and $s+s$-wave (multigap) model
inappropriate for this system. A two-gap $s$+$s$-wave model of multiband
superconductivity has been shown to be compatible with the temperature
dependence of magnetic penetration depth of LaRu4As12. The higher gap to
$T_{C}$ ratio computed in the $s+s$-wave scenario,
2$\Delta_{1}(0)/k_{\mathrm{B}}T_{C}$ = 3.73, is fairly comparable to the value
of 3.53 for BCS superconductor in case of LaRu4As12 [23]. For LaRu4As12, 2 K
specific phonon modes exhibit modest softening when compared to 20 K,
demonstrating that the electron–phonon interactions causing the
superconductivity have an audible impact on the vibrational eigenstates [23].
Using McMillan’s relation, it is also possible to determine the
electron–phonon coupling constant ($\lambda_{\mathrm{e-ph}}$) [40]:

$\lambda_{\mathrm{e-ph}}=\frac{1.04+\mu^{*}\ln(\Theta_{\mathrm{D}}/1.45T_{\mathrm{C}})}{(1-0.62\mu^{*})\ln(\Theta_{\mathrm{D}}/1.45T_{\mathrm{C}})-1.04}.$
(4)

where $\mu^{*}$ is the repulsive screened Coulomb parameter usually assigned
as $\mu^{*}$ = 0.13. The calculated value of the $\lambda_{\mathrm{e-ph}}$ is
0.534. The London model is described as
$\lambda_{\mathrm{L}}^{2}=m^{*}c^{2}/4\pi n_{\mathrm{s}}e^{2}$. It connects
the effective mass enhancement m∗ [$=(1+\lambda_{e-ph})*m_{e}]$,
superconducting carrier density $n_{\mathrm{s}}$ [$=m^{*}c^{2}/4\pi
e^{2}\lambda_{L}(0)^{2}$], and London penetration depth. By employing the
$s$-wave model, we determined the London penetration depth of $\lambda_{L}(0)$
= 168 nm. The effective mass enhancement is calculated to be
$m^{*}=1.53\leavevmode\nobreak\ m_{\mathrm{e}}$, and the superconducting
carrier density is predicted to be $n_{\mathrm{s}}=1.53\times 10^{27}$
carriers m-3. References [41, 42, 24] include a description of the
computations in detail. The calculated values of $\lambda_{L}(0)$ = 240
nm,$n_{\mathrm{s}}=8.6\times 10^{27}$ carriers m-3 and
$m^{*}=1.749\leavevmode\nobreak\ m_{\mathrm{e}}$ for LaRu4As12 [23]. The
fitted parameters for LaOs4As12 and LaRu4As12 (for comparison) are shown in
Table 1. To explain the observed nature of the superconducting gap structures,
it is important to comprehend the electronic structures of these compounds,
which have been carried [28] and the results suggest that the single-band
order parameter in LaOs4As12 seems to be associated with the hybridized As-p
and Os-d electronic character of the Fermi surface. On the other hand, the
lack of hybridization for the disjointed Fermi surface of LaRu4As12, may
explain its multiband superconducting nature.

Table 1: Fitted parameters for LaOs4As12 and LaRu4As12. Model |  $\Delta_{i}(0)$ (meV) | $2\Delta_{i}(0)/k_{B}T_{C}$ |  $T_{C}$ (K) |  $\lambda_{L}(0)$ (nm)
---|---|---|---|---
LaOs4As12 (s-wave) | 0.45 | 3.26 | 3.2 | 168
LaRu4As12 (s + s-wave) | 1.656, 0.064 | 3.73, 0.144 | 10.3 | 240

###  III.3 Preserved Time Reversal Symmetry: ZF-$\mu$SR

In order to determine if there is a spontaneous magnetic field present in the
superconducting ground state, we conducted the ZF-$\mu$SR experiment. Figure
4b shows the time evolution of the asymmetry spectra for $T$ = 0.3 K
$<T_{\mathrm{C}}$ and $T$ = 3.5 K $>T_{\mathrm{C}}$. The ZF-$\mu$SR spectra
recorded in the normal and superconducting states show the same relaxations
that can be found in overlapping ZF-$\mu$SR spectra, indicating that the
superconducting state does not shows any spontaneous magnetic field or spin
fluctuations. This result suggests that the time-reversal symmetry is
preserved in LaOs4As12 superconducting state. The strong resemblance of the
ZF-$\mu$SR spectra (above and below TC) suggests that the time-reversal
symmetry is also retained in the superconducting state of LaRu4As12. In order
to fit the ZF data, a Lorentzian function was used [43],

$G_{\mathrm{ZF}}(t)=A_{\mathrm{sc}}(t)\exp{(-\lambda_{ZF}t)}+A_{\mathrm{bg}},$
(5)

where $\lambda_{ZF}$ is the electronic relaxation rate, $A_{\mathrm{sc}}$
stands for the sample asymmetry, $A_{\mathrm{bg}}$ for the constant
nondecaying background signal. The red line in Figure 4b indicates the fits to
the ZF-$\mu$SR data. The ZF-$\mu$SR asymmetry data fitting parameters are
$\lambda_{ZF}$ = 0.754(4) $\mu\mathrm{s}^{-1}$ at 0.3 K and $\lambda_{ZF}$ =
0.744(5) $\mu\mathrm{s}^{-1}$ at 3.5 K. No conclusive evidence of TRS breaking
can be found since the relaxation rate change is within the error bar.

##  IV Summary

We employed TF-$\mu$SR to determine the gap symmetry of the superconducting
state of LaOs4As12. An isotropic BCS-type $s$-wave gap model explains the
temperature dependence of the superfluid density. The gap to $T_{\mathrm{C}}$
ratio, which was determined from the $s$-wave gap fit to the superfluid
density, is 3.26; nonetheless, this is smaller than 3.53 expected for
conventional BCS systems. The ZF-$\mu$SR spectra at 0.3 K and 3.5 K are
strikingly similar, indicating that the time-reversal symmetry is intact.
These results open up the possibility of using the compounds LaRu4As12 and
LaOs4As12 as special research platforms for investigating filled skutterudites
for the interplay between single- and multiband superconducting order
parameters in conventional systems.

### Acknowledgements

We thank T. Cichorek and J. Juraszek for providing LaOs4As12 sample and the
ascii heat capacity data. We would like to thank T. Cichorek, P. P. Ferreira,
R. Lucrezi, J. Juraszek, C. Heil and L. T. F. Eleno for interesting
discussions. AB expresses gratitude to the Science and Engineering Research
Board for the CRG Research Grant (CRG/2020/000698 & CRG/2022/008528) and CRS
Project Proposal at UGC-DAE CSR (CRS/2021-22/03/549). DTA appreciates the
support provided by the Royal Society of London for the Newton Advanced
Fellowship between the UK and China, the International Exchange between the UK
and Japan, and EPSRC-UK (Grant number EP/W00562X/1). We thanks the ISIS
Facility for the beam time, RB1520431 [44].

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