# The Leray transform: distinguished measures, symmetries and polygamma
inequalities

Luke D. Edholm & Yonatan Shelah  Department of Mathematics
Universität Wien, Vienna, Austria<EMAIL_ADDRESS>Department
of Mathematics
University of Ljubljana, Slovenia<EMAIL_ADDRESS>
###### Abstract.

New symmetries, norm computations and spectral data are obtained for the Leray
transform on a class of unbounded hypersurfaces in ${\mathbb{C}}^{2}$. Focus
is placed on certain key measures where monotonicity results and related
polygamma inequalities are established, with Bernstein-Widder and Euler-
Maclaurin formulae playing critical roles. Underpinning this work is a
projective geometric theory of duality, manifesting here as Hölder invariance.

The first author was supported by FWF Grants DOI 10.55776/P36884 and DOI
10.55776/I4557.
The second author was supported by ERC Advanced Grant 101053085
2020 Mathematics Subject Classification: 32A26 (Primary); 26D15; 33B15; 47A30;
65B15 (Secondary)

##  1\. Introduction

The Leray transform $\bm{L}$ is a higher dimensional analogue of the familiar
Cauchy transform on domains in the complex plane. A body of recent work [5, 6,
7, 8, 9, 16, 17, 18, 19] has investigated the mapping and invariance
properties of this operator, with the construction of interesting new
holomorphic function spaces on domains in ${\mathbb{C}}^{n}$ and
${\mathbb{C}}\mathbb{P}^{n}$ seen as an important application. In [7], Barrett
and Edholm studied the Leray transform on unbounded ${\mathbb{C}}$-convex
hypersurfaces given by

(1.1)
$M_{\gamma}=\\{(\zeta_{1},\zeta_{2})\in{\mathbb{C}}^{2}:\text{Im}{(\zeta_{2})}=|\zeta_{1}|^{\gamma}\\},\qquad\gamma>1.$

In the present paper, further analysis of the Leray transform on $M_{\gamma}$
is carried out and many new analytic quantities (some of which were discussed,
but not calculated, in the previous work) are computed explicitly. Thoughout
the paper, it is always assumed that $\gamma>1$.

###  1.1. Two distinguished measures

Let us parameterize $M_{\gamma}$ by
$(\zeta_{1},\zeta_{2})=(re^{i\theta},s+ir^{\gamma})$ and consider measures of
the form $\mu_{d}=r^{d}\,dr\wedge d\theta\wedge ds$, $d\in{\mathbb{R}}$. The
rotational symmetry of $M_{\gamma}$ in $\zeta_{1}$ yields an orthogonal
decomposition of the Leray transform
$\bm{L}=\bigoplus_{k=0}^{\infty}\bm{L}_{k}$; see Section 2.3. The behavior of
$\bm{L}$ in the spaces $L^{2}(M_{\gamma},\mu_{d})$ can be understood
explicitly, with surprising connections to projective geometry and special
function theory. Many precise results are obtained in [7] for each
$\bm{L}_{k}$; this is reviewed in Sections 1.2 and 2.3.

Two measures are of particular importance to higher-dimensional Cauchy-Leray
theory: the pairing measure $\sigma$ and the preferred measure $\nu$; see [5,
6, 7]. On $M_{\gamma}$, they take the form

(1.2) $\displaystyle\sigma=r^{\gamma-1}\,dr\wedge d\theta\wedge ds,$ (1.3)
$\displaystyle\nu=r^{\frac{\gamma+1}{3}}\,dr\wedge d\theta\wedge ds.$

The pairing measure $\sigma$ arises as (a constant multiple of) the Leray-Levi
measure corresponding to the natural defining function
$\rho(\zeta)=|\zeta_{1}|^{\gamma}-\text{Im}(\zeta_{2})$; see (2.3b). This
measure also appears in an alternative description of the Leray transform,
facilitating a natural bilinear pairing between projectively invariant dual
Hardy spaces; see [7, Proposition 6.18].

The preferred measure $\nu$ admits a projective transformation law; see (2.4).
This leads to a corresponding Leray transformation law in (2.6), and to the
definition of the aforementioned Hardy spaces in (2.7). These notions were
introduced by Barrett [5, Sections 7 and 8] and developed further in [6, 7];
see [7, Section 2] for a detailed discussion in the $M_{\gamma}$ setting.

######  Remark 1.4.

Note that in the Heisenberg case ($\gamma=2$), the pairing and preferred
measures are equal and coincide with Lebesgue measure $\mu_{1}$ on the
parameter space ${\mathbb{R}}^{3}$. $\lozenge$

###  1.2. Review of past results on $M_{\gamma}$

In [7], Barrett and Edholm completely characterize those $d$ for which the
Leray transform and each of its constituent pieces $\bm{L}_{k}$ (sub-Leray
operators) are bounded on $L^{2}(M_{\gamma},\mu_{d})$, $\gamma>1$:

######  Proposition 1.5 (Barrett-Edholm [7]).

The Leray transform $\bm{L}$ is a bounded operator from
$L^{2}(M_{\gamma},\mu_{d})$ to itself if and only if
$d\in(-1,2\gamma-1):={\mathcal{I}}_{0}(\gamma)$.

From the forms of (1.2) and (1.3) notice that $\bm{L}$ is bounded on
$L^{2}(M_{\gamma},\sigma)$ and $L^{2}(M_{\gamma},\nu)$.

We now recall for non-negative integers $k$ the Leray symbol function of
$L^{2}(M_{\gamma},\mu_{d})$:

(1.6)
$J(d,\gamma,k)=\frac{\Gamma\big{(}\frac{2k+1+d}{\gamma}\big{)}\Gamma\big{(}2k+2-\frac{2k+1+d}{\gamma}\big{)}}{\Gamma(k+1)^{2}}\left(\frac{\gamma}{2}\right)^{2k+2}(\gamma-1)^{-\left(2k+2-\frac{2k+1+d}{\gamma}\right)}.$

Due to the prominence of $\sigma$ and $\nu$ in this work, we often use
alternate notation when the symbol function corresponds to $d=\gamma-1$ and
$d=\tfrac{\gamma+1}{3}$, as indicated by (1.2) and (1.3). Set

(1.7a) $\displaystyle C_{\sigma}(\gamma,k)$
$\displaystyle=J(\gamma-1,\gamma,k),$ (1.7b) $\displaystyle C_{\nu}(\gamma,k)$
$\displaystyle=J\big{(}\tfrac{\gamma+1}{3},\gamma,k\big{)},$

and call $C_{\sigma}$ and $C_{\nu}$ the pairing symbol and preferred symbol
functions, respectively.

The square-root of the Leray symbol function measures the norm of each
$\bm{L}_{k}$:

######  Proposition 1.8 (Barrett-Edholm [7]).

$\bm{L}_{k}$ is bounded on $L^{2}(M_{\gamma},\mu_{d})$ if and only if

(1.9) $d\in(-2k-1,(2k+2)(\gamma-1)+1):={\mathcal{I}}_{k}(\gamma).$

When $d$ is taken in this range, the norm

(1.10)
$\left\|\bm{L}_{k}\right\|_{L^{2}(M_{\gamma},\mu_{d})}=\sqrt{J(d,\gamma,k)}.$

These intervals satisfy
${\mathcal{I}}_{k}(\gamma)\subsetneq{\mathcal{I}}_{k+1}(\gamma)$, and they
exhaust the real line as $k\to\infty$. This means for any given $d$, at most a
finite number of the $\bm{L}_{k}$ are unbounded on
$L^{2}(M_{\gamma},\mu_{d})$.

It also happens that for every $d$, the symbol $J(d,\gamma,k)$ (and thus the
corresponding norm of the sub-Leray operators) stabilizes to the same value at
high $k$-frequencies:

######  Proposition 1.11 (Barrett-Edholm [7]).

For every $d\in{\mathbb{R}}$, the following limit holds

$\lim_{k\to\infty}\left\|\bm{L}_{k}\right\|_{L^{2}(M_{\gamma},\mu_{d})}=\sqrt{\frac{\gamma}{2\sqrt{\gamma-1}}}.$

See Figure 1 for an illustration when $\gamma=5$. There is an intriguing
relationship between the limit in Proposition 1.11 and a projective geometric
invariant on $M_{\gamma}$. This is related the Leray essential norm
conjecture; see [7, Section 2.3].

###  1.3. New symmetries, norms and symbol function monotonicity

While the results summarized in Section 1.2 shed considerable light on the
Cauchy-Leray theory of $M_{\gamma}$, many questions are left open. In
particular,

  * $(i)$

For $d\in{\mathcal{I}}_{0}(\gamma)$, what is the norm in
$L^{2}(M_{\gamma},\mu_{d})$ of the full Leray transform?

In [7] the $L^{2}$-norm of $\bm{L}$ was calculated only in the case of the
pairing measure $\sigma$ (where $d=\gamma-1$). It is of particular interest to
compute this norm with respect to the preferred measure $\nu$ (where
$d=\tfrac{\gamma+1}{3}$), as this quantifies the efficiency of a natural
duality pairing of projectively invariant Hardy spaces; see Proposition 2.8.

  * $(ii)$

What is the relationship between the norms of the $\bm{L}_{k}$ as $k$ varies?
Given $\gamma>1$, are there conditions on $d$ guaranteeing that the function
$k\mapsto J(d,\gamma,k)$ is monotone?

In [7] it is shown that for $\gamma\neq 2$, the pairing symbol function
$k\mapsto C_{\sigma}(\gamma,k)$ is strictly decreasing. Monotonicity, however,
fails for other choices of $d$, while yet other $d$-values correspond to a
strictly increasing symbol function; see Figure 1.

  * $(iii)$

To what extent is there a relationship between the Cauchy-Leray theories of
$M_{\gamma}$ and $M_{\gamma^{*}}$, where $\gamma^{*}=\frac{\gamma}{\gamma-1}$
is the Hölder conjugate of $\gamma$?

In [7] the pairing symbol function is shown to be Hölder symmetric, i.e.,
$C_{\sigma}(\gamma,k)=C_{\sigma}(\gamma^{*},k)$. This observation meshes well
with our understanding of Cauchy-Leray theory from a projective dual point of
view, as $M_{\gamma^{*}}$ is known to be the projective dual hypersurface of
$M_{\gamma}$; see [7, Section 6.1]. But it was not clarified in the previous
work the extent to which a more general version of this correspondence holds,
and in particular, whether the preferred symbol function $C_{\nu}(\gamma,k)$
is Hölder invariant.

####  1.3.1. Norm computations

We answer question $(i)$ above for a wide range of measures.

In the $\gamma=2$ case, we obtain the norm of $\bm{L}$ in
$L^{2}(M_{\gamma},\mu_{d})$ for all $d$ for which the operator is bounded.
When $\gamma\neq 2$, our results below include as special cases the explicit
$L^{2}$-norms with respect to Lebesgue measure $\mu_{1}$ on
${\mathbb{R}}^{3}$, the pairing measure $\sigma$ (previously calculated in [7]
by a different argument), and most importantly, the preferred measure $\nu$.

We now summarize these results, emphasizing that in the special case when
$\gamma=2$ and $d=1$, the measures $\mu_{1}=\sigma=\nu$ coincide, and the norm
of $\left\|\bm{L}\right\|_{L^{2}(M_{2},\mu_{1})}=1$. In this setting, the
Leray transform coincides with the Szegő projection.

######  Theorem 1.12.

The norm of the Leray transform is calculated in many settings, including

  * $(a)$

The norm of the Leray transform on $L^{2}(M_{2},\mu_{d})$ is given by

$\left\|\bm{L}\right\|_{L^{2}(M_{2},\mu_{d})}=\begin{cases}\sqrt{\frac{\pi}{2}(1-d)\sec{\big{(}\frac{d\pi}{2}\big{)}}},&d\in(-1,1)\cup(1,3)\\\
1,&d=1.\end{cases}$

  * $(b)$

The norm of the Leray transform on $L^{2}(M_{\gamma},\mu_{1})$ is given by

$\left\|\bm{L}\right\|_{L^{2}(M_{\gamma},\mu_{1})}=\begin{cases}(\gamma-1)^{\frac{1}{\gamma}-1}\sqrt{\frac{\pi}{4}(\gamma-2)\gamma\csc\big{(}\frac{2\pi}{\gamma}\big{)}},&\gamma\in(1,2)\cup(2,\infty)\\\
1,&\gamma=2.\end{cases}$

  * $(c)$

The norm of the Leray transform on $L^{2}(M_{\gamma},\sigma)$ is given by

$\left\|\bm{L}\right\|_{L^{2}(M_{\gamma},\sigma)}=\frac{\gamma}{2\sqrt{\gamma-1}}.$

  * $(d)$

The norm of the Leray transform on $L^{2}(M_{\gamma},\nu)$ is given by

$\left\|\bm{L}\right\|_{L^{2}(M_{\gamma},\nu)}=\sqrt{\frac{\gamma}{2\sqrt{\gamma-1}}}.$

Proofs of the parts of Theorem 1.12 appear throughout the paper. Part $(a)$ is
found in Corollary 3.16. Parts $(b)$ and $(c)$ are found in Section 4, where
the Bernstein-Widder theorem and the notion of complete montonicity play
important roles; see Corollaries 4.25 and 4.27, which themselves are “endpoint
cases” of the more general Theorem 4.23. The preferred measure result, part
$(d)$, is the most difficult to prove. In Section 5 and the Appendix, we start
from the Euler-Maclaurin summation formula and carry out lengthy and precise
analysis, reducing the original problem to concrete questions about two
explicit polynomials (of degree 16 and 22) that are answered using Descartes’
Rule of Signs.

Figure 1. Behavior of $k\mapsto J(d,5,k)$ for certain $1\leq d\leq 4$.

####  1.3.2. Monotonicity

Question $(ii)$ above is also answered for a wide range of measures.

Figure 1 illustrates how the behavior of $k\mapsto
J(d,\gamma,k)=\left\|\bm{L}_{k}\right\|^{2}_{L^{2}(M_{\gamma},\mu_{d})}$ can
change as $d$ varies in the boundedness interval
${\mathcal{I}}_{0}(\gamma)=(-1,2\gamma-1)$. In our work below, we see that for
many measures (including $\mu_{1}$ and $\sigma$), this function strictly
decreases in $k$, while for others the function is not monotone. The preferred
measure $\nu$, on the other hand, corresponds to a strictly increasing symbol
function. This is now summarized.

We again highlight the case $\gamma=2$ and $d=1$: by (1.6), for each $k$,
$\left\|\bm{L}_{k}\right\|^{2}_{L^{2}(M_{2},\mu_{1})}=1$.

######  Theorem 1.13.

The sub-Leray operators $\bm{L}_{k}$ exhibit the following monotone behavior.

  * $(a)$

If $\gamma=2$ and $d\in(-1,1)\,\cup\,(1,3)$, the function $k\mapsto
J(d,\gamma,k)$ is strictly decreasing on the non-negative integers. Thus

$\left\|\bm{L}_{k}\right\|_{L^{2}(M_{2},\mu_{d})}>\left\|\bm{L}_{k+1}\right\|_{L^{2}(M_{2},\mu_{d})}.$

  * $(b)$

If $\gamma>2$ and $d\in(-1,1]\cup[\gamma-1,2\gamma-1)$, the function $k\mapsto
J(d,\gamma,k)$ is strictly decreasing on the non-negative integers. Thus

$\left\|\bm{L}_{k}\right\|_{L^{2}(M_{\gamma},\mu_{d})}>\left\|\bm{L}_{k+1}\right\|_{L^{2}(M_{\gamma},\mu_{d})}.$

  * $(c)$

If $\gamma<2$ and $d\in(-1,\gamma-1]\cup[1,2\gamma-1)$, the function $k\mapsto
J(d,\gamma,k)$ is strictly decreasing on the non-negative integers. Thus

$\left\|\bm{L}_{k}\right\|_{L^{2}(M_{\gamma},\mu_{d})}>\left\|\bm{L}_{k+1}\right\|_{L^{2}(M_{\gamma},\mu_{d})}.$

  * $(d)$

Let $\gamma\neq 2$. The preferred symbol function $k\mapsto
C_{\nu}(\gamma,k)=J\big{(}\frac{\gamma+1}{3},\gamma,k\big{)}$ is strictly
increasing on the non-negative integers. Thus,

$\left\|\bm{L}_{k}\right\|_{L^{2}(M_{\gamma},\nu)}<\left\|\bm{L}_{k+1}\right\|_{L^{2}(M_{\gamma},\nu)}.$

Theorems 1.12 and 1.13 are closely related. The monotonicity result in part
$(a)$ is shown in Theorem 3.13, parts $(b)$ and $(c)$ are shown in Theorem
4.23, and part $(d)$ in Theorem 5.22. The results here depend on careful
analysis of certain polygamma functions, previewed in Section 1.4 below. The
montonicity and norm results above have immediate implications for the spectra
of the related self-adjoint and anti-self-adjoint operators
$\bm{L}^{*}\bm{L}$, $\bm{L}\bm{L}^{*}$ and $\bm{L}^{*}-\bm{L}$; see [7,
Section 5.3].

####  1.3.3. Hölder invariance

We answer question $(iii)$ above by proving that a form of Hölder invariance
holds for every pair $\gamma\neq 2$, $d\in{\mathbb{R}}$. Let
$\gamma^{*}=\tfrac{\gamma}{\gamma-1}$ be the Hölder conjugate of $\gamma$.

In [7] it was shown that the pairing symbol function $(d=\gamma-1)$ remains
the same when $\gamma$ is replaced with $\gamma^{*}$. To obtain a
corresponding symmetry for more general measures, we parameterize the exponent
of $\mu_{d}$ by setting $d$ to be equal to

$\delta_{a}(\gamma)=a(\gamma-2)+1,\qquad a\in{\mathbb{R}}.$

######  Theorem 1.14.

The symbol function $J(\delta_{a}(\gamma),\gamma,k)$ is Hölder symmetric, in
that

$J(\delta_{a}(\gamma),\gamma,k)=J(\delta_{a}(\gamma^{*}),\gamma^{*},k).$

This theorem is proved in Section 3.1. Taking $a=\tfrac{1}{3}$ corresponds to
$d=\tfrac{\gamma+1}{3}$, implying the Hölder symmetry of the preferred symbol
function, i.e., $C_{\nu}(\gamma,k)=C_{\nu}(\gamma^{*},k)$. Taking $a=0$ shows
that $J(1,\gamma,k)=J(1,\gamma^{*},k)$, verifying the analogous invariance
property holds for the symbol functions assigned to Lebesgue measure
$\mu_{1}$. Hölder invariance plays an important role in the proofs of both
Theorems 1.12 and 1.13 above and can be viewed as a manifestation of the
projective dual nature of Cauchy-Leray theory; see Section 2.1.

###  1.4. Polygamma inequalities and complete monotonicity

Recall the digamma function $\psi$, which is the logarithmic derivative of the
familiar $\Gamma$-function:

(1.15) $\psi(r)=\frac{\Gamma^{\prime}(r)}{\Gamma(r)}.$

Further derivatives of $\psi$ are called polygamma functions. These functions
satisfy interesting functional identities and naturally arise in many analytic
and number theoretic problems; see [1]. The polygamma functions admit
practical representations, both in series form (3.17) and integral form (4.5).

The following combination of polygamma functions plays a central role in the
paper.

$\Phi(r,q)=r^{2}\psi^{\prime\prime}(r+1-q)+2r\psi^{\prime}(r+1-q).$

Figure 2 shows the function $r\mapsto\Phi(r,q)$ for certain fixed $q$. In
Lemma 3.21, we show the behavior of the symbol function is closely tied to the
behavior of $\Phi(r,q)$:

  * •

if $\Phi(r,q)<1$ for $r>q$, then $k\mapsto J(\delta_{1-q}(\gamma),\gamma,k)$
is strictly decreasing.

  * •

if $\Phi(r,q)>1$ for $r>q$, then $k\mapsto J(\delta_{1-q}(\gamma),\gamma,k)$
is strictly increasing.

Figure 2. Behavior of $r\mapsto\Phi(r,q)$ for certain $0\leq q\leq 1$

######  Theorem 1.16.

Sharp polygamma inequalities related to the symbol function are obtained.

  * $(a)$

If $q\in(-\infty,0]\cup[1,\infty)$, the following holds for $r>q$

$\Phi(r,q)=r^{2}\psi^{\prime\prime}(r+1-q)+2r\psi^{\prime}(r+1-q)<1.$

  * $(b)$

When $q=\frac{2}{3}$, the following holds for $r>\tfrac{2}{3}$

$\Phi\big{(}r,\tfrac{2}{3}\big{)}=r^{2}\psi^{\prime\prime}(r+\tfrac{1}{3})+2r\psi^{\prime}(r+\tfrac{1}{3})>1.$

The proof of Theorem 1.16 part $(a)$ is found in Corollary 4.21, where the
inequality follows by proving the complete monotonicity of a closely related
function in Theorem 4.15. The $q=\tfrac{2}{3}$ result in part $(b)$
corresponds directly to the preferred symbol function. We prove this
inequality in Theorem 5.5 using a series representation for
$\Phi(r,\tfrac{2}{3})$.

There is a large body of literature on inequalities involving combinations of
polygamma functions; see, e.g., [2, 3, 11, 12, 13, 14, 15, 20] and the
references therein. The inequalities in Theorem 1.16 appear to be new and we
suspect that mathematicians working in special function theory will take
interest in results found in Sections 4, 5, and the Appendix.

The paper is organized as follows. Section 2 provides background material on
the Leray transform. In Section 3 we establish new symmetries for the symbol
function, which is then related to $\Phi(r,q)$. In Section 4 we introduce the
Bernstein-Widder theorem and prove several polygamma and symbol function
inequalities. In Section 5 the Euler-Maclaurin formula is used to study the
preferred symbol function and the corresponding $\Phi(r,\tfrac{2}{3})$. The
proof of one especially difficult estimate used in Section 5 is postponed
until the Appendix.

##  2\. The Leray transform

Recall that a domain is ${\mathbb{C}}$-convex if every (non-empty)
intersection of $\Omega$ with a complex line is both connected and simply
connected; see [4]. Let $\Omega$ be a ${\mathbb{C}}$-convex domain in
${\mathbb{C}}^{n}$ (or ${\mathbb{C}}\mathbb{P}^{n}$), with a $C^{2}$-smooth
boundary $b\Omega={\mathcal{S}}$, and let $\rho$ be a defining function for
$\Omega$. The Leray transform $\bm{L}$ of a function $f$ defined on
${\mathcal{S}}$ is a higher dimensional analogue of the familiar Cauchy
transform in the complex plane. For $z\in\Omega$ it is given by the formula

(2.1a) $\displaystyle\bm{L}f(z)$
$\displaystyle=\int_{\zeta\in{\mathcal{S}}}f(\zeta){\mathscr{L}}(z,\zeta),$
(2.1b) $\displaystyle{\mathscr{L}}(z,\zeta)$ $\displaystyle=\frac{1}{(2\pi
i)^{n}}\frac{\partial\rho(\zeta)\wedge(\overline{\partial}\partial\rho(\zeta))^{n-1}}{\langle\partial\rho(\zeta),\zeta-z\rangle^{n}}.$

The $(n,n-1)$-form ${\mathscr{L}}$ is called the Leray kernel and
$\langle\cdot,\cdot\rangle$ denotes the bilinear pairing of $(1,0)$-forms and
vectors. The Leray transform reproduces holomorphic functions from their
boundary values, and generates holomorphic functions from more general
boundary data.

######  Remark 2.2.

It is straightforward to check that the formula for the Leray transform is
independent of the choice of defining function $\rho$; see [21, Section
IV.3.2]. $\lozenge$

It is often useful to decompose the Leray kernel into two pieces:

(2.3a) $\displaystyle\ell(z,\zeta)$
$\displaystyle=\langle\partial\rho(\zeta),\zeta-z\rangle^{-n},$ (2.3b)
$\displaystyle\lambda_{\rho}(\zeta)$ $\displaystyle=\frac{1}{(2\pi
i)^{n}}\partial\rho(\zeta)\wedge(\overline{\partial}\partial\rho(\zeta))^{n-1}.$

We refer to $\lambda_{\rho}$ as the Leray-Levi measure. By itself,
$\lambda_{\rho}$ clearly depends on the choice of defining function, though
natural choices of $\rho$ often correspond to important measures. See [16] for
a survey on $\bm{L}$ and related operators within the Cauchy-Fantappiè
framework.

Barrett and Lanzani [8] study the $L^{2}$-theory of $\bm{L}$ on smoothly
bounded, strongly convex Reinhardt domains in ${\mathbb{C}}^{2}$. They obtain
detailed spectral information on $\bm{L}^{*}\bm{L}$, $\bm{L}\bm{L}^{*}$ and
$\bm{L}^{*}-\bm{L}$, as well as the essential norm of $\bm{L}$ in terms of a
geometric invariant. Several articles by Lanzani and Stein have investigated
other aspects the Leray transform. In [17] they prove that $\bm{L}$ preserves
$L^{p}$ spaces ($1<p<\infty$) whenever the hypersurface ${\mathcal{S}}$ is
bounded, strongly ${\mathbb{C}}$-linearly convex and $C^{1,1}$ smooth. In [18,
19] they show that counter-examples to $L^{p}$-boundedness exist when either
the smoothness or convexity hypothesis is dropped.

###  2.1. Projective invariance

In [5, Section 5], Barrett defines a projective invariant associated to any
smoothly bounded, strongly ${\mathbb{C}}$-convex hypersurface ${\mathcal{S}}$
and uses it [5, Section 8] to construct a projectively invariant measure
$\nu_{\mathcal{S}}$: If $\Phi$ is an automorphism of
${\mathbb{C}}\mathbb{P}^{n}$, the measure transforms as

(2.4)
$\Phi^{*}\big{(}\nu_{\Phi({\mathcal{S}})}\big{)}=|\det\Phi^{\prime}|^{\frac{2n}{n+1}}\,\nu_{\mathcal{S}}.$

We refer to $\nu_{\mathcal{S}}$ as the preferred measure. Equation (2.4) shows
that the operator

(2.5) $f\mapsto(\det\Phi^{\prime})^{\frac{n}{n+1}}\cdot(f\circ\Phi)$

maps $L^{2}\big{(}\Phi({\mathcal{S}}),\nu_{\Phi({\mathcal{S}})}\big{)}$
isometrically to $L^{2}({\mathcal{S}},\nu_{\mathcal{S}})$. If we use the
Radon-Nikodym Theorem to re-express the Leray kernel (2.1b) in terms of the
preferred measure, then the Leray transform admits the transormation law

(2.6)
$\bm{L}_{\mathcal{S}}\left((\det\Phi^{\prime})^{\frac{n}{n+1}}(f\circ\Phi)\right)=(\det\Phi^{\prime})^{\frac{n}{n+1}}\left(\bm{L}_{\Phi({\mathcal{S}})}(f)\circ\Phi\right);$

see [5, Section 9] and [9, Section 5]. When the Leray transform is bounded on
$L^{2}({\mathcal{S}},\nu_{\mathcal{S}})$ (e.g., when Lanzani-Stein conditions
in [17] hold), it can be used to define projectively invariant Hardy spaces
consisting of boundary values of holomorphic functions:

(2.7)
$H^{2}({\mathcal{S}},\nu_{\mathcal{S}})=\bm{L}_{\mathcal{S}}\left(L^{2}({\mathcal{S}},\nu_{\mathcal{S}})\right).$

###  2.2. Projective duality

Given a smooth strongly ${\mathbb{C}}$-convex hypersurface ${\mathcal{S}}$,
there is a unique complex tangent hyperplane at each $\zeta\in{\mathcal{S}}$.
This determines a unique point in a dual copy of ${\mathbb{C}}\mathbb{P}^{n}$;
the set of all such points form the dual hypersurface ${\mathcal{S}}^{*}$.

In [5, Section 6], Barrett shows that ${\mathcal{S}}^{*}$ is smoothly bounded
and strongly ${\mathbb{C}}$-convex whenever ${\mathcal{S}}$ is, and further,
that ${\mathcal{S}}$ and ${\mathcal{S}}^{*}$ are always diffeomorphic. If
$w:{\mathcal{S}}\to{\mathcal{S}}^{*}$ is a such a diffeomorphism, it can be
used to pull back the space $H^{2}({\mathcal{S}}^{*},\nu_{{\mathcal{S}}^{*}})$
by setting

$H_{\sf
dual}^{2}({\mathcal{S}},\nu_{{\mathcal{S}}}^{*})=w^{*}\left(H^{2}({\mathcal{S}}^{*},\nu_{{\mathcal{S}}^{*}})\right),\qquad\mathrm{where}\qquad\nu_{\mathcal{S}}^{*}=w^{*}(\nu_{{\mathcal{S}}^{*}}).$

The pairing measure $\sigma_{\mathcal{S}}$ is defined as the geometric mean of
$\nu_{\mathcal{S}}$ and $\nu_{{\mathcal{S}}}^{*}$.

Given functions $f\in H^{2}({\mathcal{S}},\nu_{\mathcal{S}})$ and $g\in
H_{\sf{dual}}^{2}({\mathcal{S}},\nu_{{\mathcal{S}}}^{*})$, define their
bilinear pairing as

$\langle\\!\langle
f,g\rangle\\!\rangle=\int_{{\mathcal{S}}}f(\zeta)g(\zeta)\,\sigma_{\mathcal{S}}(\zeta).$

The map
$\widehat{\chi}_{\gamma}:g\mapsto\langle\\!\langle\cdot,g\rangle\\!\rangle$
gives a quasi-isometric identification of
$H_{\sf{dual}}^{2}({\mathcal{S}},\nu_{{\mathcal{S}}}^{*})$ with function
theoretic dual space $H^{2}({\mathcal{S}},\nu_{\mathcal{S}})^{\prime}$. In the
$M_{\gamma}$ setting, we have the following result.

######  Proposition 2.8 (Barrett-Edholm [7]).

The operator $\widehat{\chi}_{\gamma}:H_{\sf{dual}}^{2}(M_{\gamma},\nu)\mapsto
H^{2}(M_{\gamma},\nu)^{\prime}$ is an invertible map such that

$\left\|\widehat{\chi}_{\gamma}^{-1}\right\|_{H^{2}(M_{\gamma},\nu)^{\prime}}=\left\|\bm{L}\right\|_{L^{2}(M_{\gamma},\nu)}.$

In light of Theorem 1.12 part $(d)$ we can now express this as a concrete
number:

(2.9)
$\left\|\widehat{\chi}_{\gamma}^{-1}\right\|_{H^{2}(M_{\gamma},\nu)^{\prime}}=\sqrt{\frac{\gamma}{2\sqrt{\gamma-1}}}.$

###  2.3. Leray theory on $M_{\gamma}$

The following material is found in [7, Sections 3, 5]. Parameterize
$M_{\gamma}$ by $(\zeta_{1},\zeta_{2})=(re^{i\theta},s+ir^{\gamma})$ and
consider the measures

(2.10) $\mu_{d}=r^{d}\,dr\wedge d\theta\wedge ds,\qquad d\in{\mathbb{R}}.$

The pairing measure $\sigma$ ($d=\gamma-1$) arises as (a constant multiple of)
the Leray-Levi measure corresponding to the defining function
$\rho(\zeta)=|\zeta_{1}|^{\gamma}-\text{Im}{(\zeta_{2})}$.

Rotational symmetry in $\zeta_{1}$ yields a decomposition of
$L^{2}(M_{\gamma},\mu_{d})$ into an orthogonal sum of subspaces
$L^{2}_{k}(M_{\gamma},\mu_{d})$, each consisting of functions of the form
$f_{k}(r,\theta)e^{ik\theta}$. Likewise, the Leray transform decomposes into
orthogonal sub-Leray operators $\bm{L}_{k}$, each mapping
$L^{2}(M_{\gamma},\mu_{d})$ to $L^{2}_{k}(M_{\gamma},\mu_{d})$ whenever it is
bounded:

$\bm{L}=\bigoplus_{k=0}^{\infty}\bm{L}_{k}.$

Given $f\in L^{2}(M_{\gamma},\mu_{d})$, the following inequality holds; see
[7, Theorem 5.13]:

(2.11)
$\left\|\bm{L}_{k}f\right\|_{L^{2}(M_{\gamma},\mu_{d})}\leq\sqrt{J(d,\gamma,k)}\cdot\left\|f\right\|_{L^{2}(M_{\gamma},\mu_{d})},$

where $J(d,\gamma,k)$ is the symbol function defined in (1.6). It turns out
there are always functions $f_{k}(r,s)e^{ik\theta}\in
L^{2}_{k}(M_{\gamma},\mu_{d})$ achieving equality in the relation (2.11)
above.

If $f=\sum_{k}f_{k}e^{ik\theta}$ orthogonality implies the following estimate;
see [7, Corollary 5.28]:

$\left\|\bm{L}f\right\|^{2}=\sum_{k=0}^{\infty}\left\|\bm{L}_{k}f\right\|^{2}=\sum_{k=0}^{\infty}\left\|\bm{L}_{k}f_{k}\,e^{ik\theta}\right\|^{2}\leq\sum_{k=0}^{\infty}J(d,\gamma,k)\left\|f_{k}\right\|^{2}\leq
M\sum_{k=0}^{\infty}\left\|f_{k}\right\|^{2}=M\left\|f\right\|^{2},$

where $M$ denotes the supremum of $J(d,\gamma,k)$ as $k$ ranges over the non-
negative integers. But since the norm of each $\bm{L}_{k}$ is achieved by a
concrete function, it quickly follows that

(2.12)
$\left\|\bm{L}\right\|_{L^{2}(M_{\gamma},\mu_{d})}=\sup\left\\{\sqrt{J(d,\gamma,k)}:k=0,1,2,\dots\right\\}.$

##  3\. Symmetries and Monotonicity

###  3.1. Hölder symmetry, $\gamma\neq 2$

Proposition 1.8 says $\bm{L}_{k}$ is bounded on $L^{2}(M_{\gamma},\mu_{d})$ if
and only if $d\in{\mathcal{I}}_{k}(\gamma)$. Here, it is shown that for each
$d$ there is a unique $d^{*}\in{\mathbb{R}}$ such that the behavior of
$\bm{L}_{k}$ in $L^{2}(M_{\gamma},\mu_{d})$ closely parallels its behavior in
$L^{2}(M_{\gamma^{*}},\mu_{d^{*}})$. For $a\in{\mathbb{R}}$, define

(3.1) $\delta_{a}(\gamma)=a(\gamma-2)+1.$

######  Theorem 3.2.

The symbol function $J(\delta_{a}(\gamma),\gamma,k)$ is Hölder symmetric, in
that

$J(\delta_{a}(\gamma),\gamma,k)=J(\delta_{a}(\gamma^{*}),\gamma^{*},k).$

###### Proof.

From (1.6) we have

$J(\delta_{a}(\gamma),\gamma,k)=\frac{\Gamma\big{(}\frac{2k+2+(\gamma-2)a}{\gamma}\big{)}\Gamma\big{(}2k+2-\frac{2k+2+(\gamma-2)a}{\gamma}\big{)}}{\Gamma(k+1)^{2}}\left(\frac{\gamma}{2}\right)^{2k+2}(\gamma-1)^{\big{(}\frac{2k+2+(\gamma-2)a}{\gamma}-2k-2\big{)}}.$

But notice that

$\displaystyle\frac{2k+2+(\gamma^{*}-2)a}{\gamma^{*}}=\frac{2k+2-2a}{\gamma^{*}}+a$
$\displaystyle=\left(1-\frac{1}{\gamma}\right)(2k+2-2a)+a$
$\displaystyle=2k+2-a-\frac{2k+2-2a}{\gamma}$ (3.3)
$\displaystyle=2k+2-\frac{2k+2+(\gamma-2)a}{\gamma}.$

Thus, we have both

(3.4a)
$\displaystyle\Gamma\bigg{(}\frac{2k+2+(\gamma^{*}-2)a}{\gamma^{*}}\bigg{)}=\Gamma\bigg{(}2k+2-\frac{2k+2+(\gamma-2)a}{\gamma}\bigg{)},$
(3.4b)
$\displaystyle\Gamma\bigg{(}\frac{2k+2+(\gamma-2)a}{\gamma}\bigg{)}=\Gamma\bigg{(}2k+2-\frac{2k+2+(\gamma^{*}-2)a}{\gamma^{*}}\bigg{)}.$

It is also easily seen that

$\Big{(}\frac{\gamma^{*}}{2}\Big{)}^{2k+2}=\Big{(}\frac{\gamma}{2}\Big{)}^{2k+2}(\gamma-1)^{-2k-2},$

and so from (3.3) we are able to deduce

$\displaystyle(\gamma^{*}-1)^{\big{(}\frac{2k+2+(\gamma^{*}-2)a}{\gamma^{*}}-2k-2\big{)}}$
$\displaystyle=(\gamma-1)^{\big{(}2k+2-\frac{2k+2+(\gamma^{*}-2)a}{\gamma^{*}}\big{)}}=(\gamma-1)^{\big{(}\frac{2k+2+(\gamma-2)a}{\gamma}\big{)}}.$

Combining the previous two equations we have

(3.5)
$\Big{(}\frac{\gamma^{*}}{2}\Big{)}^{2k+2}(\gamma^{*}-1)^{\big{(}\frac{2k+2+(\gamma^{*}-2)a}{\gamma^{*}}-2k-2\big{)}}=\Big{(}\frac{\gamma}{2}\Big{)}^{2k+2}(\gamma-1)^{\big{(}\frac{2k+2+(\gamma-2)a}{\gamma}-2k-2\big{)}}.$

Putting together the equations in (3.4) with (3.5) gives the result. ∎

######  Remark 3.6.

The computations in the theorem above yield a Hölder symmetric reformulation
of the symbol function whenever $\gamma\neq 2$:

$J(\delta_{a}(\gamma),\gamma,k)=\frac{\Gamma\big{(}\frac{2k+2+(\gamma-2)a}{\gamma}\big{)}\Gamma\big{(}\frac{2k+2+(\gamma^{*}-2)a}{\gamma^{*}}\big{)}}{\Gamma(k+1)^{2}}\left(\frac{\gamma}{2}\right)^{\big{(}\frac{2k+2+(\gamma-2)a}{\gamma}\big{)}}\Big{(}\frac{\gamma^{*}}{2}\Big{)}^{\big{(}\frac{2k+2+(\gamma^{*}-2)a}{\gamma^{*}}\big{)}}.$

$\lozenge$

######  Corollary 3.7.

The pairing symbol function and the preferred symbol function are both Hölder
symmetric, i.e., $C_{\sigma}(\gamma,k)=C_{\sigma}(\gamma^{*},k)$ and
$C_{\nu}(\gamma,k)=C_{\nu}(\gamma^{*},k)$.

###### Proof.

The pairing measure $\sigma=\mu_{d}$ with $d=\gamma-1$. This $d$ value
coincides with $\delta_{a}(\gamma)$ with $a=1$, so Theorem 3.2 applies. The
preferred measure $\nu=\mu_{d}$ with $d=\frac{\gamma+1}{3}$. This $d$ value
coincides with $\delta_{a}(\gamma)$ with $a=\frac{1}{3}$, so Theorem 3.2 again
applies. ∎

######  Remark 3.8.

In addition to $\sigma$ and $\nu$, Barrett and Edholm [7, Section 6.2.2] also
introduce the dual preferred measure $\nu^{*}$ on a ${\mathbb{C}}$-convex
hypersurface ${\mathcal{S}}$ as the pullback of the preferred measure on the
dual hypersurface ${\mathcal{S}}^{*}$. (Recall the discussion in Section 2.1
above, where these three measures are related by the property
$\sqrt{\nu\nu^{*}}=\sigma$.)

When ${\mathcal{S}}=M_{\gamma}$,

$\nu^{*}=r^{\frac{5\gamma-7}{3}}\,dr\wedge d\theta\wedge ds,$

so $\nu^{*}=\mu_{d}$ with $d=\tfrac{5\gamma-7}{3}$. It follows from (1.9) that
$\tfrac{5\gamma-7}{3}\in{\mathcal{I}}_{0}(\gamma)$, meaning that $\bm{L}$ is
bounded on $L^{2}(M_{\gamma},\nu^{*})$. Since this $d$ value equals
$\delta_{a}(\gamma)$ with $a=\frac{5}{3}$, Theorem 3.2 implies Hölder
symmetry: if we set
$C_{\nu^{*}}(\gamma,k):=J\big{(}\frac{5\gamma-7}{3},\gamma,k\big{)}$, then
$C_{\nu^{*}}(\gamma,k)=C_{\nu^{*}}(\gamma^{*},k)$. $\lozenge$

######  Remark 3.9.

When $\gamma=2$, the function $\delta_{a}(\gamma)=a(\gamma-2)+1\equiv 1$ for
every $a$. Thus each pair of function spaces $L^{2}(M_{\gamma},\mu_{d})$ and
$L^{2}(M_{\gamma^{*}},\mu_{d^{*}})$ with parallel symbol function behavior
reduces to a pairing of $L^{2}(M_{2},\mu_{1})$ with itself when $\gamma=2$. We
note that on this space, the Leray transform coincides with the Szegő
projection. $\lozenge$

###  3.2. Heisenberg monotonicity

From (1.6), observe that the symbol function greatly simplifies when
$\gamma=2$:

(3.10)
$J(d,2,k)=\frac{\Gamma\big{(}k+\frac{1+d}{2}\big{)}\Gamma\big{(}k+\frac{3-d}{2}\big{)}}{\Gamma(k+1)^{2}}.$

A fraction involving $\Gamma$-functions in this form is sometimes called
Gurland’s ratio; see [20].

######  Proposition 3.11.

For each non-negative integer $k$,

$\left\|\bm{L}_{k}\right\|_{L^{2}(M_{2},\mu_{d})}=\left\|\bm{L}_{k}\right\|_{L^{2}(M_{2},\mu_{2-d})}.$

###### Proof.

From (3.10) it is immediate that $J(d,2,k)=J(2-d,2,k)$. Now use (1.10). ∎

######  Remark 3.12.

When $\gamma=2$ and $d=1$, we see that
$\left\|\bm{L}_{k}\right\|_{L^{2}(M_{2},\mu_{1})}=1$ for every $k$. This of
course is unsurprising, since the Leray transform is the Szegő projection on
$L^{2}(M_{2},\mu_{1})$. $\lozenge$

######  Theorem 3.13.

Let $d$ be any real number. The function $k\mapsto J(d,2,k)$ is decreasing
whenever it is finite. Consequently,

$\left\|\bm{L}_{k}\right\|_{L^{2}(M_{2},\mu_{d})}\geq\left\|\bm{L}_{k+1}\right\|_{L^{2}(M_{2},\mu_{d})}.$

The inequality is strict for $d\neq 1$.

###### Proof.

When $d=1$, Remark 3.12 says that the norm of $\bm{L}_{k}$ is equal to 1 for
all $k$.

When $d\neq 1$, we see from (1.9) that $J(d,2,k)$ is finite if and only if

$d\in{\mathcal{I}}_{k}(2)=(-2k-1,2k+3).$

Since ${\mathcal{I}}_{k}(2)\subset{\mathcal{I}}_{k+1}(2)$ and these intervals
exhaust the real line, it is clear that $J(d,2,\gamma)$ can be infinite for at
most a finite number of integers $k$.

Observe that $d\in{\mathcal{I}}_{k}(2)$ if and only if
$2-d\in{\mathcal{I}}_{k}(2)$. Proposition 3.11 says it is sufficient to
consider the case when $d>1$, which we now assume for the rest of the proof.

Taking logarithms, (3.10) becomes

$\log{J(d,2,k)}=\log{\Gamma\big{(}k+\tfrac{1+d}{2}\big{)}}+\log{\Gamma\big{(}k+\tfrac{3-d}{2}\big{)}}-2\log{\Gamma(k+1)}.$

Now treat $k$ as a continuous variable and differentiate:

(3.14) $\frac{\partial}{\partial
k}\log{J(d,2,k)}=\psi(k+\tfrac{1+d}{2})+\psi(k+\tfrac{3-d}{2})-2\psi(k+1),$

where $\psi$ is the digamma function defined in (1.15). Now consider the
right-hand side of (3.14) in two separate pieces. By the mean value theorem,

(3.15a)
$\psi\big{(}k+\tfrac{1+d}{2}\big{)}-\psi(k+1)=\psi^{\prime}(\xi_{1})\big{(}\tfrac{d-1}{2}\big{)},$
for some $\xi_{1}\in\big{(}k+1,k+\tfrac{1+d}{2}\big{)}$. Similarly, (3.15b)
$\psi\big{(}k+\tfrac{3-d}{2}\big{)}-\psi(k+1)=-\psi^{\prime}(\xi_{2})\big{(}\tfrac{d-1}{2}\big{)},$
for some $\xi_{2}\in\big{(}k+\tfrac{3-d}{2},k+1\big{)}$.

Combining (3.15a), (3.15b) and (3.14) we use the mean value theorem again,

$\displaystyle\frac{\partial}{\partial
k}\log{J(d,2,k)}=\big{(}\tfrac{d-1}{2}\big{)}(\psi^{\prime}(\xi_{1})-\psi^{\prime}(\xi_{2}))=\big{(}\tfrac{d-1}{2}\big{)}\psi^{\prime\prime}(\xi_{3})(\xi_{1}-\xi_{2}),$

for some $\xi_{3}\in(\xi_{2},\xi_{1})$. It is well-known that
$\psi^{\prime\prime}<0$ on the positive reals (see, e.g., formula (3.17)
below). This means that $k\mapsto\log{J(d,2,k)}$ is strictly decreasing for
$k\in[0,\infty)$, which implies that $k\mapsto J(d,2,k)$ is strictly
decreasing for $k\in[0,\infty)$. Now restrict $k$ to the non-negative integers
and recall (1.10) to complete the proof. ∎

By Proposition 1.5, the Leray transform is bounded on $L^{2}(M_{2},\mu_{d})$
if and only if $d\in{\mathcal{I}}_{0}(2)=(-1,3)$. When this holds, the norm of
the full Leray transform can be computed:

######  Corollary 3.16.

Let $d\in(-1,3)$. The norm of the Leray transform on $L^{2}(M_{2},\mu_{d})$ is

$\left\|\bm{L}\right\|_{L^{2}(M_{2},\mu_{d})}=\begin{cases}\sqrt{\frac{\pi}{2}(1-d)\sec{\big{(}\frac{d\pi}{2}\big{)}}},&d\in(-1,1)\cup(1,3)\\\
1,&d=1.\end{cases}$

This formula is continuous at $d=1$.

###### Proof.

The $d=1$ case can be seen directly from (3.10) (see Remark 3.12) and
L’Hôpital’s rule shows continuity at $d=1$. When $d\neq 1$, Theorem 3.13 says
$k\mapsto J(d,\gamma,k)$ is strictly decreasing in $k$. Thus by (2.12),

$\left\|\bm{L}\right\|_{L^{2}(M_{2},\mu_{d})}=\sqrt{J(d,2,0)}.$

Now compute:

$J(d,2,0)=\big{(}\tfrac{1-d}{2}\big{)}\Gamma\big{(}1-\tfrac{1-d}{2}\big{)}\Gamma\big{(}\tfrac{1-d}{2}\big{)}=\big{(}\tfrac{1-d}{2}\big{)}\pi\csc\big{(}\tfrac{\pi}{2}-\tfrac{d\pi}{2}\big{)}=\tfrac{\pi}{2}(1-d)\sec\big{(}\tfrac{d\pi}{2}\big{)}.$

In this computation we have used the factorial property of the
$\Gamma$-function along with Euler’s reflection formula:
$\Gamma(z)\Gamma(1-z)=\pi\csc(\pi z)$. ∎

###  3.3. Polygamma functions and ratios of successive symbols

In this section, once again consider $\gamma\neq 2$ and set
$\delta_{a}(\gamma)=a(\gamma-2)+1$.

The polygamma functions are the successive derivatives of the digamma function
$\psi$, defined in (1.15). For integers $m\geq 1$, these functions admit the
following series representations; see [1, Equation 6.4.10]:

(3.17) $\psi^{(m)}(r)=(-1)^{m+1}m!\sum_{j=1}^{\infty}\frac{1}{(r+j-1)^{m+1}}.$

The following polygamma combination is closely tied to Leray symbol function
and is prominently featured throughout the paper (recall Figure 2 in Section
1):

(3.18a) $\Phi(r,q)=2r\psi^{\prime}(r+1-q)+r^{2}\psi^{\prime\prime}(r+1-q).$ In
light of (3.17), it is often useful to write this function as a series (3.18b)
$\Phi(r,q)=\sum_{j=1}^{\infty}\frac{2r(j-q)}{(r+j-q)^{3}}.$

The establishment of new inequalities related to the polygamma functions has
been an active area of research for decades; see [2, 3, 11, 12, 13, 14, 15,
20] for results with similar flavor to those we prove below. A typical way in
which new polygamma inequalities are obtained is to show that some auxiliary
function is completely monotone (see Section 4.2), a checkable condition
thanks to the celebrated Bernstein-Widder theorem (see Theorem 4.7).

In Sections 4 and 5, we prove inequalities involving $\Phi(r,q)$ and related
functions, both by use of Bernstein-Widder and in situations where it fails to
apply. Our inequalities appear to be new and the methods are likely adaptable
to more general settings.

For later convenience, we frame the finiteness of the symbol function in terms
of $\delta_{1-q}$:

######  Lemma 3.19.

Let $\gamma\neq 2$. The symbol function $J(\delta_{1-q}(\gamma),\gamma,k)$ is
finite for all non-negative integers $k$ if and only if

$|q|<\frac{\gamma}{|\gamma-2|}.$

###### Proof.

By Proposition 1.8, $J(\delta_{1-q}(\gamma),\gamma,k)$ is finite for all $k$
if and only if

$\delta_{1-q}(\gamma)\in\bigcap_{k=0}^{\infty}{\mathcal{I}}_{k}(\gamma)={\mathcal{I}}_{0}(\gamma)=(-1,2\gamma-1).$

For $\gamma>2$ this means

$\displaystyle-1<(1-q)(\gamma-2)+1<2\gamma-1\qquad$
$\displaystyle\Longleftrightarrow\qquad-\frac{\gamma}{\gamma-2}<q<\frac{\gamma}{\gamma-2},$

confirming the inequality. When $\gamma<2$,

$\displaystyle-1<(1-q)(\gamma-2)+1<2\gamma-1\qquad$
$\displaystyle\Longleftrightarrow\qquad-\frac{\gamma}{2-\gamma}<q<\frac{\gamma}{2-\gamma},$

confirming the inequality. ∎

######  Remark 3.20.

Observe that the estimate appearing in Lemma 3.19 is Hölder symmetric, i.e.,
invariant under the change of variable $\gamma\mapsto\gamma^{*}$. $\lozenge$

######  Lemma 3.21.

Let $\gamma\neq 2$ and $|q|<\dfrac{\gamma}{|\gamma-2|}$, so that
$J(\delta_{1-q}(\gamma),\gamma,k)$ is finite for $k\geq 0$.

  1. (1)

If $\Phi(r,q)<1$ for $r>q$, then $k\mapsto J(\delta_{1-q}(\gamma),\gamma,k)$
is strictly decreasing on the non-negative integers.

  2. (2)

If $\Phi(r,q)>1$ for $r>q$, then $k\mapsto J(\delta_{1-q}(\gamma),\gamma,k)$
is strictly increasing on the non-negative integers.

###### Proof.

Strict increasing (resp. decreasing) behavior of the function $k\mapsto
J(\delta_{1-q}(\gamma),\gamma,k)$ would follow by showing the below ratio is
strictly greater than 1 (resp. strictly less than 1) for non-negative integers
$k$. Throughout the proof, set $a=1-q$:

$\displaystyle\frac{J(\delta_{a}(\gamma),\gamma,k+1)}{J(\delta_{a}(\gamma),\gamma,k)}$
$\displaystyle=\frac{\Gamma\left(\frac{2k+4+(\gamma-2)a}{\gamma}\right)\Gamma\left(2k+4-\frac{2k+4+(\gamma-2)a}{\gamma}\right)}{\Gamma\left(\frac{2k+2+(\gamma-2)a}{\gamma}\right)\Gamma\left(2k+2-\frac{2k+2+(\gamma-2)a}{\gamma}\right)}\left(\frac{\gamma}{2k+2}\right)^{2}(\gamma-1)^{\frac{2}{\gamma}-2}$

By the Hölder symmetry of the symbol function (Theorem 3.2), it is sufficient
to restrict analysis to $\gamma\in(1,2)$; upon making the substitution
$x=\frac{2}{\gamma}$, this is equivalent to analyzing the behavior of the
following function for $x\in(1,2)$:

$\frac{\Gamma\left((k+2-a)x+a\right)\Gamma\left((k+2-a)(2-x)+a\right)}{\Gamma\left((k+1-a)x+a\right)\Gamma\left((k+1-a)(2-x)+a\right)}\cdot\frac{x^{-x}(2-x)^{x-2}}{(k+1)^{2}}:=e^{A(k,x)}$

Upon taking a logarithm, observe that $A(k,1)\equiv 0$, and further that
$A(k,x)$ can be written

$\displaystyle A(k,x)=B(k,x)+B(k,2-x),$

where

(3.22)
$B(k,x):=\log\left[\frac{\Gamma\left((k+2-a)x+a\right)}{\Gamma\left((k+1-a)x+a\right)}\right]-x\log
x-\log(k+1).$

For the remainder of the proof we focus on case (2), noting that the same
argument with trivial modifications in appropriate places will prove case (1).

We have just seen that the statement

(3.23)
$\frac{J(\delta_{a}(\gamma),\gamma,k+1)}{J(\delta_{a}(\gamma),\gamma,k)}>1$

for all non-negative integers $k$ and $\gamma\in(1,2)$ is equivalent to the
statement that $A(k,x)>0$ for all non-negative integers $k$ and $x\in(1,2)$.

Since $A(k,1)\equiv 0$ for all non-negative integers $k$, the desired
positivity of $A(k,x)$ will follow by establishing a stronger condition,
namely, that for $x\in(1,2)$

(3.24) $\frac{\partial A}{\partial x}(k,x)=\frac{\partial B}{\partial
x}(k,x)-\frac{\partial B}{\partial x}(k,2-x)>0.$

Estimate (3.24) will in turn follow if it can be established that
$\frac{\partial B}{\partial x}(k,x)$ is strictly increasing for $x\in(1,2)$,
since $0<2-x<x$ for such $x$. It thus suffices to show

(3.25) $\frac{\partial^{2}B}{\partial x^{2}}(k,x)>0.$

Recalling the representation (3.17) of $\psi^{\prime}$ by a series,
calculation now shows

$\displaystyle\frac{\partial^{2}B}{\partial x^{2}}(k,x)$
$\displaystyle=(k+2-a)^{2}\psi^{\prime}((k+2-a)x+a)-(k+1-a)^{2}\psi^{\prime}((k+1-a)x+a)-\frac{1}{x}$
$\displaystyle=\sum_{j=1}^{\infty}\frac{(k+2-a)^{2}}{(j+(k+2-a)x+(a-1))^{2}}-\sum_{j=1}^{\infty}\frac{(k+1-a)^{2}}{(j+(k+1-a)x+(a-1))^{2}}-\frac{1}{x}$
(3.26) $\displaystyle=D(k+1,x)-D(k,x),$

where

(3.27)
$D(k,x):=\sum_{j=1}^{\infty}\frac{(k+1-a)^{2}}{(j+(k+1-a)x+(a-1))^{2}}-\frac{k}{x}.$

Temporarily regard $k$ as a continuous variable. By (3.26), it would now be
sufficient to show that $D(k,x)$ increases as a function of $k\in[0,\infty)$
in order to conclude that estimate (3.25) holds. We claim that for
$x\in(1,2)$,

$\displaystyle\frac{\partial D}{\partial k}(k,x)$
$\displaystyle=\sum_{j=1}^{\infty}\frac{\partial}{\partial
k}\left[\frac{(k+1-a)^{2}}{(j+(k+1-a)x+(a-1))^{2}}\right]-\frac{1}{x}$ (3.28)
$\displaystyle=\sum_{j=1}^{\infty}\frac{2(k+1-a)(j+a-1)}{(j+(k+1-a)x+(a-1))^{3}}-\frac{1}{x}\quad>0.$

The inequality in (3.28) is equivalent to saying that, for $k\in[0,\infty)$
and $x\in(1,2)$,

$\sum_{j=1}^{\infty}\frac{2x(k+1-a)(j+a-1)}{(j+(k+1-a)x+(a-1))^{3}}>1.$

Now substituting $q=1-a$ and $r=x(k+1-a)=x(k+q)$, the above inequality is
equivalent to showing that, for $r>q$,

(3.29) $\sum_{j=1}^{\infty}\frac{2r(j-q)}{(j+r-q)^{3}}=\Phi(r,q)>1.$

This is the condition listed in case (2) of the theorem. Retracing our steps,
we see that inequality (3.29) is equivalent to inequality (3.28). Now, by
(3.26), these inequalities imply inequality (3.25). This implies (3.24), which
in turn shows (3.23), meaning that $k\mapsto J(\delta_{1-q}(\gamma),\gamma,k)$
is an increasing function on the non-negative integers.

To prove case (1), return to (3.23) and retrace the same steps, changing “$>$”
to “$<$” and “increases” to “decreases” in all necessary places. ∎

##  4\. Polygamma inequalities and complete monotonicity

###  4.1. An initial estimate

We now examine the properties of

$\Phi(r,q)=2r\psi^{\prime}(r+1-q)+r^{2}\psi^{\prime\prime}(r+1-q).$

We start from a well-known inequality; see, e.g., [15, Theorem 3]. If $x>0$
and $m$ is a positive integer, then

(4.1)
$\frac{(m-1)!}{x^{m}}+\frac{m!}{2x^{m+1}}<(-1)^{m+1}\psi^{(m)}(x)<\frac{(m-1)!}{x^{m}}+\frac{m!}{x^{m+1}}.$

We use this to describe the behavior of $\Phi(r,q)$ as $r\to\infty$. (Compare
with Figure 2.)

######  Proposition 4.2.

Let q be a fixed real number. Then

$\lim_{r\to\infty}\Phi(r,q)=1.$

###### Proof.

Directly from (4.1), we have for $r>\max\\{q-1,0\\}$,

$\frac{2r^{2}-2rq+3r}{(r+1-q)^{2}}<2r\psi^{\prime}(r+1-q)<\frac{2r^{2}-2rq+4r}{(r+1-q)^{2}},$

and

$-\frac{(r-q+3)r^{2}}{(r+1-q)^{3}}<r^{2}\psi^{\prime\prime}(r+1-q)<-\frac{(r-q+2)r^{2}}{(r+1-q)^{3}}.$

Now combine these:

(4.3)
$\frac{r^{3}+(2-3q)r^{2}+(3-5q+2q^{2})r}{(r+1-q)^{3}}<\Phi(r,q)<\frac{r^{3}+(4-3q)r^{2}+(4-6q+2q^{2})r}{(r+1-q)^{3}}.$

Taking the limit as $r\to\infty$ gives the result. ∎

######  Remark 4.4.

Given $q$, we must show either $\Phi(r,q)>1$ for $r>q$, or $\Phi(r,q)<1$ for
$r>q$ before Lemma 3.21 can be applied. This requires better estimates than
those obtained in (4.3). This is easily seen when $q=0$, in which case

$\frac{r^{3}+2r^{2}+3r}{r^{3}+3r^{2}+3r+1}<\Phi(r,0)<\frac{r^{3}+4r^{2}+4r}{r^{3}+3r^{2}+3r+1}.$

This lower bound is always less than 1, while the upper bound is eventually
greater than 1. Similarly, neither the upper nor lower bound in (4.3) is
strong enough to warrant application of Lemma 3.21 for any $q$. Our goal is
now to improve the estimates on $\Phi(r,q)$. $\lozenge$

###  4.2. The Bernstein-Widder theorem

Let us recall another well-known formula for the polygamma functions
$\psi^{(m)}$; see [1, Equation 6.4.1]:

(4.5)
$\psi^{(m)}(r)=(-1)^{m+1}\int_{0}^{\infty}\frac{t^{m}e^{-rt}}{1-e^{-t}}\,dt.$

Let $f$ be a real-valued function defined on $(c,\infty)$, for some
$c\in{\mathbb{R}}$. We say that $f$ is strictly completely monotone on
$(c,\infty)$ if it is of class $C^{\infty}$ and

(4.6) $(-1)^{m}\frac{d^{m}}{dr^{m}}f(r)>0$

for all non-negative integers $m$ and $r>c$.

Strictly completely monotone functions are characterized by the following
theorem, which can be found in [23, page 161].

######  Theorem 4.7 (Bernstein-Widder).

A function $f$ is strictly completely monotone on $(c,\infty)$ if and only if
$f(r-c)$ is the Laplace transform of a finite positive Borel measure $\mu$ on
$(c,\infty)$. In other words,

$f(r-c)=\int_{0}^{\infty}e^{-rt}\,d\mu(t).$

Our goal is to use Bernstein-Widder to analyze $\Phi(r,q)$ and closely related
functions.

Let us define

(4.8a) $\Theta(r,q)=r^{2}\psi^{\prime}(r+1-q)$ and observe that (4.8b)
$\frac{\partial\Theta}{\partial r}(r,q)=\Phi(r,q).$

######  Lemma 4.9.

For $x>0$ we have the following

(4.10)
$\Theta(x+q,q)=x+2q-\frac{1}{2}+\int_{0}^{\infty}\frac{M(t,q)}{(e^{t}-1)^{3}}e^{-xt}\,dt,$

where

(4.11) $M(t,q)=g_{0}(t)-g_{1}(t)q+g_{2}(t)q^{2},$

and $g_{0},g_{1},g_{2}$ are positive functions on $(0,\infty)$. Explicitly,

(4.12a) $\displaystyle g_{0}(t)$ $\displaystyle=e^{t}(2-2e^{t}+t+te^{t}),$
(4.12b) $\displaystyle g_{1}(t)$ $\displaystyle=2(e^{t}-1)(1-e^{t}+te^{t}),$
(4.12c) $\displaystyle g_{2}(t)$ $\displaystyle=t(e^{t}-1)^{2}.$

###### Proof.

From definition (4.8a) and (4.5) we have

$\displaystyle\Theta(x+q,q)$
$\displaystyle=(x+q)^{2}\int_{0}^{\infty}\frac{te^{-(x+1)t}}{1-e^{-t}}\,dt=\int_{0}^{\infty}\frac{te^{qt}}{e^{t}-1}(x+q)^{2}e^{-(x+q)t}\,dt.$

Integrate by parts twice, setting

$u_{1}=\frac{te^{qt}}{e^{t}-1},\qquad dv_{1}=(x+q)^{2}e^{-(x+q)t}\,dt,$

for the first application, and $u_{2}=\frac{du_{1}}{dt}$ and
$dv_{2}=v_{1}\,dt$ for the second. This means that

$u_{2}=\frac{e^{t}-1-te^{t}-(t-te^{t})q}{(e^{t}-1)^{2}}\,e^{qt},\qquad
v_{1}=-(x+q)e^{-(x+q)t},$

and consequently,

$du_{2}=\frac{g_{0}(t)-g_{1}(t)q+g_{2}(t)q^{2}}{(e^{t}-1)^{3}}\,e^{qt}\,dt,\qquad
v_{2}=e^{-(x+q)t},$

where $g_{0},g_{1},g_{2}$ are given in the statement of the theorem. Now,

$\displaystyle\Theta(x+q,q)=\int_{0}^{\infty}u_{1}\,dv_{1}$
$\displaystyle=u_{1}v_{1}\Big{|}_{0}^{\infty}-\int_{0}^{\infty}v_{1}\,du_{1}$
$\displaystyle=u_{1}v_{1}\Big{|}_{0}^{\infty}-u_{2}v_{2}\Big{|}_{0}^{\infty}+\int_{0}^{\infty}v_{2}\,du_{2}$
(4.13)
$\displaystyle=(x+q)-\big{(}\tfrac{1}{2}-q\big{)}+\int_{0}^{\infty}\frac{M(t,q)}{(e^{t}-1)^{3}}\,e^{-xt}\,dt.$

Using the definitions of $M(t,q)$ and the $g_{j}(t)$ given above, we see that
for $t$ near 0,

$M(t,q)=\Big{(}q^{2}-q+\frac{1}{6}\Big{)}t^{3}+\Big{(}q^{2}-\frac{7q}{6}+\frac{1}{4}\Big{)}t^{4}+O\big{(}t^{5}\big{)}$

On the other hand, if $t$ is large enough (say, $t>1$), there is a constant
$C_{q}$ such that

$|M(t,q)|\leq C_{q}\,te^{2t}.$

From these estimates on $M(t,q)$ we see that the integral appearing in (4.13)
converges for $x>0$. It now only remains to show the positivity of the
$g_{j}(t)$ for $t>0$.

Let us first set $h_{0}(t)=2-2e^{t}+t+te^{t}$, so that
$g_{0}(t)=e^{t}h_{0}(t)$. Calculating derivatives,

$h_{0}^{\prime}(t)=1+(t-1)e^{t},\qquad h_{0}^{\prime\prime}(t)=te^{t},$

and so $0=h_{0}(0)=h^{\prime}_{0}(0)=h^{\prime\prime}(0)$. Also, clearly
$h_{0}^{\prime\prime}(t)>0$, for $t>0$ which now implies that
$h_{0}^{\prime\prime}(t)>0$, which in turn implies $h_{0}(t)>0$. Thus
$g_{0}(t)>0$ for $t>0$.

Now let $h_{1}(t)=1-e^{t}+te^{t}$, so that $g_{1}(t)=2(e^{t}-1)h_{1}(t)$. We
see that

$h_{1}^{\prime}(t)=te^{t},$

and so $0=h_{1}(0)=h_{1}^{\prime}(0)$. Clearly $h_{1}^{\prime}(t)>0$ for
$t>0$, which implies $h_{1}(t)>0$ and therefore $g_{1}(t)>0$ for $t>0$.

Finally, it is immediate from the formula that $g_{2}(t)>0$ for $t>0$. ∎

###  4.3. Completely monotone functions

We begin by showing that, for all but a small interval of $q\in{\mathbb{R}}$,
the following function is strictly completely monotone on $x>0$:

(4.14) $F_{q}(x)=\Theta(x+q,q)-x-2q+\frac{1}{2}.$

######  Theorem 4.15.

Let $q\in(-\infty,0]\cup[1,\infty)$. The function $x\mapsto F_{q}(x)$ is
strictly completely monotone on $x>0$.

###### Proof.

From Lemma 4.9, we have

(4.16) $F_{q}(x)=\int_{0}^{\infty}\frac{M(t,q)}{(e^{t}-1)^{3}}\,e^{-xt}\,dt,$

where $M(t,q)=g_{0}(t)-g_{1}(t)q+g_{2}(t)q^{2}$, and the $g_{j}$ are given in
(4.12).

Thought of as a quadratic in the $q$ variable, the discriminant of $M(t,q)$ is

$\displaystyle\Delta(t)$ $\displaystyle=g_{1}(t)^{2}-4g_{0}(t)g_{2}(t)$
$\displaystyle=4-16e^{t}+24e^{2t}-16e^{3t}+4e^{4t}+(-4e^{t}+8e^{2t}-4e^{3t})t^{2}$
(4.17) $\displaystyle=4(e^{t}-1)^{2}(1-2e^{t}+e^{2t}-e^{t}t^{2}).$

The roots to $M(t,q)=0$ are therefore given by taking

$q=\frac{g_{1}(t)\pm\sqrt{\Delta(t)}}{2g_{2}(t)};$

we now label these roots as

(4.18a) $\displaystyle s_{1}(t)$
$\displaystyle=\frac{te^{t}+1-e^{t}+\sqrt{1-2e^{t}+e^{2t}-t^{2}e^{t}}}{t(e^{t}-1)},$
(4.18b) $\displaystyle s_{2}(t)$
$\displaystyle=\frac{te^{t}+1-e^{t}-\sqrt{1-2e^{t}+e^{2t}-t^{2}e^{t}}}{t(e^{t}-1)}.$

Observe that the term under the square root is positive for $t>0$:

$\displaystyle
1-2e^{t}+e^{2t}-t^{2}e^{t}>0\qquad\Longleftrightarrow\qquad\cosh{t}>1+\frac{t^{2}}{2}\qquad\Longleftrightarrow\qquad\sum_{j=2}^{\infty}\frac{t^{2j}}{(2j)!}>0.$

We now claim for $t>0$ that

(4.19) $0<s_{2}(t)<s_{1}(t)<1.$

The fact that $s_{2}(t)<s_{1}(t)$ is immediate. The inequality $s_{2}(t)>0$
follows from the fact that $M(t,q)>0$ for all $q\leq 0$ and $t>0$ (since each
$g_{j}(t)>0$ for $t>0$).

For the remaining inequality we show that it is both true and sharp. Indeed,

$\displaystyle s_{1}(t)<1\qquad$ $\displaystyle\Longleftrightarrow\qquad
te^{t}-t>te^{t}+1-e^{t}+\sqrt{1-2e^{t}+e^{2t}-t^{2}e^{t}}$
$\displaystyle\Longleftrightarrow\qquad
e^{t}-1-t>\sqrt{1-2e^{t}+e^{2t}-t^{2}e^{t}}$
$\displaystyle\Longleftrightarrow\qquad(e^{t}-1)^{2}-2t(e^{t}-1)+t^{2}>(e^{t}-1)^{2}-t^{2}e^{t}$
$\displaystyle\Longleftrightarrow\qquad(t-2)e^{t}+t+2>0.$

Now set $E(t):=(t-2)e^{t}+t+2$. Then

$E^{\prime}(t)=(t-1)e^{t}+1,\qquad E^{\prime\prime}(t)=te^{t},$

and so $0=E(0)=E^{\prime}(0)=E^{\prime\prime}(0)$. Since
$E^{\prime\prime}(t)>0$ for $t>0$, we have that $E^{\prime}(t)>0$ which in
turn implies that $E(t)>0$, thus confirming that $s_{1}(t)<1$. (Notice that
the function $E$, along with the same line of reasoning given here, appeared
in Lemma 4.9, where the function was called $h_{0}$.)

For the sharpness of this inequality, observe that

$\displaystyle\lim_{t\to\infty}s_{1}(t)=\lim_{t\to\infty}\frac{te^{t}}{te^{t}}\cdot\frac{1+t^{-1}(e^{-t}-1)+\sqrt{t^{-2}(1-e^{-t})^{2}-e^{-t}})}{1-e^{-t}}=1.$

Now since $0<s_{2}(t)<s_{1}(t)<1$ for all $t>0$, we conclude that $M(t,q)>0$
for $t>0$ and $q\in(-\infty,0]\cup[1,\infty)$. The Bernstein-Widder theorem
now shows that $F_{q}(x)$ is strictly completely monotone for $q$ in this
range. ∎

######  Remark 4.20.

The proof of Theorem 4.15 shows the right endpoint $q=1$ of the interval
$0<s_{2}(t)<s_{1}(t)<1$ is sharp. No claim of sharpness is made for the left
endpoint, however. Numerical evidence suggests that $s_{2}(t)$ is a strictly
increasing function on $t>0$, which would imply that the left endpoint can be
slightly improved from $q=0$ to

$q=\lim_{t\searrow 0}s_{2}(t)=\frac{3-\sqrt{3}}{6}\approx 0.211325.$

In any case, we conclude that $F_{q}(x)$ fails to be completely monotone for
$q\in\big{(}\tfrac{3-\sqrt{3}}{6},1\big{)}$. $\lozenge$

######  Corollary 4.21.

Choose $q\in(-\infty,0]\cup[1,\infty)$. Then $\Phi(r,q)<1$ for $r>q$.

###### Proof.

Fix $q\in(-\infty,0]\cup[1,\infty)$ and set $x=r-q$. Theorem 4.15 now says the
function

$r\mapsto F_{q}(r-q)=\Theta(r,q)-r-q+\frac{1}{2}$

is strictly completely monotone for $r>q$. Now differentiate in $r$ to see
that

$\frac{\partial\Theta}{\partial r}(r,q)-1<0,$

for all $r>q$. Since $\frac{\partial\Theta}{\partial r}(r,q)=\Phi(r,q)$ by
(4.8b), we obtain the result. ∎

######  Remark 4.22.

Notice that $q=\frac{2}{3}$ (the $q$-value corresponding to the preferred
symbol function) lies in the interval for which complete monotonicity of
$F_{q}(x)$ is known to fail; see Remark 4.20. Since the Berstein-Widder
approach cannot be applied, the preferred symbol function must be handled
differently (see Section 5). $\lozenge$

###  4.4. Consequences for the Leray transform

The results in Sections 4.2 and 4.3 are now combined with Lemma 3.21.

Recall (Proposition 1.5) that the Leray transform is bounded on
$L^{2}(M_{\gamma},\mu_{d})$ if and only if
$d\in(-1,2\gamma-1)={\mathcal{I}}_{0}(\gamma)$. We now see that the norms of
the $\bm{L}_{k}$ operators strictly decrease with $k$ for more than half of
the interval ${\mathcal{I}}_{0}(\gamma)$.

######  Theorem 4.23.

Let $\bm{L}_{k}$ denote the sub-Leray operator for each non-negative integer
$k$.

  1. (1)

If $\gamma>2$ and $d\in(-1,1]\cup[\gamma-1,2\gamma-1)$, then the function
$k\mapsto J(d,\gamma,k)$ is strictly decreasing on the non-negative integers.
Thus

$\left\|\bm{L}_{k}\right\|_{L^{2}(M_{\gamma},\mu_{d})}>\left\|\bm{L}_{k+1}\right\|_{L^{2}(M_{\gamma},\mu_{d})}.$

  2. (2)

If $\gamma<2$ and $d\in(-1,\gamma-1]\cup[1,2\gamma-1)$, then the function
$k\mapsto J(d,\gamma,k)$ is strictly decreasing on the non-negative integers.
Thus

$\left\|\bm{L}_{k}\right\|_{L^{2}(M_{\gamma},\mu_{d})}>\left\|\bm{L}_{k+1}\right\|_{L^{2}(M_{\gamma},\mu_{d})}.$

Consequently, in both settings, the norm of the full Leray transform is

$\left\|\bm{L}\right\|_{L^{2}(M_{\gamma},\mu_{d})}=\left\|\bm{L}_{0}\right\|_{L^{2}(M_{\gamma},\mu_{d})}.$

###### Proof.

By Corollary 4.21, $\Phi(r,q)<1$ for $q\in(-\infty,0]\cup[1,\infty)$ and
$r>q$. On the other hand, Lemma 3.19 says that the symbol function
$J(\delta_{1-q}(\gamma),\gamma,k)$ is finite for all non-negative integers $k$
if and only if $|q|<\frac{\gamma}{|\gamma-2|}$.

Upon intersecting these two intervals, Lemma 3.21 says that for $\gamma\neq
2$, the function $k\mapsto J(\delta_{1-q}(\gamma),\gamma,k)$ is strictly
decreasing on the non-negative integers for

$q\in\Big{(}\tfrac{-\gamma}{|\gamma-2|},0\Big{]}\cup\Big{[}1,\tfrac{\gamma}{|\gamma-2|}\Big{)}.$

Let us write

(4.24) $d=\delta_{1-q}(\gamma)=(1-q)(\gamma-2)+1$

and recall that
$\sqrt{J(d,\gamma,k)}=\left\|\bm{L}_{k}\right\|_{L^{2}(M_{\gamma},\mu_{d})}$
by equation (1.10).

Now consider four separate cases:

When $\gamma>2$ and $q\in\big{[}1,\tfrac{\gamma}{\gamma-2}\big{)}$, equation
(4.24) implies that
$k\mapsto\left\|\bm{L}_{k}\right\|_{L^{2}(M_{\gamma},\mu_{d})}$ is strictly
decreasing for $d\in(-1,1]$.

When $\gamma>2$ and $q\in\big{(}\tfrac{-\gamma}{\gamma-2},0\big{]}$, equation
(4.24) implies that
$k\mapsto\left\|\bm{L}_{k}\right\|_{L^{2}(M_{\gamma},\mu_{d})}$ is strictly
decreasing for $d\in[\gamma-1,2\gamma-1)$.

When $\gamma<2$ and $q\in\big{[}1,\tfrac{\gamma}{2-\gamma}\big{)}$, equation
(4.24) implies that
$k\mapsto\left\|\bm{L}_{k}\right\|_{L^{2}(M_{\gamma},\mu_{d})}$ is strictly
decreasing for $d\in[1,2\gamma-1)$.

When $\gamma<2$ and $q\in\big{(}\tfrac{-\gamma}{2-\gamma},0\big{]}$, equation
(4.24) implies that
$k\mapsto\left\|\bm{L}_{k}\right\|_{L^{2}(M_{\gamma},\mu_{d})}$ is strictly
decreasing for $d\in(-1,\gamma-1]$.

This establishes both (1) and (2). In all settings covered by these two cases,
the $L^{2}(M_{\gamma},d)$-norm of $\bm{L}_{k}$ decreases with $k$, implying
that $\bm{L}_{0}$ is the sub-Leray operator with the biggest norm. Equation
(2.12) now says that
$\left\|\bm{L}\right\|_{L^{2}(M_{\gamma},d)}=\left\|\bm{L}_{0}\right\|_{L^{2}(M_{\gamma},d)}$.
∎

Both the pairing measure $\sigma=r^{\gamma-1}\,dr\wedge d\theta\wedge ds$
($d=\gamma-1$) and Lebesgue measure $\mu_{1}=r\,dr\wedge d\theta\wedge ds$
($d=1$) fall within the range of applicability of Theorem 4.23. We now record
the norms of the Leray transform in both settings:

######  Corollary 4.25.

Let $\sigma=r^{\gamma-1}\,dr\wedge d\theta\wedge ds$. The Leray transform is
bounded on $L^{2}(M_{\gamma},\sigma)$ with norm

$\left\|\bm{L}\right\|_{L^{2}(M_{\gamma},\sigma)}=\frac{\gamma}{2\sqrt{\gamma-1}}.$

###### Proof.

Since $\sigma$ corresponds to $d=\gamma-1$, Theorem 4.23 applies. Thus

$\left\|\bm{L}\right\|_{L^{2}(M_{\gamma},\sigma)}=\left\|\bm{L}_{0}\right\|_{L^{2}(M_{\gamma},\sigma)}=\sqrt{C_{\sigma}(\gamma,0)}=\frac{\gamma}{2\sqrt{\gamma-1}}.$

Recall that the pairing symbol function
$C_{\sigma}(\gamma,k)=J(\gamma-1,\gamma,k)$. ∎

######  Remark 4.26.

Barrett and Edholm calculated
$\left\|\bm{L}\right\|_{L^{2}(M_{\gamma},\sigma)}$ in [7, Proposition 4.18]
using a different approach. The argument given is tailored to the pairing
measure ($q=0$) setting and fails to extend to other measures. $\lozenge$

######  Corollary 4.27.

Let $\mu_{1}=r\,dr\wedge d\theta\wedge ds$. The Leray transform $\bm{L}$ is
bounded on $L^{2}(M_{\gamma},\mu_{1})$ with norm

$\left\|\bm{L}\right\|_{L^{2}(M_{\gamma},\mu_{1})}=\begin{cases}(\gamma-1)^{\frac{1}{\gamma}-1}\sqrt{\frac{\pi}{4}(\gamma-2)\gamma\csc\big{(}\frac{2\pi}{\gamma}\big{)}},&\gamma\in(1,2)\cup(2,\infty)\\\
1,&\gamma=2.\end{cases}$

The formula is continuous at $\gamma=2$.

###### Proof.

Since $\mu_{1}$ corresponds to $d=1$, Theorem 4.23 applies once again for
$\gamma\neq 2$:

$\displaystyle\left\|\bm{L}\right\|_{L^{2}(M_{\gamma},\mu_{1})}=\left\|\bm{L}_{0}\right\|_{L^{2}(M_{\gamma},\mu_{1})}=\sqrt{J(1,\gamma,0)}$
$\displaystyle=(\gamma-1)^{\frac{1}{\gamma}-1}\sqrt{\big{(}\tfrac{\gamma}{2}\big{)}^{2}\,\Gamma\big{(}\tfrac{2}{\gamma}\big{)}\Gamma\big{(}2-\tfrac{2}{\gamma}\big{)}}$
$\displaystyle=(\gamma-1)^{\frac{1}{\gamma}-1}\sqrt{\tfrac{\pi}{4}(\gamma-2)\gamma\csc\big{(}\tfrac{2\pi}{\gamma}\big{)}},$

where we’ve used the factorial property of the $\Gamma$-function as well as
Euler’s reflection formula.

The $\gamma=2$ case follows from (3.10) and L’Hôpital’s rule confirms
continuity. ∎

##  5\. The preferred symbol function

In this section we prove that the preferred symbol function $k\mapsto
C_{\nu}(\gamma,k)$, $\gamma\neq 2$, is strictly increasing, in stark contrast
with the strictly decreasing symbol function behavior seen in other parts of
the paper. Our goal is to show

$\Phi(r,\tfrac{2}{3})>1,\qquad\mathrm{for}\qquad r>\tfrac{2}{3}.$

Unlike the work in Section 4, we are unable to use complete monotonicity and
the Berstein-Widder theorem to prove such an inequality, since it is known
that the related function $F_{q}$ fails to be completely monotone for
$q=\frac{2}{3}$; see Remark 4.20.

In place of the integral representations used in the previous section, we use
here the infinite series description obtained from equation (3.18b):

(5.1)
$\Phi\big{(}r,\tfrac{2}{3}\big{)}=\sum_{j=1}^{\infty}f_{r}(j),\qquad\mathrm{where}\qquad
f_{r}(j)=\frac{18r(3j-2)}{(3r+3j-2)^{3}}.$

###  5.1. Two tools

Two classical results are crucial to the subsequent analysis.

The first is Descartes’ Rule of Signs (see [22] for a simple proof):

######  Proposition 5.2 (Descartes).

Let $p$ be a single variable polynomial with real coefficients, with monomial
terms arranged so that exponents appear in ascending order. The number of
positive roots of $p$ (counted with multiplicities) is either (i) equal to the
number of sign changes between consecutive (non-zero) coefficients, or (ii)
less than that by an even number.

The second is the Euler-Maclaurin formula relating sums and integrals; see
[10, Section 9.5]. For our purposes it is sufficient to use the following
first-order version:

######  Proposition 5.3 (Euler-Maclaurin).

Let $m<n$ be integers and $f\in C^{1}[m,n]$. Then

(5.4)
$\sum_{j=m+1}^{n}f(j)=\int_{m}^{n}f(x)\,dx+\frac{f(n)-f(m)}{2}+\int_{m}^{n}f^{\prime}(x)P_{1}(x)\,dx,$

where $P_{1}(x)=B_{1}(x-\lfloor x\rfloor)$, $B_{1}(x)=x-\tfrac{1}{2}$ is the
first Bernoulli polynomial and $\lfloor x\rfloor$ the greatest integer less
than or equal to $x$.

We refer to the rightmost integral in (5.4) as the Bernoulli integral.

###  5.2. Applying Euler-Maclaurin

We apply the Euler-Maclaurin formula to re-express the sum in (5.1), then use
carefully chosen bounds to estimate the Bernoulli integral. After a
considerable amount of analysis, the behavior of
$r\mapsto\Phi\big{(}r,\frac{2}{3}\big{)}$ is reduced to a problem about the
positive real roots of an explicit polynomial of degree 16. This in turn is
completely understood using only Descartes’ Rule of Signs and direct
polynomial evaluation.

######  Theorem 5.5.

If $r>\tfrac{2}{3}$, then
$\Phi\big{(}r,\tfrac{2}{3}\big{)}=2r\psi^{\prime}\big{(}r+\tfrac{1}{3}\big{)}+r^{2}\psi^{\prime\prime}\big{(}r+\tfrac{1}{3}\big{)}>1$.

###### Proof.

Start with the function $f_{r}$ from (5.1) and use it in the first order
Euler-Maclaurin formula (5.4) with $m=0$ and sending $n\to\infty$. Computation
yields the first two terms on the right side of (5.4):

(5.6)
$\int_{0}^{\infty}f_{r}(x)\,dx=1-\frac{4}{(3r-2)^{2}},\qquad\frac{f_{r}(\infty)-f_{r}(0)}{2}=\frac{18r}{(3r-2)^{3}},$

where $f_{r}(\infty)=\lim_{n\to\infty}f_{r}(n)$. We also calculate that

$f^{\prime}_{r}(x)=\frac{54r(4+3r-6x)}{(3x+3r-2)^{4}},$

from which the following expression for the Bernoulli integral in (5.4) is
obtained:

$\int_{0}^{\infty}f^{\prime}_{r}(x)P_{1}(x)\,dx=\sum_{N=0}^{\infty}\int_{0}^{1}f_{r}^{\prime}(x+N)B_{1}(x)\,dx=\sum_{N=0}^{\infty}\frac{81r(9N^{2}-3N-9r^{2}-2)}{(3r+3N+1)^{3}(3r+3N-2)^{3}}.$

Denote the rational function appearing in the sum above by

(5.7) $S(r,N)=\frac{81r(9N^{2}-3N-9r^{2}-2)}{(3r+3N+1)^{3}(3r+3N-2)^{3}}.$

Now inserting (5.6) and (5.7) into (5.4), we see by (5.1) that

$\displaystyle\Phi\big{(}r,\tfrac{2}{3}\big{)}=\sum_{j=1}^{\infty}f_{r}(j)$
$\displaystyle=1-\frac{4}{(3r-2)^{2}}+\frac{18r}{(3r-2)^{3}}+\sum_{N=0}^{\infty}S(r,N)$
(5.8) $\displaystyle=1+\frac{6r-1}{(3r+1)^{3}}+\sum_{N=1}^{\infty}S(r,N).$

(Note that we have peeled the $N=0$ term off of the sum.)

It thus remains to show that

(5.9) $\frac{6r-1}{(3r+1)^{3}}+\sum_{N=1}^{\infty}S(r,N)$

is strictly positive for $r>\frac{2}{3}$.

####  5.2.1. The function $S(r,x)$

We now show that $x\mapsto S(r,x)$ ($x$ is for now regarded as a non-negative
real number) has a single local extrema (a maximum) on $[0,\infty)$. Indeed,

$\displaystyle\frac{\partial S}{\partial
x}(r,x)=\frac{243r[(-4+39r-36r^{2}+162r^{3})+(18+18r+216r^{2})x+(54-54r)x^{2}-108x^{3}]}{(3r+3x+1)^{4}(3r+3x-2)^{4}}$

Write the cubic polynomial in $x$ appearing in the brackets above by

(5.10)
$p_{r}(x)=(-4+39r-36r^{2}+162r^{3})+(18+18r+216r^{2})x+(54-54r)x^{2}-108x^{3}.$

It is easily checked that the coefficients of $1$ and $x$ are positive for
$r>\frac{2}{3}$, the coefficient of $x^{2}$ changes sign at $r=1$, while the
coefficient of $x^{3}$ is a negative constant. In all cases, there is exactly
one sign change among the consecutive coefficients when the monomial terms are
considered in ascending order. Descartes’ Rule of Signs thus guarantees the
existence of a unique positive real root of the function $x\mapsto p_{r}(x)$,
where the values of $p_{r}(x)$ change from positive to negative. Denote this
root by $Q_{r}$.

We claim that
$Q_{r}\in\big{(}\frac{3r}{2}+\frac{1}{6},\frac{3r}{2}+\frac{1}{3}\big{)}$:
Indeed, for $r>\frac{2}{3}$,

$\displaystyle p_{r}\big{(}\tfrac{3r}{2}+\tfrac{1}{6}\big{)}=81r>0,$
$\displaystyle
p_{r}\big{(}\tfrac{3r}{2}+\tfrac{1}{3}\big{)}=4+66r-\tfrac{225}{2}r^{2}<0.$

(An more precise estimate on $Q_{r}$ is obtained in Lemma A.3.)

We see from (5.7) that $x\mapsto S(r,x)$ starts negative but then becomes (and
remains) positive for $x$ large enough, since the function tends to zero as
$x\to\infty$ and its derivative changes signs exactly once. In particular, the
global maximum value of $S(r,Q_{r})$ is positive.

We now trivially re-write the quantity in (5.9) as

(5.11) $\frac{6r-1}{(3r+1)^{3}}+S(r,1)+S(r,2)+\sum_{N=3}^{\infty}S(r,N).$

Separating out the $N=1$ and $N=2$ terms from the rest of the summation
simplifies the following estimate. We claim that

(5.12) $\sum_{N=3}^{\infty}S(r,N)>\int_{2}^{\infty}S(r,x)\,dx-S(r,Q_{r}).$

To see this, set $K:=\lfloor Q_{r}\rfloor$, the greatest integer less than or
equal to $Q_{r}$. From the discussion above, $S(r,\cdot)$ increases on $(1,K)$
and decreases on $(K+1,\infty)$. We now consider two cases: $K\leq 2$ and
$K\geq 3$.

When $K\leq 2$, we have that $S(r,\cdot)$ decreases on $(3,\infty)$, so

(5.13a) $\sum_{N=3}^{\infty}S(r,N)>\int_{3}^{\infty}S(r,x)\,dx.$ But also note
that (5.13b) $0>\int_{2}^{3}S(r,x)\,dx-S(r,Q_{r}),$

since any integral over an interval of unit length is overestimated by the
maximum value of its integrand. Combining (5.13a) with (5.13b) now yields
(5.12) for $K\leq 2$.

When $K\geq 3$, we are able to write

(5.14a) $\displaystyle\sum_{N=3}^{K}S(r,N)$
$\displaystyle>\int_{2}^{K}S(r,x)\,dx,$ (5.14b)
$\displaystyle\sum_{N=K+1}^{\infty}S(r,N)$
$\displaystyle>\int_{K+1}^{\infty}S(r,x)\,dx.$ Using reasoning identical to
what justified (5.13b), we see that (5.14c)
$0>\int_{K}^{K+1}S(r,x)\,dx-S(r,Q_{r}).$

Combining (5.14a), (5.14b) and (5.14c) yields (5.12) for $K\geq 3$.

From (5.12), the following quantity is a lower bound on (5.9) = (5.11):

(5.15)
$\frac{6r-1}{(3r+1)^{3}}+S(r,1)+S(r,2)+\int_{2}^{\infty}S(r,x)\,dx-S(r,Q_{r}).$

We now show (5.16) is positive by computing the above integral and estimating
$S(r,Q_{r})$.

A partial fraction decomposition of $S(r,x)$ helps to yield the following:

(5.16)
$\int_{2}^{\infty}S(r,x)\,dx=\frac{3r(108r^{3}+594r^{2}+1035r+616)}{2(3r+4)^{2}(3r+7)^{2}}-2r\log\bigg{(}\frac{3r+7}{3r+4}\bigg{)}.$

We also use the following estimate on $S(r,Q_{r})$: for $r>\frac{2}{3}$,

(5.17) $S(r,Q_{r})<\frac{16}{3125r^{3}}.$

The right hand side of (5.17) is the leading term in the Taylor expansion of
$S(r,Q_{r})$ at $\infty$, see Remark A.9. The proof of estimate (5.17) is
given in Lemma A.4 of the Appendix.

We now use the (5.16) and (5.17) to define a new function:

$\displaystyle H(r)$
$\displaystyle=\frac{6r-1}{(3r+1)^{3}}+S(r,1)+S(r,2)+\int_{2}^{\infty}S(r,x)\,dx-\frac{16}{3125r^{3}}$
(5.18)
$\displaystyle=\frac{6r-1}{(3r+1)^{3}}+\frac{81r(4-9r^{2})}{(3r+1)^{3}(3r+4)^{3}}+\frac{81r(28-9r^{2})}{(3r+4)^{3}(3r+7)^{3}}+$
$\displaystyle\qquad\qquad+\frac{3r(108r^{3}+594r^{2}+1035r+616)}{2(3r+4)^{2}(3r+7)^{2}}-2r\log\bigg{(}\frac{3r+7}{3r+4}\bigg{)}-\frac{16}{3125r^{3}}$

Upon combining (5.16) with the bound (5.17), we see that $H(r)$ is a lower
bound for (5.15). Our goal is now to show $H(r)$ is positive for
$r>\frac{2}{3}$. Note that while $H$ is itself not a rational function, its
second derivative is rational. This is crucial to the coming argument.

####  5.2.2. The function $H(r)$

We now present $H$ in a more digestible fashion. Combining all of the rational
functions in (5.18) yields

(5.19)
$H(r)=\frac{F_{1}(r)}{6250r^{3}(3r+1)^{3}(3r+4)^{3}(3r+7)^{3}}-2r\log\bigg{(}\frac{3r+7}{3r+4}\bigg{)},$

where $F_{1}$ is a polynomial of degree 12 with integer coefficients, the full
formula of which is included in Table 1 below. The leading coefficient of
$F_{1}$ (written $c_{1,12}$ in the table) is $246\,037\,500=2^{2}3^{9}5^{5}$.
Note that the denominator of the rational function in (5.19) is also a
polynomial of degree 12, with leading coefficient
$123\,018\,750=2^{1}3^{9}5^{5}$. Thus,

$\lim_{r\to\infty}H(r)=2-2=0.$

Now differentiate $H$ to obtain

(5.20)
$H^{\prime}(r)=\frac{F_{2}(r)}{3125r^{4}(3r+1)^{4}(3r+4)^{4}(3r+7)^{4}}-2\log\bigg{(}\frac{3r+7}{3r+4}\bigg{)},$

where $F_{2}$ is a polynomial of degree 15 with integer coefficients (the full
formula is also included in Table 1). Since the denominator polynomial has
degree 16, it follows that

$\lim_{r\to\infty}H^{\prime}(r)=0.$

Finally, differentiate $H^{\prime}$ to obtain

(5.21)
$H^{\prime\prime}(r)=\frac{F_{3}(r)}{3125r^{5}(3r+1)^{5}(3r+4)^{5}(3r+7)^{5}},$

where $F_{3}$ is a polynomial of degree 16 with integer coefficients, the
exact expression of which is included in Table 1. Since the denominator
polynomial has degree 20, it follows that

$\lim_{r\to\infty}H^{\prime\prime}(r)=0.$

The following statements are easily verified from Table 1 below:

  1. (1)

The coefficients of the $1,r,\cdots,r^{7}$ terms in $F_{3}$ are negative.

  2. (2)

The coefficients of the $r^{8},r^{9},\cdots,r^{16}$ terms in $F_{3}$ are
positive.

  3. (3)

Descartes’ Rule of Signs thus guarantees that $F_{3}$ has a unique positive
real root. Since the denominator of $H^{\prime\prime}(r)$ is positive for
$r>0$, we see that $H^{\prime\prime}(r)$ changes sign exactly once for $r>0$.

  4. (4)

Evaluation of the rational function $H^{\prime\prime}$ shows this sign change
occurs inside the interval $\big{(}\frac{1}{3},\frac{2}{3}\big{)}$:

$H^{\prime\prime}\left(\tfrac{1}{3}\right)=-\frac{437\,616\,243}{25\,600\,000},\qquad\mathrm{and}\qquad
H^{\prime\prime}\left(\tfrac{2}{3}\right)=\frac{49\,618}{2\,278\,125}.$

  5. (5)

We conclude that $H^{\prime\prime}>0$ on the interval
$\big{(}\frac{2}{3},\infty\big{)}$.

  6. (6)

Since $H^{\prime\prime}>0$ on $\big{(}\frac{2}{3},\infty\big{)}$, $H^{\prime}$
strictly increases on this interval. Since
$\lim\limits_{r\to\infty}H^{\prime}(r)=0$, we conclude that $H^{\prime}<0$ on
$\big{(}\frac{2}{3},\infty\big{)}$.

  7. (7)

Since $H^{\prime}<0$ on $\big{(}\frac{2}{3},\infty\big{)}$, $H$ strictly
decreases on this interval. Since $\lim\limits_{r\to\infty}H(r)=0$, we
conclude that $H>0$ on $\big{(}\frac{2}{3},\infty\big{)}$.

Since $H(r)$ is positive for $r>\frac{2}{3}$, the expression in (5.15) is
positive for this range of $r$. This in turn shows that (5.11) = (5.9) is
positive for the same range of $r$. Thus by (5.8),

$\displaystyle\Phi\big{(}r,\tfrac{2}{3}\big{)}=\sum_{j=1}^{\infty}f_{r}(j)>1+H(r)>1$

for $r>\tfrac{2}{3}$, completing the proof. ∎

$n$ |  $c_{1,n}$ |  $c_{2,n}$ |  $c_{3,n}$
---|---|---|---
$0$ |  $-702\,464$ |  $29\,503\,488$ |  $-3\,304\,390\,656$
$1$ |  $-8\,805\,888$ |  $493\,129\,728$ |  $-69\,038\,161\,920$
$2$ |  $-44\,924\,544$ |  $3\,546\,063\,360$ |  $-640\,689\,315\,840$
$3$ |  $-258\,414\,880$ |  $14\,430\,286\,080$ |  $-3\,491\,968\,112\,640$
$4$ |  $1\,018\,286\,832$ |  $77\,896\,979\,088$ |  $-12\,471\,183\,325\,440$
$5$ |  $4\,962\,569\,148$ |  $110\,838\,411\,360$ |  $-48\,684\,386\,314\,944$
$6$ |  $11\,832\,384\,015$ |  $-17\,706\,703\,248$ |  $-111\,582\,268\,515\,360$
$7$ |  $23\,240\,472\,534$ |  $244\,982\,773\,080$ |  $-78\,421\,336\,513\,920$
$8$ |  $29\,834\,360\,478$ |  $1\,512\,143\,688\,033$ |  $148\,629\,164\,640\,120$
$9$ |  $23\,154\,232\,644$ |  $2\,940\,847\,647\,885$ |  $378\,180\,897\,173\,910$
$10$ |  $10\,449\,759\,375$ |  $3\,231\,415\,617\,165$ |  $377\,142\,473\,066\,319$
$11$ |  $2\,501\,381\,250$ |  $2\,264\,445\,221\,688$ |  $224\,889\,469\,312\,590$
$12$ |  $246\,037\,500$ |  $1\,025\,079\,243\,543$ |  $92\,232\,089\,533\,215$
$13$ |  $0$ |  $290\,262\,740\,625$ |  $25\,224\,576\,030\,090$
$14$ |  $0$ |  $47\,054\,671\,875$ |  $3\,414\,213\,475\,245$
$15$ |  $0$ |  $3\,321\,506\,250$ |  $66\,996\,641\,106$
$16$ |  $0$ |  $0$ |  $14\,946\,778\,125$
Table 1. Exact values of coefficients of polynomials $F_{j}(r)=\sum
c_{j,n}r^{n}$ in Section 5.2.2.

###  5.3. Consequences for Leray transform

We now prove that for $\gamma\neq 2$, the $L^{2}(M_{\gamma},\nu)$ norm of
$\bm{L}_{k}$ is strictly increasing in $k$.

######  Theorem 5.22.

Let $\gamma\neq 2$. Then $k\mapsto C_{\nu}(\gamma,k)$ is a strictly increasing
function on the non-negative integers. Thus,

$\left\|\bm{L}_{k}\right\|_{L^{2}(M_{\gamma},\nu)}<\left\|\bm{L}_{k+1}\right\|_{L^{2}(M_{\gamma},\nu)}.$

###### Proof.

We have just seen in Theorem 5.5 that $\Phi\big{(}r,\tfrac{2}{3}\big{)}>1$ for
$r>\frac{2}{3}$. Lemma 3.21 now applies, saying the preferred symbol function
$k\mapsto C_{\nu}(\gamma,k)$ is strictly increasing on the non-negative
integers. Thus, recalling (1.7b) and (1.10), we have

(5.23)
$\left\|\bm{L}_{k}\right\|_{L^{2}(M_{\gamma},\nu)}=\sqrt{C_{\nu}(\gamma,k)}<\sqrt{C_{\nu}(\gamma,k+1)}=\left\|\bm{L}_{k+1}\right\|_{L^{2}(M_{\gamma},\nu)},$

completing the proof. ∎

We now easily deduce the $L^{2}(M_{\gamma},\nu)$ norm of the full Leray
transform.

######  Theorem 5.24.

The norm of the Leray transform on $L^{2}(M_{\gamma},\nu)$ is given by

$\left\|\bm{L}\right\|_{L^{2}(M_{\gamma},\nu)}=\sqrt{\frac{\gamma}{2\sqrt{\gamma-1}}}.$

###### Proof of Theorem 5.24.

The strictly increasing behavior seen in (5.23) combines with (2.12) and
Proposition 1.11 to give

$\left\|\bm{L}\right\|_{L^{2}(M_{\gamma},\nu)}=\lim_{k\to\infty}\left\|\bm{L}_{k}\right\|=\sqrt{\frac{\gamma}{2\sqrt{\gamma-1}}},$

completing the proof. ∎

##  Appendix A Proof of the estimate on $S(r,Q_{r})$

In this appendix we prove (5.17), the crucial upper bound on $S(r,Q_{r})$ that
was used in our proof of Theorem 5.5. Recall the definition of the rational
function

$S(r,x)=\frac{81r(9x^{2}-3x-9r^{2}-2)}{(3r+3x+1)^{3}(3r+3x-2)^{3}}.$

It was shown in Section 5.2.1 that for $r>\frac{2}{3}$, $x\mapsto S(r,x)$ has
a single local extrema (a maximum) at $x=Q_{r}$, where $Q_{r}$ is the unique
positive root of the polynomial

(A.1)
$p_{r}(x)=(-4+39r-36r^{2}+162r^{3})+(18+18r+216r^{2})x+(54-54r)x^{2}-108x^{3}.$

It was previously demonstrated that
$Q_{r}\in\big{(}\tfrac{3r}{2}+\tfrac{1}{6},\tfrac{3r}{2}+\tfrac{1}{3}\big{)}$,
but here sharper precision is needed; see Remark A.9.

###  A.1. A sharper estimate on $Q_{r}$

Let us define

(A.2a) $\displaystyle m(r)$
$\displaystyle=\frac{3r}{2}+\frac{1}{6}+\frac{2}{25r}-\frac{21}{3125r^{3}}$
(A.2b) $\displaystyle M(r)$
$\displaystyle=\frac{3r}{2}+\frac{1}{6}+\frac{2}{25r}.$

(These functions are truncated Taylor expansions of $Q_{r}$ at $\infty$; see
Remark A.9.)

######  Lemma A.3.

Let $m(r)$ and $M(r)$ be as above. The following inequality holds

$m(r)<Q_{r}<M(r).$

###### Proof.

This can be seen from direct evaluation. Indeed,

$p_{r}(m(r))=\frac{81(12348+125r^{2}(1515625r^{4}+21000r^{2}-5292))}{5^{15}\,r^{9}},$

and it is easily checked that $1515625r^{4}+21000r^{2}-5292>0$ for
$r>\frac{2}{3}$. Since all other terms in the formula are clearly positive, we
conclude $p_{r}(m(r))>0$.

Now calculate

$p_{r}(M(r))=-\frac{81\left(36+875r^{2}\right)}{5^{6}\,r^{3}},$

which is clearly negative for $r>\frac{2}{3}$. Since $Q_{r}$ is the unique
positive root of $p_{r}$, we conclude it must lie in the interval
$(m(r),M(r))$. ∎

With these improved bounds on $Q_{r}$ we are ready to prove the desired
estimate:

######  Lemma A.4.

The following estimate holds for $r>\frac{2}{3}$

(A.5) $S(r,Q_{r})<\frac{16}{3125r^{3}}.$

###### Proof.

First define the following two variable polynomial

(A.6)
$W(r,Q)=2^{4}(3r+3Q+1)^{3}(3r+3Q-2)^{3}-3^{4}5^{5}r^{4}(9Q^{2}-3Q-9r^{2}-2).$

This function is closely tied to the inequality in (A.5). Indeed,

$W(r,Q_{r})>0\qquad\Longleftrightarrow\qquad S(r,Q_{r})<\frac{16}{3125r^{3}}.$

Now expand (A.6) and collect like-terms to express $W$ as

$W(r,Q)=\sum\nolimits_{j,k}a_{j,k}\,r^{j}Q^{k}.$

Let ${\mathcal{H}}$ be the Heaviside step function (the indicator function of
the positive real numbers) and define two closely related polynomials with
positive integer coefficients:

(A.7a) $\displaystyle U(r,Q)$
$\displaystyle=\sum\nolimits_{j,k}{\mathcal{H}}(a_{j,k})\,a_{j,k}\,r^{j}Q^{k}$
(A.7b) $\displaystyle V(r,Q)$
$\displaystyle=-\sum\nolimits_{j,k}{\mathcal{H}}(-a_{j,k})\,a_{j,k}\,r^{j}Q^{k}.$

Clearly, we have that

$W(r,Q)=U(r,Q)-V(r,Q).$

Since both $U$ and $V$ have only positive coefficients in their expansions,
Lemma A.3 implies

$\displaystyle U(r,Q_{r})>U(r,m(r)),\qquad\qquad V(r,M(r))>V(r,Q_{r}).$

Our goal will be to prove that

$U(r,m(r))>V(r,M(r)),$

which will imply $W(r,Q_{r})>0$ and thereby give the result.

###  A.2. Analysis of the polynomials $U(r,Q)$ and $V(r,Q)$

The polynomials $U$ and $V$ are obtained from $W$ by expanding (A.6) and
separating the monomial terms by the signs of their coefficients. After
collecting terms in this way, we may re-write (A.7a) and (A.7b) as polynomials
in the $Q$ variable:

(A.8) $U(r,Q)=\sum_{k=0}^{6}u_{k}(r)Q^{k},\qquad\qquad
V(r,Q)=\sum_{k=0}^{5}v_{k}(r)Q^{k},$

where the coefficient functions $u_{k}(r)$ and $v_{k}(r)$ are given in Table
2.

$k$ |  $u_{k}(r)$ |  $v_{k}(r)$
---|---|---
$0$ |  $2289789r^{6}+502362r^{4}+4752r^{3}+864r^{2}$ |  $11664r^{5}+576r+128$
$1$ |  $69984r^{5}+701055r^{4}+14256r^{2}+1728r$ |  $15552r^{3}+576$
$2$ |  $14256r+864$ |  $2103165r^{4}+116640r^{3}+23328r^{2}$
$3$ |  $233280r^{3}+4752$ |  $116640r^{2}+15552r$
$4$ |  $174960r^{2}$ |  $58320r+3888$
$5$ |  $69984r$ |  $11664$
$6$ |  $11664$ |  $0$
Table 2. Coefficients polynomials in $U(r,Q)$ and $V(r,Q)$

Now insert $m(r)$ and $M(r)$ into (A.8) and expand $U(r,m(r))-V(r,M(r))$ out
as a rational function of the form

$\sum b_{n}r^{n}.$

Using Table 2 together with degree considerations, we see that $b_{n}=0$ for
$n<-18$ and $n>6$. Some of the remaining $b_{j}$, including $b_{5}$ and
$b_{6}$, are also equal to zero; see Table 4, noting that
$\beta_{n}=b_{n-18}$. But it is easily checked from (A.8) and Table 2 that
$b_{-18}\neq 0$, which leads to the definition of the related polynomial

$\displaystyle P(r)$
$\displaystyle=r^{18}(U(r,m(r))-V(r,M(r)))=\sum_{n=0}^{22}\beta_{n}r^{n}$

The exact values of the $\beta_{n}$ are listed in Table 4. But we can explain
the next step of the argument just by considering the signs of the $\beta_{n}$
in Table 3.

$0$ |  $1$ |  $2$ |  $3$ |  $4$ |  $5$ |  $6$ |  $7$ |  $8$ |  $9$ |  $\\!10$ |  $\\!11$ |  $\\!12$ |  $\\!13$ |  $\\!14$ |  $\\!15$ |  $\\!16$ |  $\\!17$ |  $\\!18$ |  $\\!19$ |  $\\!20$ |  $\\!21$ |  $\\!22$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\\!+$ |  $0$ |  $\\!-$ |  $\\!-$ |  $\\!+$ |  $\\!+$ |  $\\!+$ |  $\\!-$ |  $\\!-$ |  $-$ |  $-$ |  $+$ |  $+$ |  $+$ |  $-$ |  $-$ |  $-$ |  $-$ |  $-$ |  $-$ |  $-$ |  $0$ |  $+$
Table 3. Signs of coefficients $\beta_{n}$ in $P(r)$

Table 3 shows that the coefficients of $P(r)$ change signs six times when its
terms are listed in ascending order; Descartes’ Rule of Signs thus says that
the number of positive real roots of $P$ is either $0$, $2$, $4$ or $6$. This
is not precise enough for our purposes, but we can cut down the number of sign
changes by repeatedly differentiating. After fourteen derivatives, Table 3
shows the resulting polynomial has exactly one coefficient sign change when
the coefficients are listed in ascending order. Consequently, $P^{(14)}$ has a
unique positive real root.

The remaining computations are nothing more than differentiation and direct
polynomial evaluation: $P(r)$ and its first thirteen derivatives are evaluated
at $r=\tfrac{2}{3}$, while $P^{(14)}(r)$ is evaluated at $r=0$ and
$\tfrac{2}{3}$. These calculations are tedious by hand, but they can be worked
out in totality from Table 4.

Evaluation shows that the unique root of $P^{(14)}(r)$ lies in the interval
$(0,\tfrac{2}{3})$. Indeed,

$\displaystyle
P^{(14)}(0)=-\frac{2\,159\,106\,379\,702\,272}{5^{7}}\approx-2.76366\cdot
10^{10},$ $\displaystyle
P^{(14)}\Big{(}\frac{2}{3}\Big{)}=\frac{18\,441\,535\,745\,869\,667\,168\,145\,408}{5^{7}}\approx
2.36052\cdot 10^{20}.$

We thus conclude that

$P^{(14)}(r)>0,\qquad r>\frac{2}{3}.$

It can also be seen by direct evaluation that for integers $0\leq n\leq 13$,

$P^{(n)}\Big{(}\frac{2}{3}\Big{)}>0.$

Decreasing the number of derivatives one step at a time, we see that for
integers $0\leq n\leq 13$, it also holds that

$P^{(n)}(r)>0,\qquad r>\frac{2}{3}.$

In particular, $P(r)>0$ for $r>\tfrac{2}{3}$, so the difference
$U(r,m(r))-V(r,M(r))>0$. This implies $U(r,Q_{r})-V(r,Q_{r})=W(r,Q_{r})>0$,
thereby proving (A.5). ∎

$n$ |  $\beta_{n}$ |  $7$ |  $-\dfrac{162\,030\,456}{5^{17}}$ |  $15$ |  $-\dfrac{3\,195\,801}{5^{6}}$
---|---|---|---|---|---
$0$ |  $\dfrac{1\,000\,376\,035\,344}{5^{30}}$ |  $8$ |  $-\dfrac{3\,421\,928\,916}{5^{17}}$ |  $16$ |  $-\dfrac{2\,065\,794\,597}{5^{9}}$
$1$ |  $0$ |  $9$ |  $-\dfrac{84\,873\,096}{5^{14}}$ |  $17$ |  $-\dfrac{91\,854}{5^{2}}$
$2$ |  $-\dfrac{857\,465\,173\,152}{5^{27}}$ |  $10$ |  $-\dfrac{922\,948\,992}{5^{15}}$ |  $18$ |  $-\dfrac{629\,807\,157}{2^{2}5^{6}}$
$3$ |  $-\dfrac{47\,636\,954\,064}{5^{25}}$ |  $11$ |  $\dfrac{17\,635\,968}{5^{11}}$ |  $19$ |  $-\dfrac{12\,267\,612}{5^{4}}$
$4$ |  $\dfrac{163\,326\,699\,648}{5^{24}}$ |  $12$ |  $\dfrac{657\,460\,071}{5^{12}}$ |  $20$ |  $-\dfrac{71\,827\,641}{2^{2}5^{4}}$
$5$ |  $\dfrac{6\,805\,279\,152}{5^{21}}$ |  $13$ |  $\dfrac{10\,471\,356}{5^{9}}$ |  $21$ |  $0$
$6$ |  $\dfrac{9\,694\,822\,284}{5^{20}}$ |  $14$ |  $-\dfrac{619\,164}{5^{9}}$ |  $22$ |  $\dfrac{455\,625}{2}$
Table 4. Exact values of coefficients in $P(r)=\sum\beta_{n}r^{n}$

######  Remark A.9.

Applying the cubic formula to the polynomial $p_{r}(x)$ given in (A.1), we
have

$\displaystyle
Q_{r}=\frac{1}{6\alpha_{r}}\left(3+25r^{2}+(1-r)\alpha_{r}+\alpha_{r}^{2}\right),\qquad\mathrm{where}$
$\displaystyle\alpha_{r}=\left(125r^{3}+36r-3\sqrt{375r^{4}+69r^{2}-3}\right)^{1/3}.$

The bounds $M(r)$ and $m(r)$ in Lemma A.3 come from the Taylor expansion of
$Q_{r}$ at $\infty$:

$Q_{r}=\frac{3r}{2}+\frac{1}{6}+\frac{3}{25r}-\frac{21}{3125r^{3}}+O\Big{(}\frac{1}{r^{4}}\Big{)}.$

It should be emphasized that the negative degree terms in $M(r)$ and $m(r)$
are crucial to the proof of Lemma A.4; simply using affine functions to bound
$Q_{r}$ is not sufficient to prove the estimate on $S(r,Q_{r})$.

Similarly, the bound on $S(r,Q_{r})$ also comes from its the Taylor expansion
at $\infty$:

$S(r,Q_{r})=\frac{16}{3125r^{3}}+O\Big{(}\frac{1}{r^{5}}\Big{)}$

This method of generating candidates for sufficiently sharp bounds is likely
to have application for many other polygamma inequalities, and more generally,
in other problems in which the Euler-Maclaurin formula is utilized. $\lozenge$

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