# Bernstein inequality on conic domains and triangles

Yuan Xu  Department of Mathematics, University of Oregon, Eugene, OR
97403–1222, USA<EMAIL_ADDRESS>
###### Abstract.

We establish weighted Bernstein inequalities in $L^{p}$ space for the doubling
weight on the conic surface
${\mathbb{V}}_{0}^{d+1}=\\{(x,t):\|x\|=t,x\in{\mathbb{R}}^{d},t\in[0,1]\\}$ as
well as on the solid cone bounded by the conic surface and the hyperplane
$t=1$, which becomes a triangle on the plane when $d=1$. While the
inequalities for the derivatives in the $t$ variable behave as expected, there
are inequalities for the derivatives in the $x$ variables that are stronger
than what one may have expected. As an example, on the triangle
$\\{(x_{1},x_{2}):x_{1}\geq 0,\,x_{2}\geq 0,\,x_{1}+x_{2}\leq 1\\}$, the usual
Bernstein inequality for the derivative $\partial_{1}$ states that
$\|\phi_{1}\partial_{1}f\|_{p,w}\leq cn\|f\|_{p,w}$ with
$\phi_{1}(x_{1},x_{2}):=x_{1}(1-x_{1}-x_{2})$, whereas our new result gives

$\|(1-x_{2})^{-1/2}\phi_{1}\partial_{1}f\|_{p,w}\leq cn\|f\|_{p,w}.$

The new inequality is stronger and points out a phenomenon unobserved hitherto
for polygonal domains.

###### Key words and phrases:

Bernstein inequality, polynomials, doubling weight, conic domain, triangle

###### 2010 Mathematics Subject Classification:

41A10, 41A63, 42C10, 42C40

The author is partially supported by Simons Foundation Grant #849676 and by an
Alexander von Humboldt award.

##  1\. Introduction

The Bernstein inequalities are fundamental in approximation theory, as seen in
the inverse estimate in the characterization of best approximation and
numerous other applications (see, for example, [10, 12]). Starting from
algebraic polynomials on $[0,1]$, the Bernstein or Bernstein-Markov
inequalities have been refined and generalized extensively by many authors.
Most notable recent extensions are the inequalities in $L^{p}$ norm with the
doubling weight, initiated in [20], and inequalities for multivariable
polynomials on various domains, such as polytopes, convex domains, and domains
with smooth boundaries; see [1, 2, 4, 5, 6, 7, 8, 9, 14, 15, 16, 17, 18, 19,
22, 23, 24] and reference therein. The purpose of this paper is to establish
the Bernstein inequalities, in uniform norm and $L^{p}$ norm with doubling
weight, for polynomials on the conic surface

${\mathbb{V}}_{0}^{d+1}=\left\\{(x,t):\|x\|=t,\quad
x\in{\mathbb{R}}^{d},\,0\leq t\leq 1\right\\},$

and on the solid cone ${\mathbb{V}}^{d+1}$ bounded by ${\mathbb{V}}_{0}^{d+1}$
and the hyperplane $t=1$, as well as on the planar triangle. The domain has a
singularity at the vertex. How singularity affects the Bernstein inequalities
motivates our study.

We will establish several inequalities that demonstrate the impact of the
singularity at the vertex for the conic domains. Our results also lead to new
inequalities on triangle domains that are stronger than those known in the
literature, which are somewhat unexpected and reveal new phenomena hitherto
unnoticed. To be more precise, let us first recall the Bernstein inequalities
in $L^{p}$ norm with doubling weight on the interval [20] and on the unit
sphere [7, 9]. Let $\ell$ be a positive integer. For a doubling weight $w$ on
$[0,1]$ and $1\leq p<\infty$, it is known [20, Theorem 7.3] that

(1.1) $\left\|f^{(\ell)}\right\|_{L^{p}([0,1],w)}\leq
cn^{2\ell}\|f\|_{L^{p}([0,1],w)},\qquad\deg f\leq n;$

moreover, setting $\varphi(x)=\sqrt{x(1-x)}$, then [20, Theorem 7.4]

(1.2) $\left\|\varphi^{\ell}f^{(\ell)}\right\|_{L^{p}([0,1],w)}\leq
cn^{\ell}\|f\|_{L^{p}([0,1],w)},\qquad\deg f\leq n.$

For polynomials on the unit sphere ${\mathbb{S}^{d-1}}$ and a doubling weight
$w$ on ${\mathbb{S}^{d-1}}$, the Bernstein inequalities are of the form [9,
Theorem 5.5.2]

(1.3) $\left\|D_{i,j}^{\ell}f\right\|_{L^{p}({\mathbb{S}^{d-1}},w)}\leq
cn^{\ell}\|f\|_{L^{p}({\mathbb{S}^{d-1}},w)},\qquad\deg f\leq n,$

where the derivatives $D_{i,j}$ are the angular derivatives on the unit sphere
defined by

(1.4) $D_{i,j}=x_{i}\frac{\partial}{\partial
x_{j}}-x_{j}\frac{\partial}{\partial x_{i}},\quad 1\leq i<j\leq d.$

These inequalities also hold for the uniform norm in the respective domain.

The Bernstein inequalities on the conic surface are closely related to the
above inequalities. Indeed, parametrizing ${\mathbb{V}}_{0}^{d+1}$ as
$\\{(t\xi,t):\xi\in{\mathbb{S}^{d-1}},0\leq t\leq 1\\}$, we see that the space
of polynomials on ${\mathbb{V}}_{0}^{d+1}$ contains the subspace of
polynomials in the $t$ variable as well as spherical polynomials on the unit
sphere ${\mathbb{S}^{d-1}}$. Moreover, writing
$(x,t)=(t\xi,t)\in{\mathbb{V}}_{0}^{d+1}$ with $\xi\in{\mathbb{S}^{d-1}}$, it
follows that the derivatives on the surface ${\mathbb{V}}_{0}^{d+1}$ are the
partial derivative in the $t$ variable and the angular derivatives $D_{i,j}$
in the $x$ variable. The conic surface, however, is not a smooth surface
because of its vertex at the origin of ${\mathbb{R}}^{d+1}$. Our result shows
that the inequality for the partial derivative in the $t$ variable behaves as
expected, but the Bernstein inequality for $D_{i,j}$ on
${\mathbb{V}}_{0}^{d+1}$ turns out to satisfy

$\left\|\frac{1}{\sqrt{t}^{\ell}}D_{i,j}^{\ell}f\right\|_{L^{p}({\mathbb{V}}_{0}^{d+1},{\mathsf{w}})}\leq
cn^{\ell}\|f\|_{L^{p}({\mathbb{V}}_{0}^{d+1},{\mathsf{w}})},\quad\ell=1,2,\quad\deg
f\leq n,$

where ${\mathsf{w}}$ is a doubling weight on the conic surface, but not for
$\ell>2$ in general. The factor $1/\sqrt{t}$ reflects the geometry of the
conic surface. A similar result will also be established on the solid cone
${\mathbb{V}}^{d+1}$. For $d=1$, the cone ${\mathbb{V}}^{2}$ is a triangle and
it is mapped, by an affine transform, to the standard triangle

${\mathbb{T}}^{2}=\\{(y_{1},y_{2}):y_{1}\geq 0,y_{2}\geq 0,y_{1}+y_{2}\leq
1\\}$

in ${\mathbb{R}}^{2}$, so that we obtain new Bernstein inequalities on the
triangle. In the literature, a typical Bernstein inequality on the triangle
takes the form

(1.5) $\|\phi_{1}\partial_{1}f\|_{L^{p}({\mathbb{T}}^{2},w)}\leq
c\|f\|_{L^{p}({\mathbb{T}}^{2},w)},\quad\phi_{1}(y)=\sqrt{y_{1}(1-y_{1}-y_{2})},\quad\deg
f\leq n,$

for $\partial_{1}=\frac{\partial}{\partial y_{1}}$ for example (cf. [3, 11,
12]), and it holds also for the uniform norm. The inequality (1.5) is accepted
as an appropriate generalization of the Bernstein inequality (1.2) on the
triangle, since $\phi_{1}$ takes into account the boundary of
${\mathbb{T}}^{2}$, just like $\varphi$ on $[0,1]$ in (1.1). Nevertheless, we
obtain a stronger inequality:

(1.6)
$\left\|\frac{1}{\sqrt{1-y_{2}}}\phi_{1}\partial_{1}f\right\|_{L^{p}({\mathbb{T}}^{2},w)}\leq
c\|f\|_{L^{p}({\mathbb{T}}^{2},w)},\quad\deg f\leq n.$

The inequality (1.6) is somewhat surprising. Indeed, the additional factor
$\frac{1}{\sqrt{1-y_{2}}}$ appears to be a new phenomenon that has not been
observed before. In hindsight, the inequality (1.5) takes into account the
boundary of the triangle, whereas the new ones take into account the
singularity at the corners as well.

On several regular domains, including the conic domains, there exist second-
order differential operators that have orthogonal polynomials as
eigenfunctions. For the unit sphere, it is the Laplace-Beltrami operator. On
the conic surface ${\mathbb{V}}_{0}^{d+1}$, it is given by

$\Delta_{0,{\gamma}}=t(1-t)\partial_{t}^{2}+\big{(}d-1-(d+{\gamma})t\big{)}\partial_{t}+t^{-1}\sum_{1\leq
i<j\leq d}\left(D_{i,j}^{(x)}\right)^{2},$

and it has the orthogonal polynomials with respect to $t^{-1}(1-t)^{\gamma}$
on ${\mathbb{V}}_{0}^{d+1}$ as the eigenfunctions. The Bernstein inequality
for such an operator can be established easily. For example, it was shown in
[27, Theorem 3.1.7] that

$\left\|(-\Delta_{0,{\gamma}})^{\ell}\right\|_{L^{p}({\mathbb{V}}_{0}^{d+1},{\mathsf{w}})}\leq
cn^{2\ell}\|f\|_{L^{p}({\mathbb{V}}_{0}^{d+1},{\mathsf{w}})},\quad\ell=1,2,\ldots.$

The proof relies only on the eigenvalues of $\Delta_{0,{\gamma}}$ and follows
from a general result for all such operators on localizable homogeneous
spaces. As it is, the dependence on the domain is opaque and hidden in the
proof. This inequality nevertheless motivates our study on the Bernstein
inequalities for the first-order derivatives. It can also be used to show the
sharpness of our new result for special values of $\ell$ when $p=2$.

Our analysis relies on orthogonal structures on the conic domains [25]. It is
part of an ongoing program that aims at extending the results in approximation
theory and harmonics analysis from the unit sphere to quadratic surfaces of
revolution [25, 26, 27, 28]. Our main tools are the closed-form formula for
the reproducing kernels in [25] and the highly localized kernel studied in
[27]. The approach follows the analysis on the unit sphere [9], but the conic
domain has its intrinsic complexity. For example, the distance function on
${\mathbb{V}}_{0}^{d+1}$ is not intuitively evident and satisfies a formula
much more involved than the geodesic distance on the unit sphere (see (3.6)
below). The study in [27] establishes a framework, assuming the existence of
highly localized kernels, for approximation and localized frames on a
localizable homogenous space and, along the way, provides a toolbox for
carrying out analysis on localizable homogeneous spaces. For the conic
domains, the highly localized kernels are established by delicate estimates.
For our proof of the Bernstein inequalities, we shall establish sharp
estimates for the derivatives of these kernels.

The paper is organized as follows. The main results are stated and discussed
in Section 2. The proofs of the main results are in Section 3 for the conic
surface and Section 4 for the solid cone.

Throughout the paper, we let $c,c_{1},c_{2},\ldots$ as positive constants that
depend only on fixed parameters and their values could change from line to
line. We write $A\sim B$ if $c_{1}B\leq A\leq c_{2}B$.

##  2\. Main results

We state and discuss our main results in this section. The Bernstein
inequalities on the conic surface are stated and discussed in the first
subsection and those on the solid cone are stated in the second subsection.
The inequalities on the triangle follow from those on the solid cone and will
be discussed in the third subsection.

###  2.1. Main results on the conic surface

Parametrizing ${\mathbb{V}}_{0}^{d+1}$ as
$\\{(t\xi,t):\xi\in{\mathbb{S}^{d-1}},0\leq t\leq 1\\}$, we see that the first
order derivative on the conic surfaces are
$\partial_{t}=\frac{\partial}{\partial t}$ in the $t$ variable and the angular
derivatives $D_{i,j}$, $1\leq i,j\leq d$, in the $x$ variables, which we
denote by $D_{i,j}^{(x)}$. The latter are called angular derivatives since if
${\theta}_{i,j}$ is the angle of polar coordinates in the
$(x_{i},x_{j})$-plane, defined by
$(x_{i},x_{j})=r_{i,j}(\cos{\theta}_{i,j},\sin{\theta}_{i,j})$, where
$r_{i,j}\geq 0$ and $0\leq{\theta}_{i,j}\leq 2\pi$, then a quick computation
shows that

$D_{i,j}=x_{i}\frac{\partial}{\partial x_{j}}-x_{j}\frac{\partial}{\partial
x_{i}}=\frac{\partial}{\partial{\theta}_{i,j}},\qquad 1\leq i,j\leq d.$

In particular, they are independent of the scale of $x$. We state our
Bernstein inequalities for the derivatives $\partial_{t}$ and $D_{i,j}^{(x)}$
on the conic surface.

Let ${\mathsf{w}}$ be a doubling weight on ${\mathbb{V}}_{0}^{d+1}$; see
Subsection 3.1 for the definition. We denote by $\|\cdot\|_{p,{\mathsf{w}}}$
the weighted $L^{p}$ norm on ${\mathbb{V}}_{0}^{d+1}$

$\|f\|_{p,{\mathsf{w}}}=\left(\int_{{\mathbb{V}}_{0}^{d+1}}|f(x,t)|^{p}{\mathsf{w}}(x,t)\mathrm{d}{\mathsf{m}}(x,t)\right)^{1/p},\qquad
1\leq p<\infty,$

where ${\mathsf{m}}$ denotes the Lebesgue measure on the conic surface. For
$p=\infty$, we denote by $\|\cdot\|_{\infty}$ the uniform norm on
${\mathbb{V}}_{0}^{d+1}$. For $n=0,1,2,\ldots$, let
$\Pi_{n}({\mathbb{V}}_{0}^{d+1})$ be the space of polynomials of degree at
most $n$ on ${\mathbb{V}}_{0}^{d+1}$.

######  Theorem 2.1.

Let ${\mathsf{w}}$ be a doubling weight on ${\mathbb{V}}_{0}^{d+1}$ and let
$f\in\Pi_{n}({\mathbb{V}}_{0}^{d+1})$. For $\ell\in{\mathbb{N}}$ and $1\leq
p<\infty$,

(2.1) $\|\partial_{t}^{\ell}f\|_{p,{\mathsf{w}}}\leq
c_{p}n^{2\ell}\|f\|_{p,{\mathsf{w}}},$

and, with $\varphi(t)=\sqrt{t(1-t)}$,

(2.2) $\|\phi^{\ell}\partial_{t}^{\ell}f\|_{p,{\mathsf{w}}}\leq
c_{p}n^{\ell}\|f\|_{p,{\mathsf{w}}}.$

Moreover, for $D_{i,j}=D_{i,j}^{(x)}$, $1\leq i<j\leq d$, and
$\ell\in{\mathbb{N}}$,

(2.3) $\left\|D_{i,j}^{\ell}f\right\|_{p,{\mathsf{w}}}\leq
c_{p}n^{\ell}\|f\|_{p,{\mathsf{w}}},$

and, furthermore, for $\ell=1,2$,

(2.4)
$\left\|\frac{1}{\sqrt{t}^{\ell}}D_{i,j}^{\ell}f\right\|_{p,{\mathsf{w}}}\leq
c_{p}n^{\ell}\|f\|_{p,{\mathsf{w}}},$

but not for $\ell\geq 3$ in general. Finally, these inequalities also hold
when the norm is the uniform norm $\|\cdot\|_{\infty}$ on
${\mathbb{V}}_{0}^{d+1}$.

The inequality (2.4) is stronger than (2.3) when $\ell=1,2$, since $0\leq
t\leq 1$. Moreover, for $\ell>2$, the inequality (2.3) follows from iteration.
It is worth mentioning that the inequality (2.4) for $\ell=1$ cannot be
iterated to obtain the inequality for $\ell=2$, since the presence of
$1/\sqrt{t}$ means that the function involved in the iteration is no longer a
polynomial.

We now discuss these inequalities in light of the spectral operator
$\Delta_{0,{\gamma}}$ on the conic surface, which is defined for ${\gamma}>-1$
by

(2.5)
$\Delta_{0,{\gamma}}:=t(1-t)\partial_{t}^{2}+\big{(}d-1-(d+{\gamma})t\big{)}\partial_{t}+t^{-1}\Delta_{0}^{(\xi)},$

where $\xi=x/t$ and $\Delta_{0}^{(\xi)}$ denote the Laplace-Beltrami operator
$\Delta_{0}$ on the unit sphere ${\mathbb{S}^{d-1}}$ in $\xi$ variables. For
${\gamma}>-1$, the operator $-\Delta_{0,{\gamma}}$ is non-negative and has
orthogonal polynomials with respect to the weight function
$t^{-1}(1-t)^{\gamma}$ on ${\mathbb{V}}_{0}^{d+1}$ as eigenfunctions; see
Theorem 3.1 below. The latter property allows us to define
$(-\Delta_{0,r})^{{\alpha}}$ for ${\alpha}\in{\mathbb{R}}$. It is shown in
[27, Theorem 3.1.7] that the following Bernstein inequality holds.

######  Theorem 2.2.

Let ${\mathsf{w}}$ be a doubling weight on ${\mathbb{V}}_{0}^{d+1}$. Let
${\gamma}>-1$. Then, for $r>0$ and $1\leq p\leq\infty$,

(2.6) $\|(-\Delta_{0,{\gamma}})^{\frac{r}{2}}f\|_{p,{\mathsf{w}}}\leq
cn^{r}\|f\|_{p,{\mathsf{w}}},\qquad\forall
f\in\Pi_{n}({\mathbb{V}}_{0}^{d+1}).$

The proof of this theorem, however, does not use the explicit formula of the
operator $\Delta_{0,{\gamma}}$ and relies only on the fact that the operator
has orthogonal polynomials as eigenfunctions. The operator
$\Delta_{0,{\gamma}}$ in (2.5) contains a factor $t^{-1}\Delta_{0}^{(\xi)}$.
It is known that the Laplace-Beltrami operator $\Delta_{0}$ can be decomposed
in terms of angular derivatives (cf. [9, Theorem 1.8.2]),

(2.7) $\Delta_{0}=\sum_{1\leq i<j\leq d}D_{i,j}^{2}.$

Hence, the factor $t^{-1}\Delta_{0}^{(\xi)}$ in the operator
$\Delta_{0,{\gamma}}$ can be decomposed as a sum of
$\frac{1}{\sqrt{t}^{2}}(D_{i,j}^{(\xi)})^{2}$. Thus, the inequality (2.6) with
$r=2$ follows from inequalities in Theorem 2.1. Notice, however, that (2.4)
holds only for $\ell=1$ and $2$. This makes the higher order Bernstein
inequality for the spectral operator $\Delta_{0,{\gamma}}$ that much more
special.

The operator $-\Delta_{0,{\gamma}}$ on the conic surface
${\mathbb{V}}_{0}^{d+1}$ is self-adjoint, which is evident from the following
identity that relies on the first-order derivatives on the conic surface.

######  Proposition 2.3.

For ${\gamma}>-1$, let ${\mathsf{w}}_{-1,{\gamma}}(t)=t^{-1}(1-t)^{\gamma}$,
$0\leq t\leq 1$. Then

$\displaystyle-\int_{{\mathbb{V}}_{0}^{d+1}}\Delta_{0,{\gamma}}f(x,t)\cdot
g(x,t){\mathsf{w}}_{-1,{\gamma}}(x,t)\mathrm{d}{\mathsf{m}}(x,t)=\int_{{\mathbb{V}}^{d+1}_{0}}t(1-t)\frac{\mathrm{d}f}{\mathrm{d}t}\frac{\mathrm{d}g}{\mathrm{d}t}{\mathsf{w}}_{-1,{\gamma}}(t)\mathrm{d}{\mathsf{m}}(x,t)$
$\displaystyle+\sum_{1\leq i<j\leq
d}\int_{{\mathbb{V}}_{0}^{d+1}}t^{-2}D_{i,j}^{(x)}f(x,t)D_{i,j}^{(x)}g(x,t){\mathsf{w}}_{-1,{\gamma}}(t)\mathrm{d}{\mathsf{m}}(x,t)$
,

where
$\frac{\mathrm{d}f}{\mathrm{d}t}=\frac{\mathrm{d}}{\mathrm{d}t}f(t\xi,t)$. In
particular, $-\Delta_{0,{\gamma}}$ is self-adjoint in
$L^{2}({\mathbb{V}}^{d+1},{\mathsf{w}}_{-1,{\gamma}})$.

The proof of these results will be given in Subsection 3.5. It is used to
deduce the following corollary.

######  Corollary 2.4.

Let $d\geq 2$ and ${\gamma}>-1$. Then, for
${\mathsf{w}}={\mathsf{w}}_{-1,{\gamma}}$ and any polynomial $f$,

$\left\|\varphi\partial_{t}f\right\|_{2,{\mathsf{w}}}^{2}+\sum_{1\leq i<j\leq
d}\left\|\frac{1}{\sqrt{t}}D_{i,j}^{(x)}f\right\|_{2,{\mathsf{w}}}^{2}\leq\left\|f\right\|_{2,{\mathsf{w}}}\left\|\Delta_{0,{\gamma}}f\right\|_{2,{\mathsf{w}}}.$

In particular, for $f\in\Pi_{n}({\mathbb{V}}_{0}^{d+1})$, the inequality (2.6)
with $r=2$ and $p=2$ implies the inequalities (2.2) and (2.4) with $\ell=1$
and $p=2$ when ${\mathsf{w}}={\mathsf{w}}_{-1,{\gamma}}$.

In the other direction, the explicit formula of (2.5) and (2.7) shows
immediately that $\|-\Delta_{0,{\gamma}}f\|_{p,{\mathsf{w}}}$ is bounded by
the sum of $\|\varphi^{2}\partial_{t}^{2}f\|_{p,{\mathsf{w}}}$,
$\|\partial_{t}f\|_{p,{\mathsf{w}}}$ and
$\|(\frac{1}{\sqrt{t}}D_{i,j})^{2}f\|_{p,{\mathsf{w}}}$. In particular, the
inequalities in the Theorem 2.1 imply (2.6) with $r=2$. By the spectral
property, the inequality (2.6) is sharp. The corollary and the discussion
above provides some assurance that the inequalities (2.2) and (2.4) are sharp
when $p=2$ and ${\mathsf{w}}={\mathsf{w}}_{-1,{\gamma}}$.

###  2.2. Main results on the cone

Here the domain is the solid cone in ${\mathbb{R}}^{d+1}$ for $d\geq 1$,

${\mathbb{V}}^{d+1}=\left\\{(x,t):\|x\|\leq t,\,0\leq t\leq
1,\,x\in{\mathbb{R}}^{d}\right\\}.$

Writing ${\mathbb{V}}^{d+1}$ as $\\{(t\xi,t):\xi\in{\mathbb{S}^{d-1}},0\leq
t\leq 1\\}$, we see that the first order derivative on the conic surfaces are
$\partial_{t}=\frac{\partial}{\partial t}$ in the $t$ variable and the angular
derivatives $D_{i,j}$, $1\leq i,j\leq d$, in the $x$ variables, which we
denote by $D_{i,j}^{(x)}$, as well as one more partial derivative, denoted by
$D_{x_{j}}$ and defined by

(2.8) $D_{x_{j}}=\sqrt{t^{2}-\|x\|^{2}}\frac{\partial}{\partial x_{j}},\qquad
1\leq j\leq d,\quad(x,t)\in{\mathbb{V}}^{d+1}.$

Let $W$ be a doubling weight on ${\mathbb{V}}^{d+1}$; see 4.1 for the
definition. We denote by $\|\cdot\|_{p,W}$ the weighted $L^{p}$ norm on
${\mathbb{V}}^{d+1}$

$\|f\|_{p,W}=\left(\int_{{\mathbb{V}}^{d+1}}|f(x,t)|^{p}W(x,t)\mathrm{d}x\mathrm{d}t\right)^{1/p},\qquad
1\leq p<\infty.$

######  Theorem 2.5.

Let $W$ be a doubling weight on ${\mathbb{V}}^{d+1}$ and let $f\in\Pi_{n}$.
For $\ell\in{\mathbb{N}}$ and $1\leq p<\infty$,

(2.9) $\left\|\partial_{t}^{\ell}f\right\|_{p,W}\leq
c_{p}n^{2\ell}\|f\|_{p,W}\quad\hbox{and}\quad\left\|\varphi^{\ell}\partial_{t}^{\ell}f\right\|_{p,W}\leq
c_{p}n^{\ell}\|f\|_{p,W},$

where $\varphi(t)=\sqrt{t(1-t)}$; moreover, for $1\leq i,j\leq d$,

(2.10) $\left\|D_{i,j}^{\ell}f\right\|_{p,W}\leq c_{p}n^{\ell}\|f\|_{p,W},$

and, for $\ell=1,2$,

(2.11) $\left\|\frac{1}{\sqrt{t}^{\ell}}D_{i,j}^{\ell}f\right\|_{p,W}\leq
c_{p}n^{\ell}\|f\|_{p,W},$

which does not hold for $\ell=3,4,\ldots$ in general. Furthermore, for
$\ell\in{\mathbb{N}}$ and $1\leq j\leq d$,

(2.12) $\left\|\partial_{x_{j}}^{\ell}f\right\|_{p,W}\leq
c_{p}n^{2\ell}\|f\|_{p,W}\quad\hbox{and}\quad\left\|\Phi^{\ell}\partial_{x_{j}}^{\ell}f\right\|_{p,W}\leq
c_{p}n^{\ell}\|f\|_{p,W},$

where $\Phi(x,t)=\sqrt{t^{2}-\|x\|^{2}}$, and, for $\ell=1,2$,

(2.13)
$\left\|\frac{1}{\sqrt{t}^{\ell}}\Phi^{\ell}\partial_{x_{j}}^{\ell}f\right\|_{p,W}\leq
c_{p}n^{\ell}\|f\|_{p,W}.$

Finally, these inequalities also hold when the norm is the uniform norm
$\|\cdot\|_{\infty}$ on ${\mathbb{V}}^{d+1}$.

In contrast to the inequality (2.12), we do not know if (2.13) holds for
$\ell\geq 3$. As it will be shown in Section 4, the two cases are closely
related, but the example that works for (2.12) does not work for (2.13).

Like the case of conic surface, there is also a spectral operator on the cone
${\mathbb{V}}^{d+1}$. For $\mu>-\frac{1}{2}$ and ${\gamma}>-1$, define the
second order differential operator

(2.14) $\displaystyle{\mathfrak{D}}_{\mu,{\gamma}}:=$
$\displaystyle\,t(1-t)\partial_{t}^{2}+2(1-t){\langle}x,\nabla_{x}{\rangle}\partial_{t}+t\Delta_{x}-\langle
x,\nabla_{x}\rangle^{2}$
$\displaystyle+(2\mu+d)\partial_{t}-(2\mu+{\gamma}+d+1)({\langle}x,\nabla_{x}{\rangle}+t\partial_{t})+\langle
x,\nabla_{x}\rangle.$

It is proved in [27] that orthogonal polynomials with respect to the weight
function

$W_{\mu,{\gamma}}(x,t)=(t^{2}-\|x\|^{2})^{\mu-\frac{1}{2}}(1-t)^{\gamma},\qquad\mu>-\tfrac{1}{2},{\gamma}>-1$

on ${\mathbb{V}}^{d+1}$ are the eigenfunctions of the operator
${\mathfrak{D}}_{\mu,{\gamma}}$ (see, Theorem 4.1 below). In particular, the
operator satisfies the following Bernstein inequality [27, Theorem 3.1.7].

######  Theorem 2.6.

Let $W$ be a doubling weight on ${\mathbb{V}}^{d+1}$. Let ${\gamma}>-1$ and
$\mu>-\frac{1}{2}$. For $r>0$ and $1\leq p\leq\infty$,

(2.15) $\|(-{\mathfrak{D}}_{\mu,{\gamma}})^{\frac{r}{2}}f\|_{p,W}\leq
cn^{r}\|f\|_{p,W},\qquad\forall f\in\Pi_{n}.$

The operator ${\mathfrak{D}}_{\mu,{\gamma}}$ is self-adjoint, as can be seen
in the following theorem.

######  Theorem 2.7.

For $\mu>-\frac{1}{2}$ and ${\gamma}>-1$,

(2.16)
$\displaystyle\int_{{\mathbb{V}}^{d+1}}-{\mathfrak{D}}_{\mu,{\gamma}}f(x,t)$
$\displaystyle\cdot
g(x,t)W_{\mu,{\gamma}}(x,t)\mathrm{d}x\mathrm{d}t=\int_{{\mathbb{V}}^{d+1}}t\frac{\mathrm{d}f}{\mathrm{d}t}\frac{\mathrm{d}g}{\mathrm{d}t}W_{\mu,{\gamma}+1}(x,t)\mathrm{d}x\mathrm{d}t$
$\displaystyle+\sum_{i=1}^{d}\int_{{\mathbb{V}}^{d+1}}D_{x_{i}}f(x,t)\cdot
D_{x_{i}}g(x,t)t^{-1}W_{\mu,{\gamma}}(x)\mathrm{d}x\mathrm{d}t$
$\displaystyle+\int_{{\mathbb{V}}^{d+1}}\sum_{i<j}D_{i,j}^{(x)}f(x,t)\cdot
D_{i,j}^{(x)}g(x,t)t^{-1}W_{\mu,{\gamma}}(x)\mathrm{d}x\,\mathrm{d}t,$

where
$\frac{\mathrm{d}}{\mathrm{d}t}f=\frac{\mathrm{d}}{\mathrm{d}t}[f(ty,t)]$ for
$y\in{\mathbb{B}}^{d}$.

The identity (2.16) is proved in Subsection 4.4 and it implies immediately the
following corollary.

######  Corollary 2.8.

Let $d\geq 2$, $\mu>-\frac{1}{2}$ and ${\gamma}>-1$. Let $W=W_{\mu,{\gamma}}$.
Then for any polynomial $f$

$\left\|\varphi\partial_{t}f\right\|_{2,W}^{2}+\sum_{j=1}^{d}\left\|\frac{1}{\sqrt{t}}D_{x_{j}}f\right\|_{2,W}^{2}+\sum_{1\leq
i<j\leq
d}\left\|\frac{1}{\sqrt{t}}D_{i,j}^{(x)}f\right\|_{2,W}^{2}\leq\left\|f\right\|_{2,W}\left\|{\mathfrak{D}}_{\mu,{\gamma}}f\right\|_{2,W}.$

In particular, the inequality (2.15) for ${\mathfrak{D}}_{\mu,{\gamma}}$ with
$r=2$ implies the inequalities (2.12) for $\varphi\partial_{t}$ and (2.4) for
$D_{i,j}^{(x)}$ and $D_{x_{j}}$ with $\ell=1$ and $p=2$ for
$W=W_{\mu,{\gamma}}$.

The corollary follows as in Corollary 2.4. The Bernstein inequalities for the
first order derivatives in Theorem 2.5 imply (2.15) when $r=2$. Together, they
provide some assurance that the inequalities in Theorem 2.5 are sharp.

###  2.3. Bernstein inequality on the triangle

For $d=1$, the cone becomes the triangle ${\mathbb{V}}^{2}=\\{(x,t):0\leq
t\leq 1,\,|x|\leq t\\}$ of ${\mathbb{R}}^{2}$. Making an affine change of
variable $(x,t)=(y_{1}-y_{2},y_{1}+y_{2})$, the triangle ${\mathbb{V}}^{2}$
becomes the standard triangle domain

${\mathbb{T}}^{2}=\\{(y_{1},y_{2}):y_{1}\geq 0,\,y_{2}\geq 0,\,y_{1}+y_{2}\leq
1\\}.$

The triangle ${\mathbb{T}}^{2}$ is symmetric under permutation of
$\\{y_{1},y_{2},1-y_{1}-y_{2}\\}$ and it is customary to consider the
derivatives (cf. [3, 13])

$\partial_{1}=\partial_{y_{1}},\quad\partial_{2}=\partial_{y_{2}},\quad\partial_{3}=\partial_{y_{2}}-\partial_{y_{1}}.$

We further define, for $y=(y_{1},y_{2})\in{\mathbb{T}}^{2}$,

$\phi_{1}(y)=\sqrt{y_{1}(1-y_{1}-y_{2})},\quad\phi_{2}(y)=\sqrt{y_{2}(1-y_{1}-y_{2})},\quad\phi_{3}(y)=\sqrt{y_{1}y_{2}}.$

Let $\|\cdot\|_{p,w}$ be the $L^{p}$ norm on the triangle ${\mathbb{T}}^{2}$
for $1\leq p<\infty$ and the uniform norm on ${\mathbb{T}}^{2}$ if $p=\infty$.

######  Theorem 2.9.

Let $w$ be a doubling weigh on ${\mathbb{T}}^{2}$. Let $f\in\Pi_{n}^{2}$.
Then, for $1\leq p<\infty$, $\ell\in{\mathbb{N}}$ and $i=1,2,3$,

(2.17) $\left\|\partial_{i}^{\ell}f\right\|_{p,w}\leq
cn^{2\ell}\|f\|_{p,w}\quad\hbox{and}\quad\left\|\phi_{i}^{\ell}\partial_{i}^{\ell}f\right\|_{p,w}\leq
cn^{\ell}\|f\|_{p,w}.$

Furthermore, for $\ell=1,2$,

(2.18)
$\displaystyle\left\|\frac{1}{(1-y_{2})^{\ell/2}}\phi_{1}^{\ell}\partial_{1}^{\ell}f\right\|_{p,w}\leq
cn^{\ell}\|f\|_{p,w},$ (2.19)
$\displaystyle\left\|\frac{1}{(1-y_{1})^{\ell/2}}\phi_{2}^{\ell}\partial_{2}^{\ell}f\right\|_{p,w}\leq
cn^{\ell}\|f\|_{p,w},$ (2.20)
$\displaystyle\left\|\frac{1}{(y_{1}+y_{2})^{\ell/2}}\phi_{3}^{\ell}\partial_{3}^{\ell}f\right\|_{p,w}\leq
cn^{\ell}\|f\|_{p,w}.$

Moreover, these inequalities hold when the norm is replaced by the uniform
norm.

###### Proof.

Under the change of variables ${\mathbb{V}}^{2}\mapsto{\mathbb{T}}^{2}$:
$(x,t)\mapsto(y_{1}-y_{2},y_{1}+y_{2})$, we see that

$\partial_{x}=\frac{1}{2}(\partial_{y_{2}}-\partial_{y_{1}})=\frac{1}{2}\partial_{3}\quad\hbox{and}\quad\sqrt{t^{2}-|x|^{2}}=2\sqrt{y_{1}y_{2}}=2\phi_{3}(y).$

Hence, for $i=3$, the two inequalities in (2.17) follow from (2.12), where
$d=1$, and the inequality (2.20) follows from (2.13) in Theorem 2.5. The
inequalities for $\partial_{1}$ and $\partial_{2}$ follow from those for
$\partial_{3}$ by a change of variables that amounts to permute the variables
$(y_{1},y_{2},1-y_{1}-y_{2})$. ∎

The inequality (2.17) is known when $w$ is a constant weight (cf. [3, 12, 22])
or the Jacobi weight. It is accepted as a natural generalization of the
Bernstein inequality (1.2) of one variable, since the left-hand side of the
inequality in the uniform norm gives a pointwise inequality that reduces to
(1.2) when restricted to the boundary of the triangle. The latter property,
however, is satisfied by (2.18)–(2.20) as well. Moreover, the factors in the
left-hand side remains bounded; for example, $1-y_{1}-y_{2}\leq 1-y_{2}$ in
(2.18). The inequalities (2.18)–(2.20) are surprising and they are new as far
as we are aware. It is also suggestive and may be worthwhile to ask if the
same phenomenon appears on other domains, such as polytopes [22, 23].

##  3\. Bernstein inequalities on conic surface

We work on the conic surface in this section. In the first subsection, we
recall what is needed for our analysis on the domain, based on orthogonal
polynomials and highly localized kernels. Several auxiliary inequalities are
recalled or proved in the second subsection, to be used in the rest of the
section. The proofs of the main results are based on the estimates of the
kernel functions, which are carried out in the third subsection for the
derivatives in the $t$-variable and the fourth subsection for the angular
derivatives. Finally, the proof of Theorem 2.7 and its corollary is given in
the fifth subsection.

###  3.1. Orthogonal polynomials and localized kernels

Let $\Pi({\mathbb{V}}_{0}^{d+1})$ denote the space of polynomials restricted
on the conic surface ${\mathbb{V}}_{0}^{d+1}$ and, for $n=0,1,2,\ldots$, let
$\Pi_{n}({\mathbb{V}}_{0}^{d+1})$ be the subspace of polynomials in
$\Pi({\mathbb{V}}_{0}^{d+1})$ of total degree at most $n$. Since
${\mathbb{V}}_{0}^{d+1}$ is a quadratic surface, it is known that

$\dim\Pi_{n}({\mathbb{V}}_{0}^{d+1})=\binom{n+d}{n}+\binom{n+d-1}{n-1}.$

For ${\beta}>-d$ and ${\gamma}>-1$, we define the weight function
${\mathsf{w}}_{{\beta},{\gamma}}$ by

${\mathsf{w}}_{{\beta},{\gamma}}(t)=t^{\beta}(1-t)^{\gamma},\qquad 0\leq t\leq
1.$

Orthogonal polynomials with respect to ${\mathsf{w}}_{{\beta},{\gamma}}$ on
${\mathbb{V}}_{0}^{d+1}$ are studied in [25]. Let

${\langle}f,g{\rangle}_{{\mathsf{w}}_{{\beta},{\gamma}}}={\mathsf{b}}_{{\beta},{\gamma}}\int_{{\mathbb{V}}_{0}^{d+1}}f(x,t)g(x,t){\mathsf{w}}_{{\beta},{\gamma}}\mathrm{d}{\mathsf{m}}(x,t),$

where $\mathrm{d}{\mathsf{m}}$ denotes the Lebesgue measure on the conic
surface, which is a well defined inner product on
$\Pi({\mathbb{V}}_{0}^{d+1})$. Let
${\mathcal{V}}_{n}({\mathbb{V}}_{0}^{d+1},{\mathsf{w}}_{{\beta},{\gamma}})$ be
the space of orthogonal polynomials of degree $n$. Then
$\dim{\mathcal{V}}_{0}({\mathbb{V}}_{0}^{d+1},{\mathsf{w}}_{{\beta},{\gamma}})=1$
and

$\dim{\mathcal{V}}_{n}({\mathbb{V}}_{0}^{d+1},{\mathsf{w}}_{{\beta},{\gamma}})=\binom{n+d-1}{n}+\binom{n+d-2}{n-1},\quad
n=1,2,3,\ldots.$

Let ${\mathcal{H}}_{m}({\mathbb{S}^{d-1}})$ be the space of spherical
harmonics of degree $m$ in $d$ variables. Let
$\\{Y_{\ell}^{m}:1\leq\ell\leq\dim{\mathcal{H}}_{m}({\mathbb{S}^{d-1}})\\}$
denote an orthonormal basis of ${\mathcal{H}}_{m}({\mathbb{S}^{d-1}})$. Then
the polynomials

${\mathsf{S}}_{m,\ell}^{n}(x,t)=P_{n-m}^{(2m+{\beta}+d-1,{\gamma})}(1-2t)Y_{\ell}^{m}(x),\quad
0\leq m\leq n,\,\,1\leq\ell\leq\dim{\mathcal{H}}_{m}({\mathbb{S}^{d-1}}),$

consist of an orthogonal basis of
${\mathcal{V}}_{n}({\mathbb{V}}_{0}^{d+1},{\mathsf{w}}_{{\beta},{\gamma}})$.
Let $\Delta_{0,{\gamma}}$ be the differential operator defined in (2.5). It
has orthogonal polynomials as eigenfunctions [25, Theorem 7.2].

######  Theorem 3.1.

Let $d\geq 2$ and ${\gamma}>-1$. The orthogonal polynomials in
${\mathcal{V}}_{n}({\mathbb{V}}_{0}^{d+1},{\mathsf{w}}_{-1,{\gamma}})$ are
eigenfunctions of $\Delta_{0,{\gamma}}$; more precisely,

(3.1) $\Delta_{0,{\gamma}}u=-n(n+{\gamma}+d-1)u,\qquad\forall
u\in{\mathcal{V}}_{n}({\mathbb{V}}_{0}^{d+1},{\mathsf{w}}_{-1,{\gamma}}).$

The reproducing kernel of the space
${\mathcal{V}}_{n}({\mathbb{V}}_{0}^{d+1},{\mathsf{w}}_{{\beta},{\gamma}})$ is
denoted by ${\mathsf{P}}_{n}({\mathsf{w}}_{{\beta},{\gamma}};\cdot,\cdot)$,
which can be written as

${\mathsf{P}}_{n}\big{(}{\mathsf{w}}_{{\beta},{\gamma}};(x,t),(y,s)\big{)}=\sum_{m=0}^{n}\sum_{k=1}^{\dim{\mathcal{H}}_{m}^{d}}\frac{{\mathsf{S}}_{m,\ell}^{n}(x,t){\mathsf{S}}_{m,\ell}^{n}(y,s)}{{\langle}{\mathsf{S}}_{m,\ell}^{n},{\mathsf{S}}_{m^{\prime},\ell^{\prime}}^{n^{\prime}}{\rangle}_{{\mathsf{w}}_{{\beta},{\gamma}}}}.$

Let
$\operatorname{proj}_{n}({\mathsf{w}}_{{\beta},{\gamma}}):L^{2}({\mathbb{V}}_{0}^{d+1},{\mathsf{w}}_{{\beta},{\gamma}})\to{\mathcal{V}}_{n}({\mathbb{V}}_{0}^{d+1},{\mathsf{w}}_{{\beta},{\gamma}})$
be the orthogonal projection operator. It is an integral operator with
${\mathsf{P}}_{n}\big{(}{\mathsf{w}}_{{\beta},{\gamma}};\cdot,\cdot\big{)}$ as
its kernel,

$\operatorname{proj}_{n}({\mathsf{w}}_{{\beta},{\gamma}};f)=\int_{{\mathbb{V}}_{0}^{d+1}}f(y,s){\mathsf{P}}_{n}\big{(}{\mathsf{w}}_{{\beta},{\gamma}};\,\cdot,(y,s)\big{)}{\mathsf{w}}_{{\beta},{\gamma}}(s)\mathrm{d}{\mathsf{m}}(y,s).$

The kernel
${\mathsf{P}}_{n}\big{(}{\mathsf{w}}_{{\beta},{\gamma}};\,\cdot,\cdot\big{)}$
satisfies a closed form formula, called the addition formula since it is akin
to the classical addition formula for the spherical harmonics. The addition
formula is given in terms of the Jacobi polynomial
$P_{n}^{({\alpha},{\beta})}$, the orthogonal polynomial with respect to
$w_{{\alpha},{\beta}}(t)=(1-t)^{\alpha}(1+t)^{\beta}$ on $[-1,1]$. Let

$Z_{n}^{({\alpha},{\beta})}(t)=\frac{P_{n}^{(a,{\beta})}(t)P_{n}^{({\alpha},{\beta})}(1)}{h_{n}^{({\alpha},{\beta})}},\quad{\alpha},{\beta}>-1,$

where $h_{n}^{({\alpha},{\beta})}$ is the $L^{2}$ norm of
$P_{n}^{({\alpha},{\beta})}$ in $L^{2}([-1,1],w_{{\alpha},{\beta}})$. The
addition formula is of the simplest form when ${\gamma}=-1$.

######  Theorem 3.2.

Let $d\geq 2$ and ${\gamma}\geq-\frac{1}{2}$. Then, for
$(x,t),(y,s)\in{\mathbb{V}}_{0}^{d+1}$,

(3.2)
$\displaystyle{\mathsf{P}}_{n}\big{(}{\mathsf{w}}_{-1,{\gamma}};(x,t),(y,s)\big{)}=b_{{\gamma},d}\int_{[-1,1]^{2}}$
$\displaystyle
Z_{n}^{({\gamma}+d-\frac{3}{2},-\frac{1}{2})}\big{(}2\zeta(x,t,y,s;v)^{2}-1\big{)}$
$\displaystyle\times(1-v_{1}^{2})^{\frac{d-4}{2}}(1-v_{2}^{2})^{{\gamma}-\frac{1}{2}}\mathrm{d}v,$

where $b_{{\gamma},d}$ is a constant so that
${\mathsf{P}}_{0}\big{(}{\mathsf{w}}_{-1,{\gamma}};(x,t),(y,s)\big{)}=1$ and

(3.3)
$\zeta(x,t,y,s;v)=v_{1}\sqrt{\tfrac{st+{\langle}x,y{\rangle}}{2}}+v_{2}\sqrt{1-t}\sqrt{1-s};$

moreover, the identity holds under limit when ${\gamma}=-\frac{1}{2}$ and/or
$d=2$.

Our main tool for establishing polynomial inequalities is the highly localized
kernel defined via a smooth cur-off function $\widehat{a}\in
C^{\infty}({\mathbb{R}})$, which is a non-negative function and satisfies
$\mathrm{supp}\,\widehat{a}\subset[0,2]$ and $\widehat{a}(t)=1$, $t\in[0,1]$.
The kernel is defined by

(3.4)
${\mathsf{L}}_{n}({\mathsf{w}}_{-1,{\gamma}};(x,t),(y,s))=\sum_{k=0}^{\infty}\widehat{a}\left(\frac{k}{n}\right){\mathsf{P}}_{n}({\mathsf{w}};(x,t),(y,s)).$

Since $\widehat{a}$ is supported on $[0,2]$, this is a kernel of polynomials
of degree at most $2n$ in either the $x$ or the $y$ variable. Using the closed
form of the reproducing kernel, we can write
${\mathsf{L}}({\mathsf{w}}_{{\beta},{\gamma}})$ in terms of the kernel of the
Jacobi polynomials defined by

$L_{n}^{({\lambda},-\frac{1}{2})}(t)=\sum_{k=0}^{\infty}\widehat{a}\left(\frac{k}{n}\right)Z_{n}^{({\lambda},-\frac{1}{2})}(t).$

Indeed, it follows immediately from (3.2) that

(3.5)
$\displaystyle{\mathsf{L}}_{n}({\mathsf{w}}_{-1,{\gamma}};(x,t),(y,s))=b_{{\gamma},d}\int_{[-1,1]^{2}}$
$\displaystyle
L_{n}^{({\gamma}+d-\frac{3}{2},-\frac{1}{2})}\big{(}2\zeta(x,t,y,s;v)^{2}-1\big{)}$
$\displaystyle\times(1-v_{1}^{2})^{\frac{d-2}{2}-1}(1-v_{2}^{2})^{{\gamma}-\frac{1}{2}}\mathrm{d}v.$

To show that this kernel is highly localized, we need the distance on the
conic surface. For $(x,t)$ and $(y,s)$ on ${\mathbb{V}}_{0}^{d+1}$, the
distance function ${\mathsf{d}}_{{\mathbb{V}}_{0}}$ on
${\mathbb{V}}_{0}^{d+1}$ is defined by

(3.6)
${\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s)):=\arccos\left(\sqrt{\frac{{\langle}x,y{\rangle}+ts}{2}}+\sqrt{1-t}\sqrt{1-s}\right).$

For $r>0$ and $(x,t)$ on ${\mathbb{V}}_{0}^{d+1}$, we let
${\mathsf{c}}((x,t),r)$ be the ball centered at $(x,t)$ with radius $r$ in
terms of this distance function; that is,

${\mathsf{c}}((x,t),r):=\left\\{(y,s)\in{\mathbb{V}}_{0}^{d+1}:{\mathsf{d}}_{{\mathbb{V}}_{0}}\big{(}(x,t),(y,s)\big{)}\leq
r\right\\}.$

Let $E$ be a subset in ${\mathbb{V}}_{0}^{d+1}$ and ${\mathsf{m}}$ denotes the
Lebesgue measure in ${\mathbb{V}}_{0}^{d+1}$. We define

${\mathsf{w}}(E)=\int_{E}{\mathsf{w}}(x,t)\mathrm{d}{\mathsf{m}}(x,t).$

A weight function ${\mathsf{w}}$ is a doubling weight if there is a constant
$L>0$ such that

${\mathsf{w}}\big{(}{\mathsf{c}}((x,t),2r)\big{)}\leq
L\,{\mathsf{w}}\big{(}{\mathsf{c}}((x,t),r)\big{)},\quad r>0.$

The least constant $L$ is called a doubling constant and the doubling index
${\alpha}({\mathsf{w}})$ is the least index for which
$\sup_{{\mathsf{c}}((x,t),r}{\mathsf{w}}({\mathsf{c}}((x,t),2^{m}r)))/{\mathsf{w}}({\mathsf{c}}((x,t),r))\leq
c_{L({\mathsf{w}})}2^{m{\alpha}({\mathsf{w}})}$, $m=1,2,\ldots$. As an
example, the weight ${\mathsf{w}}_{{\beta},{\gamma}}$ is a doubling weight on
${\mathbb{V}}_{0}^{d+1}$ [27, Proposition 4.6] and, for $n=1,2,\ldots$,
${\mathsf{w}}_{-1,{\gamma}}\big{(}{\mathsf{c}}((x,t),n^{-2})\big{)}\sim{\mathsf{w}}_{{\gamma},d}(n;t)$
with

(3.7)
${\mathsf{w}}_{{\gamma},d}(n;t)=\big{(}1-t+n^{-2}\big{)}^{{\gamma}+\frac{1}{2}}\big{(}t+n^{-2}\big{)}^{\frac{d-2}{2}}.$

It is shown in [27, Theorem 4.10] that, for $d\geq 2$,
${\gamma}\geq-\frac{1}{2}$, and any ${\kappa}>0$, the kernel
${\mathsf{L}}_{n}({\mathsf{w}}_{-1,{\gamma}};\cdot,\cdot)$ satisfies the
estimate

(3.8)
$\left|{\mathsf{L}}_{n}({\mathsf{w}}_{-1,{\gamma}};(x,t),(y,s))\right|\leq\frac{c_{\kappa}n^{d}}{\sqrt{{\mathsf{w}}_{{\gamma},d}(n;t)}\sqrt{{\mathsf{w}}_{{\gamma},d}(n;s)}}\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))\big{)}^{-{\kappa}}.$

This shows, in particular, that the kernel decays away from $(x,t)=(y,s)$
faster than any polynomial rate. The decaying estimate is established using
the case of $m=0$ in the following lemma [9, Theorem 2.6.7].

######  Lemma 3.3.

Let $\ell$ be a positive integer and let $\eta$ be a function that satisfy,
$\eta\in C^{3\ell-1}({\mathbb{R}})$, $\mathrm{supp}\,\eta\subset[0,2]$ and
$\eta^{(j)}(0)=0$ for $j=0,1,2,\ldots,3\ell-2$. Then, for
${\alpha}\geq{\beta}\geq-\frac{1}{2}$, $t\in[-1,1]$ and $n\in{\mathbb{N}}$,

(3.9) $\left|\frac{d^{m}}{dt^{m}}L_{n}^{({\alpha},{\beta})}(t)\right|\leq
c_{\ell,m,{\alpha}}\left\|\eta^{(3\ell-1)}\right\|_{\infty}\frac{n^{2{\alpha}+2m+2}}{(1+n\sqrt{1-t})^{\ell}},\quad
m=0,1,2,\ldots.$

The proof of the localization of the kernel requires further properties of the
distance function. Let ${\mathsf{d}}_{[-1,1]}(\cdot,\cdot)$ be the distance
function on the interval $[0,1]$ defined by

${\mathsf{d}}_{[-1,1]}(t,s)=\arccos\left(\sqrt{t}\sqrt{s}+\sqrt{1-t}\sqrt{1-s}\right),\qquad
t,s\in[0,1]$

and let ${\mathsf{d}}_{\SS}(\cdot,\cdot)$ denote the geodesic distance on the
unit sphere ${\mathbb{S}^{d-1}}$ defined by

${\mathsf{d}}_{\SS}(\xi,\eta)=\arccos{\langle}\xi,\eta{\rangle},\qquad\xi,\eta\in{\mathbb{S}^{d-1}}.$

Then, for $(x,t),(y,s)\in{\mathbb{V}}_{0}^{d+1}$ and setting $x=t\xi$ and
$y=s\eta$, the distance ${\mathsf{d}}_{{\mathbb{V}}_{0}}(\cdot,\cdot)$
satisfies, as shown in [27, Proposition 4.3],

(3.10)
$c_{1}{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))\leq{\mathsf{d}}_{[0,1]}(t,s)+(ts)^{\frac{1}{4}}{\mathsf{d}}_{\SS}(\xi,\eta)\leq
c_{2}{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s)).$

Another two inequalities that we shall need are

(3.11)
$\big{|}\sqrt{t}-\sqrt{s}\big{|}\leq{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))\quad\hbox{and}\quad\big{|}\sqrt{1-t}-\sqrt{1-s}\big{|}\leq{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))$

for $(x,t),(y,s)\in{\mathbb{V}}_{0}^{d+1}$, established in [27, Lemma 4.4].

###  3.2. Auxiliary inequalities

We state several auxiliary results in this subsection. First of all, we will
need the maximal function $f_{{\beta},n}^{\ast}$ defined by

(3.12)
$f_{{\beta},n}^{\ast}(x,t)=\max_{(y,s)\in{\mathbb{V}}_{0}^{d+1}}\frac{|f(y,s)|}{\left(1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))\right)^{\beta}}.$

This maximal function satisfies the following property [27, Corollary 2.11].

######  Proposition 3.4.

If $0<p\leq\infty$, $f\in\Pi_{n}({\mathbb{V}}_{0}^{d+1})$ and
${\beta}>{\alpha}({\mathsf{w}})/p$, then

(3.13)
$\|f\|_{p,{\mathsf{w}}}\leq\|f_{{\beta},n}^{\ast}\|_{p,{\mathsf{w}}}\leq
c\|f\|_{p,{\mathsf{w}}},$

where $c$ depends also on $L({\mathsf{w}})$ and ${\beta}$ when ${\beta}$ is
either large or close to ${\alpha}({\mathsf{w}})/p$.

For our next result, we need the concept of a maximal
${\varepsilon}$-separated set on the conic surface. However, the concept will
be needed for the unit ball and the solid cone later in the paper, so we give
its definition on a domain $\Omega$ equipped with a distance function
${\mathsf{d}}(\cdot,\cdot)$. Let
$B(x,{\varepsilon})=\\{y\in\Omega:{\mathsf{d}}(x,y)\leq{\varepsilon}\\}$ be
the ball centered at $x$ and with radius ${\varepsilon}$ in $\Omega$.

######  Definition 3.5.

Let $\Xi$ be a discrete set in $\Omega$.

  1. (a)

Let ${\varepsilon}>0$. A discrete subset $\Xi$ of $\Omega$ is called
${\varepsilon}$-separated if ${\mathsf{d}}(x,y)\geq{\varepsilon}$ for every
two distinct points $x,y\in\Xi$.

  2. (b)

$\Xi$ is called maximal if there is a constant $c_{d}>1$ such that

$1\leq\sum_{z\in\Xi}\chi_{B(z,{\varepsilon})}(x)\leq c_{d},\qquad\forall
x\in\Omega,$

where $\chi_{E}$ denotes the characteristic function of the set $E$.

A maximal ${\varepsilon}$-separated subset is constructed in [27, Proposition
4.17] using the separated sets in the $t$-variable on $[0,1]$ and in the $\xi$
variable on ${\mathbb{S}^{d-1}}$, where
$(x,t)=(t\xi,t)\in{\mathbb{V}}_{0}^{d+1}$. We recall what is necessary for our
purpose. Let ${\varepsilon}>0$ and let
$N=\lfloor\frac{\pi}{2}{\varepsilon}^{-1}\rfloor$. We define

(3.14)
$t_{j}=\sin^{2}\frac{(2j-1)\pi}{4N}\quad\hbox{and}\quad{\varepsilon}_{j}=\frac{\pi{\varepsilon}}{2\sqrt{t_{j}}},\quad
1\leq j\leq N.$

Let $\Xi_{\SS}({\varepsilon}_{j})$ be the maximal
${\varepsilon}_{j}$-separated set of ${\mathbb{S}^{d-1}}$, so that there is a
family of sets
$\\{\SS_{\xi}({\varepsilon}_{j}):\xi\in\Xi_{\SS}({\varepsilon}_{j})\\}$ which
forms a partition
${\mathbb{S}^{d-1}}=\bigcup_{\eta\in\Xi_{\SS}({\varepsilon}_{j})}\SS_{\eta}({\varepsilon}_{j})$.
Then

(3.15)
$\Xi_{{\mathbb{V}}_{0}}=\big{\\{}(t_{j}\xi,t_{j}):\,\xi\in\Xi_{\SS}({\varepsilon}_{j}),\,1\leq
j\leq N\big{\\}}$

defined a maximal ${\varepsilon}$-separated subset on
${\mathbb{V}}_{0}^{d+1}$. Such a set is used to establish the Marcinkiewicz-
Zygmund inequality on the conic surface, which we need below.

Out next result is of interest in itself. For polynomials of one variable, it
is known [20, (7.1.7)]

$\int_{-1}^{1}|f(t)|^{p}w(t)\mathrm{d}t\leq c_{\delta}\int_{-1+\delta
n^{-2}}^{1-\delta n^{-2}}|f(t)|^{p}w(t)\mathrm{d}t,\quad\deg f\leq n,$

where $w$ is a doubling weight and $\delta$ is a positive constant. The
following proposition is an analog of the above inequality on the conic
surface.

######  Proposition 3.6.

Let ${\mathsf{w}}$ be a doubling weight function on ${\mathbb{V}}_{0}^{d+1}$.
For $n\in{\mathbb{N}}$, let $\chi_{n,\delta}(t)$ denote the characteristic
function of the interval $[\frac{\delta}{n^{2}},1-\frac{\delta}{n^{2}}]$.
Then, for $f\in\Pi_{n}({\mathbb{V}}_{0}^{d+1})$, $1\leq p<\infty$, and every
$\delta>0$,

(3.16)
$\int_{{\mathbb{V}}_{0}^{d+1}}|f(x,t)|^{p}{\mathsf{w}}(x,t)\mathrm{d}{\mathsf{m}}(x,t)\leq
c_{\delta}\int_{{\mathbb{V}}_{0}^{d+1}}|f(x,t)|^{p}\chi_{n,\delta}(t){\mathsf{w}}(x,t)\mathrm{d}{\mathsf{m}}(x,t).$

Moreover, when $p=\infty$,

(3.17) $\|f\|_{\infty}\leq c\|f\chi_{n,\delta}(t)\|_{\infty}.$

###### Proof.

The proof uses the Marcinkiewicz-Zygmund inequality on the conic surface. Let
$\Xi_{{\mathbb{V}}_{0}}$ be the ${\varepsilon}$-maximal separated set in
(3.15). For ${\varepsilon}={\beta}n^{-1}$ with ${\beta}>0$, the Marcinkiewicz-
Zygmund inequality states [27, Theorem 4.18] that, for $1\leq p<\infty$,

  1. $(i)$

for $f\in\Pi_{m}({\mathbb{V}}_{0}^{d+1})$ with $n\leq m\leq cn$,

$\sum_{(z,r)\in\Xi_{{\mathbb{V}}_{0}}}\Big{(}\max_{(x,t)\in{\mathsf{c}}\big{(}(z,r),n^{-1}\big{)}}|f(x,t)|^{p}\Big{)}{\mathsf{w}}\\!\left({\mathsf{c}}\big{(}(z,r),n^{-1}\big{)}\right)\leq
c_{{\mathsf{w}}}\|f\|_{p,{\mathsf{w}}}^{p};$

  2. $(ii)$

for $f\in\Pi_{n}({\mathbb{V}}_{0}^{d+1})$,

$\displaystyle\|f\|_{p,{\mathsf{w}}}^{p}\leq
c_{{\mathsf{w}}}\sum_{(z,r)\in\Xi_{{\mathbb{V}}_{0}}}\Big{(}\min_{(x,t)\in{\mathsf{c}}\bigl{(}(z,r),n^{-1}\bigr{)}}|f(x,t)|^{p}\Big{)}{\mathsf{w}}\bigl{(}{\mathsf{c}}\big{(}(z,r),n^{-1}\big{)}\bigr{)}.$

Clearly $t_{0}=\sin^{2}\frac{\pi}{4N}\sim n^{-2}$ and $1-t_{N}\sim n^{-2}$ and
the constant is proportional to ${\beta}^{-1}$. Hence, for a fixed $\delta>0$,
by choosing ${\beta}$ sufficiently large, we see that
$f(z,r)=f(z,r)\chi_{n,\delta}(r)$ for all $(z,r)\in\Xi_{{\mathbb{V}}_{0}}$.
Consequently, it follows from (ii) and (i) that

$\displaystyle\|f\|_{p,{\mathsf{w}}}^{p}\,$ $\displaystyle\leq
c_{{\mathsf{w}}}\sum_{(z,r)\in\Xi_{{\mathbb{V}}_{0}}}|f(z,r)\chi_{n,\delta}(r)|^{p}{\mathsf{w}}\bigl{(}{\mathsf{c}}\big{(}(z,r),n^{-1}\big{)}\bigr{)}$
$\displaystyle\leq
c\int_{{\mathbb{V}}_{0}^{d+1}}|f(x,t)|^{p}\chi_{n,\delta}(t){\mathsf{w}}(x,t)\mathrm{d}{\mathsf{m}}(x,t)$

for all $f\in\Pi_{n}({\mathbb{V}}_{0}^{d+1})$, which is the desired inequality
(3.16). For the uniform norm,

$|f(x,t)|\leq c\max_{(z,r)\in\Xi_{{\mathbb{V}}_{0}}}|f(z,r)|$

by [27, Theorem 2.14], from which the inequality (3.17) follows readily. ∎

######  Remark 3.1.

Setting $x=t\xi$, $\xi\in{\mathbb{S}^{d-1}}$ shows that the space
$\Pi_{n}^{d}({\mathbb{V}}_{0}^{d+1})$ is a subspace of $\Pi_{n}^{*}$, which
contains all polynomials of degree $n$ in $t$ and in $\xi$. Examining the
proof in [27] carefully, it is not difficult to see that the Marcinkiewicz-
Zygmund inequalities in [27, Theorem 4.18] holds for $\Pi_{n}^{*}$ and, as a
consequence, so does (3.16).

Our next lemma is technical and reduces to [27, Lemma 4.14] when $\rho=0$ and
$\tau_{1}=\tau_{2}=0$. The general version is needed to handle the derivatives
of the highly localized kernels.

######  Lemma 3.7.

Let $d\geq 2$ and ${\gamma}>-1$. Let $\rho$ satisfies $0\leq\rho\leq(d-2)/2$,
$\tau_{1}$ satisfies $0\leq\tau_{1}\leq 1/2$ and $\tau_{2}$ satisfies
$0\leq\tau_{2}\leq d/2$. For $0<p<\infty$, assume
${\kappa}>\frac{2d}{p}+({\gamma}+\frac{d-1}{2})|\frac{1}{p}-\frac{1}{2}|$.
Then for $x=t\xi$ and $y=s\eta$ and $(x,t)\in{\mathbb{V}}_{0}^{d+1}$,

(3.18)
$\displaystyle\int_{{\mathbb{V}}_{0}^{d+1}}\frac{s^{-\tau_{1}}(1-s)^{-\tau_{2}}(1+{\langle}\xi,\eta{\rangle})^{-\rho}{\mathsf{w}}_{-1,{\gamma}}(s)\mathrm{d}{\mathsf{m}}(y,s)}{{\mathsf{w}}_{{\gamma},d}(n;s)^{\frac{p}{2}}\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))\big{)}^{{\kappa}p}}\leq
cn^{-d+2|\tau|}{\mathsf{w}}_{{\gamma},d}(n;t)^{1-\frac{p}{2}}.$

###### Proof.

For $\rho=0$ and $\tau_{i}=0$, this estimated is established in [27, Lemma
4,1,4]. The proof follows along the same argument, we provide an account on
the modification. The first step is to show that it is sufficient to consider
only $p=2$ by using the doubling property of ${\mathsf{w}}_{-1,{\gamma}}$,
which remains valid when $\rho>0$ and $\tau_{i}>0$. Let $J_{2}$ denote the
left-hand side of (3.18) with $p=2$. Then, using

(3.19)
$\int_{{\mathbb{S}^{d-1}}}g({\langle}\xi,\eta{\rangle})d\sigma(\eta)={\omega}_{d-1}\int_{-1}^{1}g(u)(1-u^{2})^{\frac{d-3}{2}}\mathrm{d}u$

and ${\theta}\sim 2\sin\frac{{\theta}}{2}=\sqrt{1-\cos{\theta}}$ for
${\theta}\leq[0,\pi]$, we deduce that

$\displaystyle J_{2}\,$ $\displaystyle\leq
c\int_{0}^{1}\int_{-1}^{1}\frac{s^{d-1-\tau_{1}-\rho}(1-s)^{-\tau_{2}}(1+u)^{-\rho}(1-u^{2})^{\frac{d-3}{2}}{\mathsf{w}}_{-1,{\gamma}}(s)}{{\mathsf{w}}_{{\gamma},d}(n;s)\big{(}1+n\arccos\big{(}\sqrt{ts}\sqrt{\frac{1+u}{2}}+\sqrt{1-t}\sqrt{1-s}\big{)}\big{)}^{2{\kappa}}}\mathrm{d}u\mathrm{d}s$
$\displaystyle\leq
c\int_{0}^{1}\int_{0}^{1}\frac{s^{d-1-\tau_{1}-\rho}(1-s)^{-\tau_{2}}v^{d-2-2\rho}(1-v^{2})^{\frac{d-3}{2}}{\mathsf{w}}_{-1,{\gamma}}(s)}{{\mathsf{w}}_{{\gamma},d}(n;s)\left(1+n\sqrt{1-\sqrt{ts}v-\sqrt{1-t}\sqrt{1-s}}\right)^{2k}}\mathrm{d}v\mathrm{d}s,$

where the second step follows from the changing variable
$\sqrt{\frac{1+u}{2}}\mapsto v$. Making a further changing of variable
$v\mapsto z/\sqrt{s}$ gives

$\displaystyle J_{2}\,$ $\displaystyle\leq
c\int_{0}^{1}\int_{0}^{\sqrt{s}}\frac{s^{1-\tau_{1}+\rho}(1-s)^{-\tau_{2}}\,z^{d-2-2\rho}(s-z^{2})^{\frac{d-3}{2}}{\mathsf{w}}_{-1,{\gamma}}(s)}{{\mathsf{w}}_{{\gamma},d}(n;s)\left(1+n\sqrt{1-\sqrt{t}\,z-\sqrt{1-t}\sqrt{1-s}}\right)^{2{\kappa}}}\mathrm{d}z\mathrm{d}s.$

We now use $s^{1-\tau_{1}+\rho}z^{d-2-2\rho}\leq
s^{\frac{d}{2}-\tau_{1}}\leq(s+n^{-2})^{\frac{d}{2}-\tau_{1}}\leq
n^{2\tau_{1}}(s+n^{-2})^{\frac{d}{2}}$, which holds for $\rho\leq(d-2)/2$ and
$\tau_{1}\leq d/2$, and the formulas for ${\mathsf{w}}_{-1,{\gamma}}(s)$ and
${\mathsf{w}}_{{\gamma},d}(n;s)$ to deduce

$\displaystyle J_{2}\leq
cn^{2\tau_{1}}\int_{0}^{1}\int_{0}^{\sqrt{s}}\frac{(1-s)^{-\tau_{2}}(s-z^{2})^{\frac{d-3}{2}}}{(1-s+n^{-2})^{\frac{1}{2}}\left(1+n\sqrt{1-\sqrt{t}\,z-\sqrt{1-t}\sqrt{1-s}}\right)^{2{\kappa}}}\mathrm{d}z\mathrm{d}s.$

We make one more change of variable $s\mapsto 1-w^{2}$ and make use of
$1-s=w^{2}/(1+\sqrt{1-w^{2}})\leq w^{2}$ to obtain

$\displaystyle J_{2}\,$ $\displaystyle\leq
cn^{2\tau_{1}}\int_{0}^{1}\int_{0}^{\sqrt{1-w^{2}}}\frac{w^{1-2\tau_{2}}(1-w^{2}-z^{2})^{\frac{d-3}{2}}}{\left(w^{2}+n^{-2}\right)^{\frac{1}{2}}\left(1+n\sqrt{1-\sqrt{t}z-\sqrt{1-t}\,w}\right)^{2{\kappa}}}\mathrm{d}z\mathrm{d}w$
$\displaystyle\leq
cn^{2\tau_{1}+2\tau_{2}}\int_{0}^{1}\int_{0}^{\sqrt{1-w^{2}}}\frac{(1-w^{2}-z^{2})^{\frac{d-3}{2}}}{\left(1+n\sqrt{1-\sqrt{t}z-\sqrt{1-t}\,w}\right)^{2{\kappa}}}\mathrm{d}z\mathrm{d}w,$

where the second inequality follows from
$w^{1-2\tau_{2}}\leq(w^{2}+n^{-2})^{\frac{1}{2}-\tau_{2}}\leq
n^{2\tau_{2}}(w^{2}+n^{-2})^{\frac{1}{2}}$, which holds for
$\tau\leq\frac{1}{2}$. The last integral in the right-hand side has appeared
and estimated in the proof of [27, Lemma 4.14], from which the desired
estimate follows. ∎

Finally we recall the following lemma [27, Lemma 4.11], which plays an
important role in establishing the localization of the kernel
${\mathsf{L}}_{n}({\mathsf{w}}_{-1,{\gamma}})$.

######  Lemma 3.8.

Let $d\geq 2$ and ${\gamma}>-\frac{1}{2}$. Then, for ${\beta}\geq
2{\gamma}+d+1$,

$\displaystyle\int_{[-1,1]^{2}}$
$\displaystyle\frac{(1-v_{1})^{\frac{d-2}{2}-1}(1-v_{2})^{{\gamma}-\frac{1}{2}}}{\big{(}1+n\sqrt{1-\zeta(x,t,y,s;v)}\,\big{)}^{{\beta}}}\mathrm{d}v$
$\displaystyle\qquad\leq\frac{cn^{-(2{\gamma}+d-1)}}{\sqrt{{\mathsf{w}}_{{\gamma},d}(n;t)}\sqrt{{\mathsf{w}}_{{\gamma},d}(n;s)}\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))\big{)}^{{\beta}-3{\gamma}-\frac{3d+1}{2}}}.$

###  3.3. Bernstein inequality for $\partial_{t}$ on the conic surface

To prove the Bernstein inequality for the first-order derivatives, we need to
understand the action of these derivatives on the highly localized kernels. We
start with the case $\partial_{t}$. The key ingredient lies in the estimates
below.

######  Lemma 3.9.

Let $d\geq 2$ and ${\gamma}\geq-\frac{1}{2}$. Then for any ${\kappa}>0$,

$\left|\partial_{t}{\mathsf{L}}_{n}({\mathsf{w}}_{-1,{\gamma}};(x,t),(y,s))\right|\leq\frac{c_{\kappa}n^{d+1}}{\varphi(t)\sqrt{{\mathsf{w}}_{{\gamma},d}(n;t)}\sqrt{{\mathsf{w}}_{{\gamma},d}(n;s)}\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))\big{)}^{{\kappa}}}.$

###### Proof.

Taking derivative on the integral expression (3.5) of the kernel and writing
$\zeta(v)$ in place of $\zeta(x,t,y,s;v)$, we obtain with
${\lambda}={\gamma}+d-1$,

(3.20)
$\displaystyle\partial_{t}{\mathsf{L}}_{n}({\mathsf{w}}_{-1,{\gamma}};(x,t),(y,s))$
$\displaystyle=c_{{\gamma}}\int_{[-1,1]^{2}}\partial
L_{n}^{({\lambda}-\frac{1}{2},-\frac{1}{2})}\big{(}2\zeta(v)^{2}-1\big{)}$
$\displaystyle\times
4\zeta(v)\partial_{t}\zeta(v)(1-v_{1}^{2})^{\frac{d-2}{2}-1}(1-v_{2}^{2})^{{\gamma}-\frac{1}{2}}\mathrm{d}v,$

where $\partial
L_{n}^{({\lambda}-\frac{1}{2},-\frac{1}{2})}(z)=\frac{{\mathsf{d}}}{{\mathsf{d}}z}L_{n}^{({\lambda}-\frac{1}{2},-\frac{1}{2})}(z)$.
We first prove the following estimate

(3.21)
$\displaystyle\left|2\varphi(t)\partial_{t}\zeta(x,t,y,s;v)\right|\leq\Sigma_{1}+\Sigma_{2}(v_{1})+\Sigma_{3}(v_{2}),$

where

$\Sigma_{1}={\mathsf{d}}_{{\mathbb{V}}_{0}}\big{(}(x,t),(y,s)\big{)},\quad\Sigma_{2}(v_{1})=(1-v_{1})\sqrt{s},\quad\Sigma_{3}(v_{2})=(1-v_{2})\sqrt{1-s}.$

Recall
$\zeta(x,t,y,s;v)=v_{1}\sqrt{\tfrac{st+{\langle}x,y{\rangle}}{2}}+v_{2}\sqrt{1-t}\sqrt{1-s}$
and $|\zeta(x,t,y,s;v)|\leq 1$. For simplicity, we write
$\Psi=\sqrt{\frac{1+{\langle}\xi,\eta{\rangle}}{2}}$. Since
${\langle}\xi,\eta{\rangle}=\cos{\mathsf{d}}_{\SS}(\xi,\eta)$ in terms of the
geodesic distance on ${\mathbb{S}^{d-1}}$, it follows that
$\Psi=\cos\frac{{\mathsf{d}}_{\SS}(\xi,\eta)}{2}$. Taking derivative of
$\zeta$,

$\displaystyle\partial_{t}\zeta(x,t,y,s;v)=v_{1}\frac{\sqrt{s}}{2\sqrt{t}}\,\Psi-
v_{2}\frac{\sqrt{1-s}}{2\sqrt{1-t}},$

and writing $v_{i}=1-(1-v_{i})$, it follows readily that

$|2\varphi(t)\partial_{t}\zeta(x,t,y,s;v)|\leq(1-v_{1})\sqrt{s}+(1-v_{2})\sqrt{1-s}+\left|\sqrt{1-t}\sqrt{s}\,\Psi-\sqrt{t}\sqrt{1-s}\right|.$

If $t\geq s$, we write the last term as

$\sqrt{1-t}\sqrt{s}\,\Psi-\sqrt{t}\sqrt{1-s}=-\sqrt{1-t}\sqrt{s}(1-\Psi)+\sqrt{1-t}\sqrt{s}-\sqrt{t}\sqrt{1-s},$

whereas if $t\leq s$, we write

$\sqrt{1-t}\sqrt{s}\,\Psi-\sqrt{t}\sqrt{1-s}=\left(\sqrt{1-t}\sqrt{s}-\sqrt{t}\sqrt{1-s}\right)\Psi-\sqrt{t}\sqrt{1-s}(1-\Psi).$

From these identities, we deduce the estimate

$\left|\sqrt{1-t}\sqrt{s}\,\Psi-\sqrt{t}\sqrt{1-s}\right|\leq\left|\sqrt{1-t}\sqrt{s}-\sqrt{t}\sqrt{1-s}\right|+(ts)^{\frac{1}{4}}(1-\Psi).$

Since $\Psi=\cos\frac{{\mathsf{d}}_{\SS}(\xi,\eta)}{2}$, we obtain an upper
bound $1-\Psi\leq\frac{1}{2}{\mathsf{d}}_{\SS}(\xi,\eta)$, which implies by
(3.10) that $(ts)^{\frac{1}{4}}{\mathsf{d}}_{\SS}(\xi,\eta)\leq
c\,{\mathsf{d}}_{{\mathbb{V}}_{0}^{d+1}}\big{(}(x,t),(y,s)\big{)}$.
Furthermore, we also have

$\displaystyle|\sqrt{1-t}\sqrt{s}-\sqrt{t}\sqrt{1-s}|\,$
$\displaystyle\leq|\sqrt{1-t}-\sqrt{1-s}|\sqrt{s}+|\sqrt{s}-\sqrt{t}|\sqrt{1-s}$
$\displaystyle\leq 2{\mathsf{d}}_{{\mathbb{V}}_{0}}\big{(}(x,t),(y,s)\big{)},$

using (3.11). Putting these inequalities together, we have proved (3.21).
Using this estimate, we can then use (3.9) in (3.20) to obtain for a positive
number ${\beta}$,

$\displaystyle\left|\varphi(t)\partial_{t}{\mathsf{L}}_{n}\big{(}{\mathsf{w}}_{-1,{\gamma}};(x,t),(y,s)\big{)}\right|\leq\,c\int_{[-1,1]^{2}}\frac{n^{2{\lambda}+3}}{\left(1+n\sqrt{1-\zeta(x,t,y,s;v)}\right)^{\beta}}$
$\displaystyle\qquad\qquad\qquad\times\big{(}\Sigma_{1}+\Sigma_{2}(v_{1})+\Sigma_{3}(v_{2})\big{)}(1-v_{1}^{2})^{\frac{d-2}{2}-1}(1-v_{2}^{2})^{{\gamma}-\frac{1}{2}}\mathrm{d}v.$

The integral with $\Sigma_{1}$ term is bounded by applying Lemma 3.8 and
choosing $\beta$ appropriately. The integral with $\Sigma_{2}(v_{1})$ term is
bounded by applying the same lemma but with $(1-v_{1})^{\frac{d-2}{2}-1}$
replaced by $(1-v_{1})^{\frac{d}{2}-1}$ and then using
$n^{-1}\sqrt{s}\leq(\sqrt{t}+n^{-1})(\sqrt{s}+n^{-1})$, whereas the integral
with $\Sigma_{3}(v_{2})$ can be handled similarly. In fact, the estimate of
the last two integrals are already appeared in the proof of [27, Theorem
4.13]. This completes the proof. ∎

We are now ready to prove the Bernstein inequality for $\partial_{t}$ on the
conic surface.

######  Theorem 3.10.

Let ${\mathsf{w}}$ be a doubling weight on ${\mathbb{V}}_{0}^{d+1}$. Let
$\ell$ be a positive integer and $1\leq p<\infty$. Then

(3.22) $\|\partial_{t}f\|_{p,{\mathsf{w}}}\leq
cn^{2\ell}\|f\|_{p,{\mathsf{w}}},\quad\forall
f\in\Pi_{n}({\mathbb{V}}_{0}^{d+1}),$

and, with $\varphi(t)=\sqrt{t(1-t)}$,

(3.23) $\left\|\varphi^{\ell}\partial_{t}^{\ell}f\right\|_{p,{\mathsf{w}}}\leq
cn^{\ell}\|f\|_{p,{\mathsf{w}}},\quad\forall
f\in\Pi_{n}({\mathbb{V}}_{0}^{d+1}).$

###### Proof.

By the definition of
${\mathsf{L}}_{n}\left({\mathsf{w}}_{-1,{\gamma}};\cdot,\cdot\right)$, every
$f\in\Pi_{n}({\mathbb{V}}_{0}^{d+1})$ satisfies

(3.24)
$f(x,t)=\int_{{\mathbb{V}}_{0}^{d+1}}f(y,s){\mathsf{L}}_{n}\left({\mathsf{w}}_{-1,{\gamma}};(x,t),(y,s)\right){\mathsf{w}}_{-1,{\gamma}}(s)\mathrm{d}\sigma(y,s).$

Taking $\partial_{t}$ derivative and using the maximal function
$f_{{\beta},n}^{\ast}$ in (3.12), we obtain

$\displaystyle\left|\partial_{t}f(x,t)\right|\leq
cf_{{\beta},n}^{\ast}(x,t)\int_{{\mathbb{V}}_{0}^{d+1}}\frac{\left|\partial_{t}{\mathsf{L}}_{n}\left({\mathsf{w}}_{-1,{\gamma}};(x,t),(y,s)\right)\right|}{\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))\big{)}^{\beta}}{\mathsf{w}}_{-1,{\gamma}}(s)\mathrm{d}{\mathsf{m}}(y,s).$

Using the first estimate in Lemma 3.9 and choosing
${\beta}=2{\alpha}({\mathsf{w}})/p$ and
${\kappa}>{\beta}+2d+\frac{1}{2}({\gamma}+\frac{d-2}{2})$, we see that the
integral in the right-hand side is bounded by

$\int_{{\mathbb{V}}_{0}^{d+1}}|f(y,s)|\frac{n^{d+1}{\mathsf{w}}_{-1,{\gamma}}(s)\mathrm{d}{\mathsf{m}}(y,s)}{\varphi(t)\sqrt{{\mathsf{w}}_{{\gamma},d}(n;t)}\sqrt{{\mathsf{w}}_{{\gamma},d}(n;s)}\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))\big{)}^{{\kappa}-{\beta}}}\leq
c\frac{n}{\varphi(t)}$

where the last step follows from (3.18) with $p=1$ and $\rho=0$. Consequently,
we obtain

(3.25) $\displaystyle\left|\partial_{t}f(x,t)\right|\leq
c\frac{n}{\varphi(t)}f_{{\beta},n}^{\ast}(x,t).$

In particular, it follows by Proposition 3.6 and $\varphi(t)\geq\frac{1}{n}$
for $\delta n^{-2}\leq t\leq 1-\delta n^{-2}$ that

$\displaystyle\|\partial_{t}f\|_{p,{\mathsf{w}}}\leq\|\partial_{t}f\chi_{n,\delta}(t)\|_{p,{\mathsf{w}}}\leq
cn^{2}\|f_{{\beta},n}^{\ast}\|_{p,{\mathsf{w}}}\leq cn\|f\|_{p,{\mathsf{w}}},$

where the last step follows from (3.13), which proves the inequality (3.22)
for $\ell=1$. Iterating the inequality for $\ell=1$ proves (3.22) for
$\ell>1$. In the case of $p=\infty$, we use $\|f\|_{\infty}$ instead of
$f_{{\beta},n}^{*}$ in (3.25), the resulted estimate immediately implies
(3.23) and, with the help of (3.17), also (3.22).

Since $\left|\varphi(t)\partial_{t}f(x,t)\right|\leq
cnf_{{\beta},n}^{\ast}(x,t)$ by (3.25), we obtain immediately, by (3.13), that
$\left\|\varphi\partial_{t}f\right\|_{p,{\mathsf{w}}}\leq
cn\|f\|_{p,{\mathsf{w}}}$, which proves (3.23) for $\ell=1$. Moreover, since

$\left(\varphi(t)\partial_{t}\right)^{2}=(1-2t)\partial_{t}+\varphi(t)^{2}\partial^{2}_{t},$

we obtain immediately that

$\left\|\varphi^{2}\partial_{t}^{2}f\right\|_{p,{\mathsf{w}}}\leq\|\partial_{t}f\|_{p,{\mathsf{w}}}+\|\left(\varphi(t)\partial_{t}\right)^{2}f\|_{p,{\mathsf{w}}}\leq
cn^{2}\|f\|_{p,{\mathsf{w}}},$

where we have applyed the inequality (3.22) with $\ell=1$ and the inequality
(3.23) with $\ell=1$ twice in the second step. This proves the inequality
(3.23) for $\ell=2$. Since $\varphi^{2}\partial_{t}^{2}f$ is a polynomial in
$\Pi_{n}({\mathbb{V}}_{0}^{d+1},{\mathsf{w}})$, we can iterate the inequality
(3.23) with $\ell=2$ to establish the inequality for all even $\ell$ and,
similarly, establish the inequality for odd $\ell$ after applying the
inequality with $\ell=1$ once. This completes the proof. ∎

###  3.4. Bernstein inequality for $D_{i,j}$ on the conic surface

We start with the estimate of the derivative of the localized kernel.

######  Lemma 3.11.

Let $d\geq 2$ and ${\gamma}\geq-\frac{1}{2}$. Let $D_{i,j}$ be the operator
acting on $x$ variable. Then for any ${\kappa}>0$ and
$(x,t),(y,s)\in{\mathbb{V}}_{0}^{d+1}$,

$\left|D_{i,j}{\mathsf{L}}_{n}({\mathsf{w}}_{-1,{\gamma}};(x,t),(y,s))\right|\leq\frac{c_{\kappa}n^{d+1}\sqrt{t}}{\sqrt{{\mathsf{w}}_{{\gamma},d}(n;t)}\sqrt{{\mathsf{w}}_{{\gamma},d}(n;s)}\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))\big{)}^{{\kappa}}}$

for $1\leq i<j\leq d$ and, furthermore,

$\left|D_{i,j}^{2}{\mathsf{L}}_{n}({\mathsf{w}}_{-1,{\gamma}};(x,t),(y,s))\right|\leq\frac{c_{\kappa}n^{d+2}\big{(}t+\sqrt{t}\sqrt{s}(1+{\langle}\xi,\eta{\rangle})^{-\frac{1}{2}}\big{)}}{\sqrt{{\mathsf{w}}_{{\gamma},d}(n;t)}\sqrt{{\mathsf{w}}_{{\gamma},d}(n;s)}\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))\big{)}^{{\kappa}}}.$

###### Proof.

The inequality (3.20) holds with $D_{t}$ replaced by $D_{i,j}$. First, we need
an estimate of $|D_{i,j}\zeta(x,t,y,s;v)|$. If
$\xi,\eta\in{\mathbb{S}^{d-1}}$, then
$2\sqrt{\frac{1+{\langle}\xi,\eta{\rangle}}{2}}=\|\xi+\eta\|$. Hence, for
$x=t\xi$ and $y=s\eta$,

$D_{i,j}\zeta(x,t,y,s;v)=v_{1}\frac{x_{i}y_{j}-x_{j}y_{i}}{4\sqrt{\tfrac{ts+{\langle}x,y{\rangle}}{2}}}=v_{1}\sqrt{t}\sqrt{s}\,\frac{\xi_{i}\eta_{j}-\xi_{j}\eta_{i}}{2\|\xi+\eta\|}.$

Now, using the identity

$\displaystyle
2(\xi_{i}\eta_{j}-\xi_{j}\eta_{i})=(\xi_{i}-\eta_{i})(\xi_{j}+\eta_{j})-(\xi_{j}-\eta_{j})(\xi_{i}+\eta_{i})$

and $\|\xi-\eta\|\leq{\mathsf{d}}_{\SS}(\xi,\eta)$ for
$\xi,\eta\in{\mathbb{S}^{d-1}}$, it follows immediately that

$|\xi_{j}\eta_{j}-\xi_{j}\eta_{i}|\leq\|\xi-\eta\|\,\|\xi+\eta\|\leq\|\xi+\eta\|\,{\mathsf{d}}_{\SS}(\xi,\eta).$

Putting these together, we obtain the inequality

$\displaystyle|D_{i,j}\zeta(x,t,y,s;v)|\leq|v_{1}|\sqrt{t}\sqrt{s}\,{\mathsf{d}}_{\SS}(\xi,\eta)\leq\sqrt{t}\sqrt{s}\,{\mathsf{d}}_{\SS}(\xi,\eta).$

If $s\leq t$, then
$\sqrt{s}\,{\mathsf{d}}_{\SS}(\xi,\eta)\leq(ts)^{\frac{1}{4}}{\mathsf{d}}_{\SS}(\xi,\eta)\leq
c\,{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))$ by (3.10). If $s\geq t$, then
$\sqrt{s}\leq(ts)^{\frac{1}{4}}+{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))$
by (3.11), which implies, by (3.10),

$\sqrt{s}\,{\mathsf{d}}_{\SS}(\xi,\eta)\leq(ts)^{\frac{1}{4}}{\mathsf{d}}_{\SS}(\xi,\eta)+\pi{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))\leq
c\,{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s)).$

Consequently, we have proved the inequality

(3.26) $\displaystyle|D_{i,j}\zeta(x,t,y,s;v)|\leq
c\sqrt{t}{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s)).$

Using this inequality and $|\zeta(x,t,y,s;v)|\leq 1$, we then use (3.9) in
(3.20) to obtain

$\displaystyle\left|D_{i,j}{\mathsf{L}}_{n}\big{(}{\mathsf{w}}_{-1,{\gamma}};(x,t),(y,s)\big{)}\right|\leq\,$
$\displaystyle
c\int_{[-1,1]^{2}}\frac{n^{2{\lambda}+3}\sqrt{t}{\mathsf{d}}_{{\mathbb{V}}_{0}}\big{(}(x,t),(y,s)\big{)}}{\left(1+n\sqrt{1-\zeta(x,t,y,s;v)}\right)^{\kappa}}$
$\displaystyle\times(1-v_{1}^{2})^{\frac{d-2}{2}-1}(1-v_{2}^{2})^{{\gamma}-\frac{1}{2}}\mathrm{d}v$

from which the desired estimate follows from Lemma 3.8.

Now, taking the derivative $D_{i,j}$ one more time, we obtain

$\displaystyle
D_{i,j}^{2}{\mathsf{L}}_{n}({\mathsf{w}}_{-1,{\gamma}};(x,t),(y,s))$
$\displaystyle=\,$ $\displaystyle
c_{{\gamma}}\int_{[-1,1]^{2}}\partial^{2}L_{n}^{({\lambda}-\frac{1}{2},-\frac{1}{2})}\big{(}2\zeta(v)^{2}-1\big{)}\left(4\zeta(v)D_{i,j}\zeta(v)\right)^{2}(1-v_{1}^{2})^{\frac{d-2}{2}-1}(1-v_{2}^{2})^{{\gamma}-\frac{1}{2}}\mathrm{d}v$
$\displaystyle+\,$ $\displaystyle c_{{\gamma}}\int_{[-1,1]^{2}}\partial
L_{n}^{({\lambda}-\frac{1}{2},-\frac{1}{2})}\big{(}2\zeta(v)^{2}-1\big{)}4D_{i,j}\left[\zeta(v)D_{i,j}\zeta(v)\right](1-v_{1}^{2})^{\frac{d-2}{2}-1}(1-v_{2}^{2})^{{\gamma}-\frac{1}{2}}\mathrm{d}v.$

Applying (3.9) with $m=2$ and (3.26), it is easy to see that the first
integral in the right-hand side is bounded by

$\displaystyle
c\int_{[-1,1]^{2}}\frac{n^{2{\lambda}+5}t{\mathsf{d}}_{{\mathbb{V}}_{0}}\big{(}(x,t),(y,s)\big{)}^{2}}{\left(1+n\sqrt{1-\zeta(x,t,y,s;v)}\right)^{\kappa}}(1-v_{1}^{2})^{\frac{d-2}{2}-1}(1-v_{2}^{2})^{{\gamma}-\frac{1}{2}}\mathrm{d}v,$

which can be estimated as in the case of $D_{i,j}L_{n}$ to give the desired
upper bound. Moreover, the second integral is a sum of two integrals according
to

$D_{i,j}\left[\zeta(v)D_{i,j}\zeta(v)\right]=\left[D_{i,j}\zeta(v)\right]^{2}+\zeta(v)D_{i,j}^{2}\zeta(v);$

while the first one, using (3.26), is dominated by the first integral, the
second one can be bounded using the identity

$D_{i,j}^{2}\zeta(x,t,y,s;v)=-v_{1}\sqrt{t}\sqrt{s}\left(\frac{\xi_{i}\eta_{i}+\xi_{j}\eta_{j}}{2\|\xi+\eta\|}+\frac{(\xi_{i}\eta_{j}-\xi_{j}\eta_{i})^{2}}{2\|\xi+\eta\|^{3}}\right),$

which implies immediately that

$|D_{i,j}^{2}\zeta(x,t,y,s;v)|\leq\sqrt{t}\sqrt{s}\frac{1+{\mathsf{d}}_{\SS}(\xi,\eta)^{2}}{2\|\xi+\eta\|}\leq
c\frac{\sqrt{t}\sqrt{s}}{\sqrt{1+{\langle}\xi,\eta{\rangle}}}$

and leads to the upper bound

$c\frac{\sqrt{t}\sqrt{s}}{\sqrt{1+{\langle}\xi,\eta{\rangle}}}\int_{[-1,1]^{2}}\frac{n^{2{\lambda}+3}}{\left(1+n\sqrt{1-\zeta(x,t,y,s;v)}\right)^{\kappa}}(1-v_{1}^{2})^{\frac{d-2}{2}-1}(1-v_{2}^{2})^{{\gamma}-\frac{1}{2}}\mathrm{d}v$

which is again seen to be bounded by the desired upper bound. This completes
the proof. ∎

######  Theorem 3.12.

Let ${\mathsf{w}}$ be a doubling weight on ${\mathbb{V}}_{0}^{d+1}$. If
$f\in\Pi_{n}({\mathbb{V}}_{0}^{d+1})$ and $1\leq p<\infty$, then for $1\leq
i,j\leq d$,

(3.27) $\left\|\frac{1}{\sqrt{t}}D_{i,j}f\right\|_{p,{\mathsf{w}}}\leq
cn\|f\|_{p,{\mathsf{w}}}\quad\hbox{and}\quad\left\|\frac{1}{t}D_{i,j}^{2}f\right\|_{p,{\mathsf{w}}}\leq
cn^{2}\|f\|_{p,{\mathsf{w}}}.$

###### Proof.

For $f\in\Pi_{n}({\mathbb{V}}_{0}^{d+1})$, we start with the identity (3.24).
Taking $D_{i,j}$ derivative and using the maximal function
$f_{{\beta},n}^{\ast}$ in (3.12), we obtain

(3.28) $\left|D_{i,j}^{\ell}f(x,t)\right|\leq
f_{{\beta},n}^{\ast}(x,t)\int_{{\mathbb{V}}_{0}^{d+1}}\frac{\left|D_{i,j}^{\ell}{\mathsf{L}}^{n}({\mathsf{w}}_{-1,{\gamma}};(x,t),(y,s))\right|}{\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))\big{)}^{\beta}}{\mathsf{w}}_{-1,{\gamma}}(s)\mathrm{d}{\mathsf{m}}(y,s).$

Using the estimate in Lemma 3.11 and choosing
${\beta}=2{\alpha}({\mathsf{w}})/p$ and
${\kappa}>{\beta}+2d+\frac{1}{2}({\gamma}+\frac{d-2}{2})$, it follows that,
for $\ell=1$ the integral in the right-hand side is bounded by

$c\int_{{\mathbb{V}}_{0}^{d+1}}\frac{n^{d+1}\sqrt{t}{\mathsf{w}}_{-1,{\gamma}}(y,s)\mathrm{d}{\mathsf{m}}(y,s)}{\sqrt{{\mathsf{w}}_{{\gamma},d}(n;t)}\sqrt{{\mathsf{w}}_{{\gamma},d}(n;s)}\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))\big{)}^{{\kappa}-{\beta}}}\leq
cn\sqrt{t},$

where the last step follows from (3.18) with $p=1$ and $\rho=0$. Consequently,
we obtain

$\displaystyle\left|\frac{1}{\sqrt{t}}D_{i,j}f(x,t)\right|\leq
cnf_{{\beta},n}^{\ast}(x,t).$

For $1\leq p<\infty$, we can then apply (3.13) for ${\mathbb{V}}_{0}^{d+1}$ to
conclude that

$\left\|\frac{1}{\sqrt{t}}D_{i,j}f\right\|_{p,{\mathsf{w}}}\leq
cn\|f_{{\beta},n}^{\ast}\|_{p,{\mathsf{w}}}\leq cn\|f\|_{p,{\mathsf{w}}}.$

This proves (3.27) for $D_{i,j}$. For $D_{i,j}^{2}$, we first observe that,
using

$D_{i,j}^{2}=-x_{i}\partial_{i}-x_{j}\partial_{j}+x_{i}D_{i,j}\partial_{j}-x_{j}D_{i,j}\partial_{i},$

the function $t^{-1}D_{i,j}^{2}f(t\xi,t)$ is a polynomial in the variables $t$
and $\xi$. Hence, by Proposition 3.6 and Remark 3.1, there is a $\delta>0$
such that, with
$\chi_{n,\mathrm{d}}(t)=\chi_{[\frac{\delta}{n},1-\frac{\delta}{n}]}(t)$,

$\left\|\frac{1}{t}D_{i,j}^{2}f\right\|_{p,{\mathsf{w}}}\leq
c_{\delta}\left\|\frac{1}{t}D_{i,j}^{2}f\cdot\chi_{n,\delta}\right\|_{p,{\mathsf{w}}}.$

Using $\sqrt{s}\leq\sqrt{t}+{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))$ and
the estimate in Lemma 3.11, the integral in the right-hand side of (3.28) with
$\ell=2$ is bounded by

$\displaystyle
c_{\kappa}t\int_{{\mathbb{V}}_{0}^{d+1}}\frac{n^{d+2}(1+{\langle}\xi,\eta{\rangle})^{-\frac{1}{2}}{\mathsf{w}}_{-1,{\gamma}}(y,s)\mathrm{d}{\mathsf{m}}(y,s)}{\sqrt{{\mathsf{w}}_{{\gamma},d}(n;t)}\sqrt{{\mathsf{w}}_{{\gamma},d}(n;s)}\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))\big{)}^{{\kappa}-{\beta}}}$
$\displaystyle+c_{\kappa}t\int_{{\mathbb{V}}_{0}^{d+1}}\frac{n^{d+1}(1+{\langle}\xi,\eta{\rangle})^{-\frac{1}{2}}{\mathsf{w}}_{-1,{\gamma}}(y,s)\mathrm{d}{\mathsf{m}}(y,s)}{\sqrt{t}\sqrt{{\mathsf{w}}_{{\gamma},d}(n;t)}\sqrt{{\mathsf{w}}_{{\gamma},d}(n;s)}\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))\big{)}^{{\kappa}-{\beta}-1}},$

which, for $t\in[\frac{\delta}{n},1-\frac{\delta}{n}]$ so that $1/\sqrt{t}\leq
n$, is further bounded by

$c_{\kappa}t\int_{{\mathbb{V}}_{0}^{d+1}}|f(y,s)|\frac{n^{d+2}(1+{\langle}\xi,\eta{\rangle})^{-\frac{1}{2}}{\mathsf{w}}_{-1,{\gamma}}(y,s)\mathrm{d}{\mathsf{m}}(y,s)}{\sqrt{t}\sqrt{{\mathsf{w}}_{{\gamma},d}(n;t)}\sqrt{{\mathsf{w}}_{{\gamma},d}(n;s)}\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))\big{)}^{{\kappa}-{\beta}-1}}\leq
c_{\kappa}t\,n^{2},$

by applying (3.18) with $p=1$ and $\rho=\frac{1}{2}$. Thus, we obtain

$\left|\frac{1}{t}D_{i,j}^{2}f(x,t)\right|\chi_{n,\delta}(t)\leq
cn^{2}f_{{\beta},n}^{\ast}(x,t),$

from which the proof follows readily from the boundedness of
$f_{{\beta},n}^{*}$. Using (3.17) and $\|f\|_{\infty}$ instead of
$f_{{\beta},n^{*}}$, the above proof also applies to the case $p=\infty$. ∎

The inequalities in (3.27) are the cases of $\ell=1$ and $\ell=2$ of (2.4). We
now show that the inequality (2.4) does not hold in general for $\ell>2$, we
consider for example the polynomial $f_{0}(x,t)=x_{1}$. It is easy to see that

$D_{1,j}^{2\ell}f_{0}(x,t)=(-1)^{\ell}x_{1}\quad\hbox{and}\quad
D_{1,j}^{2\ell-1}f_{0}(x,t)=(-1)^{\ell}x_{j},$

so that $(\sqrt{t})^{-\ell}D_{i,j}^{\ell}f_{0}(t\xi,t)$ has a singularity
$t^{-\ell/2+1}$ for $\ell>2$. Hence, taking
${\mathsf{w}}={\mathsf{w}}_{-1,{\gamma}}$ on ${\mathbb{V}}_{0}^{3}$, for
example, it is easy to see that
$\|t^{-\ell/2}D_{i,j}^{\ell}f_{0}\|_{2,{\mathsf{w}}_{-1,{\gamma}}}$ is
infinite if $\ell\geq 3$. Thus, (2.4) does not hold for $\ell\geq 3$ in
general.

Finally, for $\ell=1,2,\ldots$ and any doubling weight ${\mathsf{w}}$, the
inequality (2.3) holds. Indeed, the case $\ell=1$ of (2.3) follows readily
from, and in fact weaker than, (3.27). The case $\ell>1$ of (2.3) follows
readily from iteration.

###  3.5. Self-adjoint of $-\Delta_{-1,{\gamma}}$

In this subsection we give a proof of Proposition 2.3 and Corollary 2.4.

###### Proof of Proposition 2.3.

First we assume ${\gamma}>0$. The differential operator can be written as

$\Delta_{0,{\gamma}}=\frac{1}{t^{d-2}(1-t)^{\gamma}}\frac{\partial}{\partial
t}\left(t^{d-1}(1-t)^{{\gamma}+1}\frac{\partial}{\partial
t}\right)+t^{-1}\Delta_{0}^{(\xi)}.$

Setting $x=t\xi$, the integral over the conic surface can be written as

$\int_{{\mathbb{V}}_{0}^{d+1}}f(x,t)\mathrm{d}{\mathsf{m}}(x,t)=\int_{0}^{1}t^{d-1}\int_{{\mathbb{S}^{d-1}}}f(t\xi,t)\mathrm{d}\sigma(\xi)\mathrm{d}t.$

Hence, we can write

$\displaystyle\int_{{\mathbb{V}}_{0}^{d+1}}$
$\displaystyle\Delta_{0,{\gamma}}f(x,t)\cdot
g(x,t){\mathsf{w}}_{-1,{\gamma}}(t)\mathrm{d}{\mathsf{m}}(x,t)$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\int_{{\mathbb{S}^{d-1}}}\frac{\partial}{\partial
t}\left(t^{d-1}(1-t)^{{\gamma}+1}\frac{\partial f}{\partial t}\right)\cdot
g(t\xi,t)\mathrm{d}\sigma(\xi)\mathrm{d}t$ $\displaystyle+\sum_{1\leq i<j\leq
d}\int_{0}^{1}t^{d-2}\int_{{\mathbb{S}^{d-1}}}D_{i,j}^{2}f(t\xi,t)\cdot
g(t\xi,t)\mathrm{d}\sigma(\xi){\mathsf{w}}_{-1,{\gamma}}(t)\mathrm{d}t.$

We then integrate by parts in $t$ variable, using the assumption that
${\gamma}>0$ and $d\geq 2$, in the first integral in the right-hand side, and
applying (cf. [9, Proposition 1.8.4])

(3.29)
$\int_{{\mathbb{S}^{d-1}}}f(\xi)D_{i,j}g(\xi)\mathrm{d}\sigma(\xi)=-\int_{{\mathbb{S}^{d-1}}}D_{i,j}f(\xi)g(\xi)\mathrm{d}\sigma(\xi),\quad
1\leq i,j\leq d,$

on the second integral in the right-hand side to establish the desired
formula. Analytic continuation shows that the identity holds for
${\gamma}>-1$. ∎

###### Proof of Corollary 2.4.

The inequality follows from setting $g=f$ in the Proposition 2.3 and the
Cauchy inequality. If $f\in\Pi_{n}$, then the inequality (2.6) shows that the
right-hand side of the stated inequality is bounded by
$cn^{2}\left\|f\right\|_{{\mathsf{w}},2}^{2}$, so that each term of the left-
hand sider is bounded by the same quantity and, hence, we obtain (2.2) and
(2.4) for $\ell=1$ and $p=2$. ∎

##  4\. Bernstein inequalities on the cone

We work with the solid cone ${\mathbb{V}}^{d+1}$ in this section. In the first
subsection, we recall the orthogonal polynomials and their kernels on the
cone. Instead of estimating the derivatives of the kernels as in the previous
section, we rely on an intimate connection between analysis on the cone and on
the conic surface, which is explained in the second subsection. The proof of
the Bernstein inequalities on the cone, based on the connection, is given in
the third subsection. Finally, the proof of Theorem 2.7 and its corollary is
in the fourth subsection.

###  4.1. Orthogonal polynomials and localized kernels

Let $\Pi_{n}^{d}$ denote the space of polynomials of degree at most $n$ in $d$
variables. For $\mu>-\frac{1}{2}$ and ${\gamma}>-1$, we define the weight
function $W_{\mu,{\gamma}}$ by

$W_{\mu,{\gamma}}(x,t)=(t^{2}-\|x\|^{2})^{\mu-\frac{1}{2}}(1-t)^{\gamma},\qquad\|x\|\leq
t,\quad 0\leq t\leq 1.$

Orthogonal polynomials with respect to $W_{\mu,{\gamma}}$ on
${\mathbb{V}}^{d+1}$ are studied in [25], which are orthogonal with respect to
the inner product

${\langle}f,g{\rangle}_{W_{\mu,{\gamma}}}={\mathsf{b}}_{\mu,{\gamma}}\int_{{\mathbb{V}}^{d+1}}f(x,t)g(x,t)W_{\mu,{\gamma}}\mathrm{d}x\mathrm{d}t.$

Let ${\mathcal{V}}_{n}({\mathbb{V}}^{d+1},W_{\mu,{\gamma}})$ be the space of
orthogonal polynomials of degree $n$. Then

$\dim{\mathcal{V}}_{n}({\mathbb{V}}^{d+1},W_{\mu,{\gamma}})=\binom{n+d+1}{n},\quad
n=0,1,2,\ldots.$

An orthogonal basis of
${\mathcal{V}}_{n}({\mathbb{V}}^{d+1},W_{\mu,{\gamma}})$ can be given in terms
of the Jacobi polynomials and the orthogonal polynomials on the unit ball. For
$m=0,1,2,\ldots$, let
$\\{P_{\mathbf{k}}^{m}(W_{\mu}):|{\mathbf{k}}|=m,{\mathbf{k}}\in{\mathbb{N}}_{0}^{d}\\}$
be an orthonormal basis of ${\mathcal{V}}_{n}({\mathbb{B}}^{d},W_{\mu})$ on
the unit ball. Let

(4.1)
${\mathbf{J}}_{m,{\mathbf{k}}}^{n}(x,t):=P_{n-m}^{(2m+2\mu+d-1,{\gamma})}(1-2t)t^{m}P_{\mathbf{k}}^{m}\left(W_{\mu};\frac{x}{t}\right),\quad$

Then $\\{{\mathbf{J}}_{m,{\mathbf{k}}}^{n}(x,t):|{\mathbf{k}}|=m,\,0\leq m\leq
n\\}$ is an orthogonal basis of
${\mathcal{V}}_{n}({\mathbb{V}}^{d+1},W_{\mu,{\gamma}})$. The following
theorem is established in [25, Theorem 3.2].

######  Theorem 4.1.

Let $\mu>-\tfrac{1}{2}$, ${\gamma}>-1$ and $n\in{\mathbb{N}}_{0}$. Let
${\mathfrak{D}}_{\mu,{\gamma}}$ be the second order differential operator
defined in (2.14). Then the polynomials in
${\mathcal{V}}_{n}({\mathbb{V}}^{d+1},W_{\mu,{\gamma}})$ are eigenfunctions of
${\mathfrak{D}}_{\mu,{\gamma}}$; more precisely,

(4.2) ${\mathfrak{D}}_{\mu,{\gamma}}u=-n(n+2\mu+{\gamma}+d)u,\qquad\forall
u\in{\mathcal{V}}_{n}({\mathbb{V}}^{d+1},W_{\mu,{\gamma}}).$

The reproducing kernel of the space
${\mathcal{V}}_{n}({\mathbb{V}}^{d+1},W_{\mu,{\gamma}})$, denoted by
${\mathbf{P}}_{n}(W_{{\gamma},\mu};\cdot,\cdot)$, can be written in terms of
the above basis,

${\mathbf{P}}_{n}\big{(}W_{\mu,{\gamma}};(x,t),(y,s)\big{)}=\sum_{m=0}^{n}\sum_{|{\mathbf{k}}|=n}\frac{{\mathbf{J}}_{m,{\mathbf{k}}}^{n}(x,t){\mathbf{J}}_{m,{\mathbf{k}}}^{n}(y,s)}{{\langle}{\mathbf{J}}_{m,{\mathbf{k}}}^{n},{\mathbf{J}}_{m,{\mathbf{k}}}^{n}{\rangle}_{W_{\mu,{\gamma}}}}.$

It is the kernel of the projection
$\operatorname{proj}_{n}(W_{\mu,{\gamma}}):L^{2}({\mathbb{V}}^{d+1},W_{\mu,{\gamma}})\to{\mathcal{V}}_{n}({\mathbb{V}}^{d+1},W_{\mu,{\gamma}})$,

$\operatorname{proj}_{n}(W_{\mu,{\gamma}};f)=\int_{{\mathbb{V}}^{d+1}}f(y,s){\mathbf{P}}_{n}\big{(}W_{\mu,{\gamma}};\,\cdot,(y,s)\big{)}W_{\mu,{\gamma}}(s)\mathrm{d}y\mathrm{d}s.$

This kernel enjoys an addition formula that can be written as a triple
integral over $[-1,1]^{3}$ for $\mu>0$; see [25, Theorem 4.3]. For our
purpose, we only need the closed-form formula for the degenerate case $\mu=0$;
see [27, Theorem 5.8].

######  Theorem 4.2.

Let $d\geq 2$, $\mu=0$ and ${\gamma}\geq-\frac{1}{2}$. Then, for
$n=0,1,2,\ldots$,

(4.3)
$\displaystyle{\mathbf{P}}_{n}\big{(}W_{0,{\gamma}};(x,t),(y,s)\big{)}=c_{{\gamma},d}$
$\displaystyle\int_{[-1,1]^{2}}\left[Z_{n}^{({\gamma}+d-\frac{1}{2},-\frac{1}{2})}(\xi(x,t,y,s;1,{\mathbf{v}}))\right.$
$\displaystyle\qquad\quad\left.+Z_{n}^{({\gamma}+d-\frac{1}{2},-\frac{1}{2})}(\xi(x,t,y,s;-1,{\mathbf{v}}))\right]$
$\displaystyle\times(1-v_{1}^{2})^{\frac{d-1}{2}-1}(1-v_{2}^{2})^{{\gamma}-\frac{1}{2}}\mathrm{d}{\mathbf{v}},$

where $c_{{\gamma},d}$ is a constant, so that ${\mathbf{P}}_{0}=1$ and
$\xi(x,t,y,s;u,{\mathbf{v}})\in[-1,1]$ is defined by

(4.4) $\displaystyle\xi(x,t,y,s;u,{\mathbf{v}})=$
$\displaystyle\,v_{1}\sqrt{\tfrac{1}{2}\left(ts+{\langle}x,y{\rangle}+\sqrt{t^{2}-\|x\|^{2}}\sqrt{s^{2}-\|y\|^{2}}\,u\right)}$
$\displaystyle+v_{2}\sqrt{1-t}\sqrt{1-s}.$

Let $\widehat{a}$ be an admissible cut-off function. For $(x,t)$,
$(y,s)\in{\mathbb{V}}^{d+1}$, the localized kernel
${\mathbf{L}}_{n}(W_{0,{\gamma}};\cdot,\cdot)$ is defined by

${\mathbf{L}}_{n}\left(W_{0,{\gamma}};(x,t),(y,s)\right)=\sum_{j=0}^{\infty}\widehat{a}\left(\frac{j}{n}\right){\mathbf{P}}_{j}\left(W_{0,{\gamma}};(x,t),(y,s)\right).$

Like the kernel ${\mathsf{L}}_{n}$ on the conic surface, the kernel
${\mathbf{L}}_{n}(W_{0,{\gamma}})$ is also highly localized via the distance
function ${\mathsf{d}}_{{\mathbb{V}}}(\cdot,\cdot)$ on the cone. There is,
however, no need to carry out the estimate for these kernels as in the case of
the conic surface. Indeed, there is a direct connection between the kernels
for $W_{0,{\gamma}}$ on ${\mathbb{V}}^{d+1}$ and those for $w_{-1,{\gamma}}$
on the conic surface ${\mathbb{V}}_{0}^{d+2}$ in ${\mathbb{R}}^{d+2}$, as will
be explained in the next subsection.

###  4.2. The analysis on ${\mathbb{V}}^{d+1}$ and on ${\mathbb{V}}_{0}^{d+2}$

Let $(x,t)$ and $(y,s)$ be the elements of the solid cone
${\mathbb{V}}^{d+1}$. We shall adopt the notation

(4.5) $X:=\big{(}x,\sqrt{t^{2}-\|x\|^{2}}\big{)},\quad
Y:=\big{(}y,\sqrt{s^{2}-\|y\|^{2}}\big{)},\quad
Y_{*}:=\big{(}y,-\sqrt{s^{2}-\|y\|^{2}}\big{)}.$

It is evident that $\|X\|=t$, $\|Y\|=\|Y_{*}\|=s$, so that $(X,t)$, $(Y,s)$
and $(Y_{*},s)$ are elements of the conic surface ${\mathbb{V}}_{0}^{d+2}$ in
${\mathbb{R}}^{d+2}$.

######  Proposition 4.3.

Let ${\mathsf{L}}_{n}(w_{-1,{\gamma}};\cdot,\cdot)$ be the kernel defined in
(3.5) for ${\mathbb{V}}_{0}^{d+2}$. Then

(4.6)
${\mathbf{L}}_{n}\left(W_{0,{\gamma}};(x,t),(y,s)\right)={\mathsf{L}}_{n}\big{(}{\mathsf{w}}_{-1,{\gamma}};(X,t),(Y,s)\big{)}+{\mathsf{L}}_{n}\big{(}{\mathsf{w}}_{-1,{\gamma}};(X,t),(Y_{*},s)\big{)}.$

###### Proof.

By (4.3), we can write

${\mathbf{L}}_{n}\left(W_{0,{\gamma}};(x,t),(y,s)\right)={\mathbf{L}}_{n}^{+}\left(W_{0,{\gamma}};(x,t),(y,s)\right)+{\mathbf{L}}_{n}^{-}\left(W_{0,{\gamma}};(x,t),(y,s)\right),$

where, with ${\lambda}={\gamma}+d$,

$\displaystyle{\mathbf{L}}_{n}^{\pm}\left(W_{0,{\gamma}};(x,t),(y,s)\right)=c_{0,{\gamma},d}\int_{[-1,1]^{2}}$
$\displaystyle
L_{n}^{({\lambda}-\frac{1}{2},-\frac{1}{2})}\left(2\xi(x,t,y,s;\pm
1,v)^{2}-1\right)$
$\displaystyle\qquad\times(1-v_{1}^{2})^{\frac{d-1}{2}-1}(1-v_{2}^{2})^{{\gamma}-\frac{1}{2}}\mathrm{d}v.$

Let $X=tX^{\prime}$ and $Y=sY^{\prime}$. Then
$X^{\prime}=(x^{\prime},\sqrt{1-\|x^{\prime}\|^{2}})\in\SS^{d}$ and
$Y^{\prime}=(y^{\prime},\sqrt{1-\|y^{\prime}\|^{2}})\in\SS^{d}$. Then, using
the explicit formula of $\xi$ in (4.4) and $\zeta$ in (3.3), it follows

$\xi(x,t,y,s;1,v)=v_{1}\sqrt{ts}\sqrt{\tfrac{1}{2}(1+{\langle}X^{\prime},Y^{\prime}{\rangle})}+v_{2}\sqrt{1-t}\sqrt{1-s}=\zeta(X,t,Y,s;1,v).$

The similar identity holds for $\xi(x,t,y,s;-1,v)$ if we replace $Y$ by
$Y_{*}$ in the right-hand side. As a consequence, comparing with (3.5), we
obtain

$\displaystyle{\mathbf{L}}_{n}^{+}\left(W_{0,{\gamma}};(x,t),(y,s)\right)$
$\displaystyle={\mathsf{L}}_{n}\left({\mathsf{w}}_{-1,{\gamma}};(X,t),(Y,s)\right),$
$\displaystyle{\mathbf{L}}_{n}^{-}\left(W_{0,{\gamma}};(x,t),(y,s)\right)$
$\displaystyle={\mathsf{L}}_{n}\left({\mathsf{w}}_{-1,{\gamma}};(X,t),(Y_{*},s)\right),$

from which the stated identity follows readily. ∎

The connection between the cone ${\mathbb{V}}^{d+1}$ and the conic surface
${\mathbb{V}}_{0}^{d+2}$ is further manifested in the following integral
relation.

######  Proposition 4.4.

Let $f:{\mathbb{R}}^{d+1}\mapsto{\mathbb{R}}$ be a continuous function. Let
$x_{d+1}=\sqrt{t^{2}-\|x\|^{2}}$ for $\|x\|\leq t$. Then

$\int_{{\mathbb{V}}_{0}^{d+2}}f(y,s)w(s)\mathrm{d}{\mathsf{m}}(y,s)=\int_{{\mathbb{V}}^{d+1}}\big{[}f\big{(}(x,x_{d+1}),t\big{)}+f\big{(}(x,-x_{d+1}),t\big{)}\big{]}tw(t)\frac{\mathrm{d}x\mathrm{d}t}{\sqrt{t^{2}-\|x\|^{2}}}.$

###### Proof.

Let $\mathrm{d}\sigma$ be the surface measure on $\SS^{d}$. We write the
integral on ${\mathbb{V}}_{0}^{d+1}$ as

$\displaystyle\int_{{\mathbb{V}}_{0}^{d+2}}f(y,s)w(s)\mathrm{d}{\mathsf{m}}(y,s)=\int_{0}^{1}s^{d}\int_{\SS^{d}}f(s\xi,s)w(s)\mathrm{d}\sigma(\xi)\mathrm{d}s$
$\displaystyle=\int_{0}^{1}s^{d}\int_{{\mathbb{B}}^{d}}\left[f\big{(}s(u,\sqrt{1-\|u\|^{2}}),s\big{)}+f\big{(}s(u,-\sqrt{1-\|u\|^{2}}),s\big{)}\right]\frac{\mathrm{d}u}{\sqrt{1-\|u\|^{2}}}\mathrm{d}xw(s)\mathrm{d}\sigma,$

where the second identity uses [9, (A.5.4)]. Rewriting the right-hand side as
the integral over ${\mathbb{V}}^{d+1}$, with $x=su$, proves the stated
identity. ∎

The doubling weight on ${\mathbb{V}}^{d+1}$ is defined as usual via the
distant function ${\mathsf{d}}_{{\mathbb{V}}}$. The latter can be defined in
terms of the distance function on the conic surface ${\mathbb{V}}_{0}^{d+2}$.
Indeed, using $X$ and $Y$ in (4.5), we define

(4.7)
${\mathsf{d}}_{{\mathbb{V}}}((x,t),(y,s)):=\arccos\left(\sqrt{\frac{{\langle}X,Y{\rangle}+ts}{2}}+\sqrt{1-t}\sqrt{1-s}\right),$

which is indeed a distance function on the solid cone ${\mathbb{V}}^{d+1}$, as
shown in [27]. In particular, we have the relation

(4.8)
${\mathsf{d}}_{{\mathbb{V}}^{d+1}}((x,t),(y,s))={\mathsf{d}}_{{\mathbb{V}}_{0}^{d+2}}((X,t),(Y,s)),\quad(x,t),(y,s)\in{\mathbb{V}}^{d+1},$

where we write ${\mathsf{d}}_{{\mathbb{V}}}={\mathsf{d}}_{{\mathbb{V}}^{d+1}}$
and ${\mathsf{d}}_{{\mathbb{V}}_{0}}={\mathsf{d}}_{{\mathbb{V}}_{0}^{d+2}}$ to
emphasis the dimension of the domains.

###  4.3. Proof of main results

We will need an analog of Proposition 3.6 that is of interest in itself.

######  Proposition 4.5.

Let $W$ be a doubling weight function on ${\mathbb{V}}^{d+1}$. For
$n\in{\mathbb{N}}$, let $\chi_{n,\delta}(x,t)$ denote the characteristic
function of the set

$\left\\{(x,t):\frac{\delta}{n^{2}}\leq t\leq
1-\frac{\delta}{n^{2}},\quad\sqrt{t^{2}-\|x\|^{2}}\geq\delta\frac{\sqrt{t}}{n}\right\\}.$

Then, for $f\in\Pi_{n}^{d}$, $1\leq p<\infty$, and every $\delta>0$,

(4.9) $\int_{{\mathbb{V}}^{d+1}}|f(x,t)|^{p}W(x,t)\mathrm{d}x\mathrm{d}t\leq
c_{\delta}\int_{{\mathbb{V}}^{d+1}}|f(x,t)|^{p}\chi_{n,\delta}(x,t)W(x,t)\mathrm{d}x\mathrm{d}t.$

###### Proof.

The proof uses the Marcinkiewicz-Zygmund inequality on the cone established in
[27, Theorem 5.6] that uses a maximal ${\varepsilon}$-separated subset
$\Xi_{{\mathbb{V}}}$ on ${\mathbb{V}}^{d+1}$. It is almost verbatim as the
proof of Proposition 3.6, once it is shown that if
$(x,t)\in\Xi_{{\mathbb{V}}}$ and ${\varepsilon}\approx n^{-1}$, then

(4.10) $\frac{\delta}{n^{2}}\leq t\leq
1-\frac{\delta}{n^{2}}\quad\hbox{and}\quad\sqrt{t^{2}-\|x\|^{2}}\geq\delta\frac{\sqrt{t}}{n}.$

The set $\Xi_{\mathbb{V}}$ is constructed in [27, Proposition 5.15] using the
separated sets in the $t$-variable on $[0,1]$ and in the $x^{\prime}$ variable
on ${\mathbb{B}}^{d}$, where $(x,t)=(tx^{\prime},t)\in{\mathbb{V}}^{d+1}$. The
construction is similar to that of $\Xi_{{\mathbb{V}}_{0}}$. In fact, for
${\varepsilon}>0$ and $N=\lfloor\frac{\pi}{2}{\varepsilon}^{-1}\rfloor$, we
choose the same $t_{j}$ and ${\varepsilon}_{j}$ as given in (3.14) and define

$\Xi_{{\mathbb{V}}}=\big{\\{}(t_{j}x^{\prime},t_{j}):\,x\in\Xi_{\mathbb{B}}({\varepsilon}_{j}),\,1\leq
j\leq N\big{\\}},$

where $\Xi_{\mathbb{B}}({\varepsilon}_{j})$ is the maximal
${\varepsilon}_{j}$-separated set of ${\mathbb{B}}^{d}$. Choose again
${\varepsilon}={\beta}n^{-1}$, ${\beta}>0$, so that
${\varepsilon}_{j}\sim\sqrt{t_{j}}/n$. By the choices of $t_{j}$, the bounds
for $t_{j}$ in (4.10) is immediate. Setting $x=tx^{\prime}$,
$x^{\prime}\in{\mathbb{B}}^{d}$, so that
$\sqrt{t^{2}-\|x\|^{2}}=t\sqrt{1-\|x^{\prime}\|^{2}}$, we see that the second
inequality in (4.10) is equivalent to, since
${\varepsilon}_{j}=\frac{\pi}{2}\frac{{\varepsilon}}{\sqrt{t_{j}}}$,

$\sqrt{1-\|x^{\prime}\|^{2}}\geq c\,{\varepsilon}_{j},\qquad
x^{\prime}\in\Xi_{{\mathbb{B}}}({\varepsilon}_{j}),\quad 1\leq j\leq n.$

Thus, the question is reduced to the existence of a maximal
${\varepsilon}$-separated set of ${\mathbb{B}}^{d}$ that satisfies the above
inequality. A construction is given below.

We show that there is a maximal ${\varepsilon}$-separated set
$\Xi_{\mathbb{B}}({\varepsilon})$ on ${\mathbb{B}}^{d}$ such that

(4.11) $\sqrt{1-\|u\|^{2}}\geq c{\varepsilon},\qquad
u\in\Xi_{\mathbb{B}}({\varepsilon})$

for any given ${\varepsilon}>0$. We choose
$N=\lfloor\frac{\pi}{2}{\varepsilon}^{-1}\rfloor$ as above. For $1\leq j\leq
N$ we define

${\theta}_{j}:=\frac{(2j-1)\pi}{2N},\qquad{\theta}_{j}^{-}:={\theta}_{j}-\frac{\pi}{2N}\quad\hbox{and}\quad{\theta}_{j}^{+}:={\theta}_{j}+\frac{\pi}{2N}.$

Let $r_{j}=\sin\frac{{\theta}_{j}}{2}$ and define $r_{j}^{-}$ and $r_{j}^{+}$
accordingly. In particular, $r_{1}^{-}=0$ and $r_{N}^{+}=1$. Then
${\theta}_{j+1}^{-}={\theta}_{j}^{+}$ and ${\mathbb{B}}^{d}$ can be
partitioned by

${\mathbb{B}}^{d}=\bigcup_{j=1}^{N}{\mathbb{B}}_{0}^{(j)},\quad\hbox{where}\quad{\mathbb{B}}_{0}^{(j)}:=\left\\{r\xi\in{\mathbb{B}}^{d}:r_{j}^{-}<r\leq
r_{j}^{+},\,\xi\in{\mathbb{S}^{d-1}}\right\\}.$

Let $\sigma_{j}:=(2r_{j})^{-1}\pi{\varepsilon}$. Let $\Xi_{\SS}(\sigma_{j})$
be the maximal $\sigma_{j}$-separated set of ${\mathbb{S}^{d-1}}$, such that
$\\{\SS_{\xi}(\sigma_{j}):\xi\in\Xi_{\SS}(\sigma_{j})\\}$ is a partition of
the sphere,
${\mathbb{S}^{d-1}}=\bigcup_{\eta\in\Xi_{\SS}(\sigma_{j})}\SS_{\eta}(\sigma_{j})$.
We define

$\Xi_{{\mathbb{B}}}({\varepsilon})=\left\\{r_{j}\xi:\xi\in\Xi_{\SS}(\sigma_{j}),\,1\leq
j\leq N\right\\}.$

By Definition 3.5, to show that $\Xi_{\mathbb{B}}({\varepsilon})$ is
${\varepsilon}$-separated, we need
${\mathsf{d}}_{\mathbb{B}}(r_{j}\xi,r_{k}\eta)\geq{\varepsilon}$ for any two
distinct points $r_{j}\xi$ and $r_{k}\eta$ in
$\Xi_{\mathbb{B}}({\varepsilon})$. It is known that the distance function
${\mathsf{d}}_{\mathbb{B}}$ of ${\mathbb{B}}^{d}$ is defined by

${\mathsf{d}}_{{\mathbb{B}}}(x,y)=\arccos\left({\langle}x,y{\rangle}+\sqrt{1-\|x\|^{2}}\sqrt{1-\|y\|^{2}}\right),\qquad
x,y\in{\mathbb{B}}^{d}.$

Let $x=t\xi$ and $y=s\eta$, for $t,s\in[0,1]$ and
$\xi,\eta\in{\mathbb{S}^{d-1}}$, it is easy to verify that

(4.12)
$1-\cos{\mathsf{d}}_{{\mathbb{B}}}(x,y)=1-\cos{\mathsf{d}}_{[0,1]}(t^{2},s^{2})+ts\left(1-\cos{\mathsf{d}}_{\SS}(\xi,\eta)\right)$

using $\cos{\mathsf{d}}_{[0,1]}(t,s)=\sqrt{t}\sqrt{s}+\sqrt{1-t}\sqrt{1-s}$.
We use (4.12) with $1-\cos\phi=2\sin^{2}\frac{\phi}{2}$ and also use
${\mathsf{d}}_{[0,1]}(t,s)=\frac{1}{2}|{\theta}-\phi|$ if
$t=\sin^{2}\frac{{\theta}}{2}$ and $s=\sin^{2}\frac{\phi}{2}$. If $j\neq k$
then, by (4.12),

${\mathsf{d}}_{\mathbb{B}}(r_{j}\xi,r_{k}\eta)\geq{\mathsf{d}}_{[0,1]}(r_{j}^{2},r_{k}^{2})=\frac{1}{2}|{\theta}_{j}-{\theta}_{k}|\geq\frac{\pi}{2N}\geq{\varepsilon}.$

If $j=k$, then $\xi$ and $\eta$ belong to the same $\Xi_{\SS}(\sigma_{j})$, so
that ${\mathsf{d}}_{\SS}(\xi,\eta)\geq\sigma_{j}$. Hence, using
$\frac{2}{\pi}\phi\leq\sin\phi\leq\phi$, we deduce from (4.12) that

${\mathsf{d}}_{\mathbb{B}}(r_{j}\xi,r_{j}\eta)\geq\frac{2}{\pi}r_{j}{\mathsf{d}}_{\SS}(\xi,\eta)\geq\frac{2}{\pi}r_{j}\sigma_{j}={\varepsilon}.$

Thus, $\Xi_{\mathbb{B}}({\varepsilon})$ is an ${\varepsilon}$-separated set.
Moreover, it is easy to see that $\Xi_{\mathbb{B}}({\varepsilon})$ is maximal.
Now, if $x\in\Xi_{{\mathbb{B}}}({\varepsilon})$, then $\|x\|\leq
r_{N}=\sin\frac{1}{2}{\theta}_{N}=\cos\frac{\pi}{4N}$, which implies that
$\sqrt{1-\|x\|^{2}}\geq\sin\frac{\pi}{4N}\geq c{\varepsilon}$. Thus, (4.11)
holds. This completes the proof. ∎

The unit ball ${\mathbb{B}}^{d}$ equipped with the weight function
$(1-\|u\|^{2})^{\mu-\frac{1}{2}}$ is also a localizable homogeneous space in
the sense of [27], since it possesses highly localized kernels [21]. As a
result, our construction of the maximal ${\varepsilon}$-separated set on the
unit ball also leads to the following proposition, which will not be used in
this paper but is recorded here for possible future use.

######  Proposition 4.6.

Let $W$ be a doubling weight function on ${\mathbb{B}}^{d}$. For
$n\in{\mathbb{N}}$, let $\chi_{n,\delta}(x,t)$ denote the characteristic
function of the set $\\{x\in{\mathbb{B}}^{d}:\|x\|\leq
1-\frac{\delta}{n^{2}}\\}$. Then for $f\in\Pi_{n}^{d}$, $1\leq p<\infty$, and
every $\delta>0$,

(4.13) $\int_{{\mathbb{B}}^{d}}|f(x)|^{p}W(x)\mathrm{d}x\leq
c_{\delta}\int_{{\mathbb{B}}^{d}}|f(x)|^{p}\chi_{n,\delta}(x)W(x)\mathrm{d}x.$

The proof of the Bernstein inequalities on the cone follows closely the proof
on the conic surface. We will use the maximal function $f_{{\beta},n}^{\ast}$
in (3.12), defined similarly by

(4.14)
$f_{{\beta},n}^{\ast}(x,t)=\max_{(y,s)\in{\mathbb{V}}^{d+1}}\frac{|f(y,s)|}{(1+n{\mathsf{d}}_{{\mathbb{V}}}((x,t),(y,s)))^{\beta}}.$

using ${\mathsf{d}}_{{\mathbb{V}}}((x,t),(y,s))$ and it remains bounded for
any doubling weight $W$ on ${\mathbb{V}}^{d+1}$. It again satisfies

(4.15) $\|f\|_{p,W}\leq\|f_{{\beta},n}^{\ast}\|_{p,W}\leq c\|f\|_{p,W}$

for the doubling weight $W$ on ${\mathbb{V}}^{d+1}$ and $1\leq p\leq\infty$.

######  Lemma 4.7.

Let $f$ be defined on ${\mathbb{V}}^{d+1}$. Define

(4.16)
$F\big{(}(x,x_{d+1}),t\big{)}=f(x,t),\qquad\big{(}(x,x_{d+1}),t\big{)}\in{\mathbb{V}}_{0}^{d+2},$

so that $f$ can be regarded as a function on ${\mathbb{V}}_{0}^{d+2}$. Then

(4.17)
$f_{{\beta},n}^{*}(x,t)=F_{{\beta},n}^{*}\big{(}(x,x_{d+1}),t\big{)},\qquad
x\in{\mathbb{V}}^{d+1},\quad(x,x_{d+1})\in{\mathbb{V}}_{0}^{d+2}.$

###### Proof.

Since $F$ is independent of $x_{d+1}$, it follows by the definition of
$F_{{\beta},n}^{*}$ in (3.12) that
$F_{{\beta},n}^{*}((x,x_{d+1}),t)=F_{{\beta},n}^{*}((x,-x_{d+1}),t)$. In
particular, we can take the maximum in the definition of $F_{{\beta},n}^{*}$
over
${\mathbb{V}}_{0,+}^{d+2}=\\{((y,y_{d+1}),t)\in{\mathbb{V}}_{0}^{d+2}:y_{d+1}\geq
0\\}$ instead of over ${\mathbb{V}}_{0}^{d+1}$. Then (4.17) follows from using
(4.8) and replacing the maximum in (4.14) as over
$(Y,s)\in{\mathbb{V}}_{0,+}^{d+2}$. ∎

###### Proof of Theorem 4.2.

Let $f$ be a polynomial of degree at most $n$. Then

$f(x,t)=\int_{{\mathbb{V}}^{d+1}}f(y,s){\mathbf{L}}_{n}\left(W_{{\gamma},0};(x,t),(y,s)\right)W_{{\gamma},0}(y,s)\mathrm{d}y\mathrm{d}s.$

Let $T_{x,t}$ be a differential operator of $(x,t)$ variables. Then, by (4.6)
and using the maximal function, we obtain from (4.17)

$\displaystyle|T_{x,t}f(x,t)|\leq f_{{\beta},n}^{*}(x,t)$
$\displaystyle\left[\int_{{\mathbb{V}}^{d+1}}\frac{\left|T_{x,t}{\mathsf{L}}_{n}\left(w_{-1,{\gamma}};(X,t),(Y,s)\right)\right|}{\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}\big{(}(X,t),(Y,s)\big{)}\big{)}^{\beta}}W_{{\gamma},0}(y,s)\mathrm{d}y\mathrm{d}s\right.$
$\displaystyle\left.+\int_{{\mathbb{V}}^{d+1}}\frac{\left|T_{x,t}{\mathsf{L}}_{n}\left(w_{-1,{\gamma}};(X,t),(Y_{*},s)\right)\right|}{\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}\big{(}(X,t),(Y_{*},s)\big{)}\big{)}^{\beta}}W_{{\gamma},0}(y,s)\mathrm{d}y\mathrm{d}s\right].$

By the identity in Proposition 4.4, we conclude that

(4.18) $\displaystyle|T_{x,t}f(x,t)|\leq
f_{{\beta},n}^{*}(x,t)\int_{{\mathbb{V}}_{0}^{d+2}}\frac{\left|T_{x,t}{\mathsf{L}}_{n}\left(w_{-1,{\gamma}};(X,t),(y,s)\right)\right|}{\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}\big{(}(X,t),(y,s)\big{)}\big{)}^{\beta}}\mathrm{d}{\mathsf{m}}(y,s).$

The integral in the right-hand side has been estimated in the proof of
Theorems 3.10 and 3.12 in the previous subsection. Hence, using (4.15) and
(4.9) when necessary, we have proved the inequalities (2.9) for
$\partial_{t}$, (2.10) and (2.11) for $D_{i,j}$, $1\leq i,j\leq d$.

For the $\partial_{x_{j}}$ we make the following observation. If
$G(X)=g(x,\sqrt{t^{2}-\|x\|^{2}})$ for $x\in{\mathbb{R}}^{d}$ and $|x|\leq t$,
then

$\sqrt{t^{2}-\|x\|^{2}}\partial_{j}G=\sqrt{t^{2}-\|x\|^{2}}\partial_{j}g-x_{j}\partial_{d+1}g=D_{j,d+1}^{(X)}G$

for $1\leq j\leq d$. Consequently, following the procedure in the previous
paragraph and using the proof of Theorem 3.12 for $D_{j,d+1}$, we conclude
that

(4.19) $\left\|(\Phi\partial_{x_{j}})^{\ell}f\right\|_{p,W}\leq
cn^{\ell}\|f\|_{p,W}\quad\hbox{and}\quad\left\|\frac{1}{\sqrt{t}^{\ell}}(\Phi\partial_{x_{j}})^{\ell}f\right\|_{p,W}\leq
cn^{\ell}\|f\|_{p,W},$

where the first inequality holds for all $\ell\in{\mathbb{N}}$ and the second
one holds for $\ell=1,2$. Furthermore, the estimate in the Lemma 3.11 shows
that

$\displaystyle\frac{\left|\partial_{x_{j}}{\mathsf{L}}_{n}\left(w_{-1,{\gamma}};(X,t),(y,s)\right)\right|}{\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}\big{(}(X,t),(y,s)\big{)}\big{)}^{\beta}}$
$\displaystyle\leq\frac{c_{\kappa}n^{d+1}\sqrt{t}\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}\big{(}(X,t),(y,s)\big{)}\big{)}^{-{\kappa}+{\beta}}}{\Phi(x,t)\sqrt{{\mathsf{w}}_{{\gamma},d}(n;t}\sqrt{{\mathsf{w}}_{{\gamma},d}(n;s}}$
$\displaystyle\leq\frac{c_{\kappa}n^{d+2}\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}\big{(}(X,t),(y,s)\big{)}\big{)}^{-{\kappa}+{\beta}}}{\sqrt{{\mathsf{w}}_{{\gamma},d}(n;t}\sqrt{{\mathsf{w}}_{{\gamma},d}(n;s}},$

if $(x,t)$ satisfies $\Phi(x,t)\geq\delta\sqrt{t}n^{-1}$. Consequently,
following the proof of Theorem 3.12, we conclude that

$\left|\partial_{x_{j}}f(x,t)\right|\leq
cn^{2}f_{{\beta},n}^{*}(x,t)\chi_{n,\delta}(x,t),\quad 1\leq j\leq d,$

where $\chi_{n,\delta}$ is defined in Proposition 4.5. Consequently, by (4.9)
and (4.15), we have proved the inequality
$\left\|\partial_{x_{j}}f\right\|_{p,W}\leq cn^{2}\|f\|_{p,W}$. Iterating this
inequality proves the first inequality in (2.12). Using the identity

$\left(\Phi(x,t)\partial_{x_{j}}\right)^{2}=(\Phi(x,t))^{2}\partial_{x_{j}}^{2}-x_{j}\partial_{x_{j}},$

the first inequality in (2.12) with $\ell=1$ and the first inequality in
(4.19), we conclude that

$\left\|\Phi^{2}\partial_{x_{j}}^{2}f\right\|_{p,W}\leq\left\|(\Phi\partial_{x_{j}})^{2}f\right\|_{p,W}+\left\|\partial_{x_{j}}f\right\|_{p,W}\leq
cn^{2}\|f\|_{p,W},$

which proves the second inequality in (2.12) with $\ell=2$. Since $\Phi^{2}$
is a polynomial of $(x,t)$, the second inequality (2.12) for $\ell>2$ can be
derived by iteration from the case $\ell=1$ and $2$. Finally, using the second
inequality of (4.19), the same argument shows that

$\left\|\frac{1}{t}\Phi^{2}\partial_{x_{j}}^{2}f\right\|_{p,W}\leq\left\|\frac{1}{t}(\Phi\partial_{x_{j}})^{2}f\right\|_{p,W}+\left\|\frac{x_{j}}{t}\partial_{x_{j}}f\right\|_{p,W}\leq
cn^{2}\|f\|_{p,W},$

where we have used $|x_{j}|\leq\|x\|\leq t$. This completes the proof. ∎

###  4.4. Self-adjoint of the spectral operator on the cone

We prove Theorem 2.7 that shows the differential operator
${\mathfrak{D}}_{\mu,{\gamma}}$ defined in (2.14) is self-adjoint. We start
with a representation ${\mathfrak{D}}_{\mu,{\gamma}}$ that is of interest in
itself. It shows, in particular, that the operator
${\mathfrak{D}}_{\mu,{\gamma}}$ can be written in terms of the first-order
differential operators appeared in Theorem 2.5.

######  Lemma 4.8.

The operator ${\mathcal{D}}_{\mu,{\gamma}}$ can be written as

(4.20) $\displaystyle{\mathfrak{D}}_{\mu,{\gamma}}$
$\displaystyle=\frac{1}{t^{d}W_{\mu,{\gamma}}(x,t)}(\partial_{t}+t^{-1}{\langle}x,\nabla_{x}{\rangle})t^{d+1}W_{\mu,{\gamma}+1}(x,t)(\partial_{t}+t^{-1}{\langle}x,\nabla_{x}{\rangle})$
$\displaystyle+t^{-1}\bigg{(}\sum_{i=1}^{d}\frac{1}{W_{\mu,{\gamma}}(x,t)}\partial_{x_{i}}\big{(}W_{\mu+1,{\gamma}}(x,t)\partial_{x_{i}}\big{)}+\sum_{1\leq
i<j\leq d}(D_{i,j}^{(x)})^{2}\bigg{)}.$

###### Proof.

We start with an observation that follows from a quick computation,

$\displaystyle
t(\partial_{t}+t^{-1}{\langle}x,\nabla_{x}{\rangle})(t^{2}-\|x\|^{2})^{\mu-\frac{1}{2}}$
$\displaystyle=t(2\mu-1)(t^{2}-\|x\|^{2})^{\mu-\frac{3}{2}}(t-t^{-1}{\langle}x,x{\rangle})$
$\displaystyle=(2\mu-1)(t^{2}-\|x\|^{2})^{\mu-\frac{1}{2}},$

from which it follows that

$\displaystyle\frac{1}{t^{d}W_{\mu,{\gamma}}(x,t)}\left((\partial_{t}+t^{-1}{\langle}x,\nabla_{x}{\rangle})t^{d+1}W_{\mu,{\gamma}+1}(x,t)(\partial_{t}+t^{-1}{\langle}x,\nabla_{x}{\rangle})\right)$
$\displaystyle\qquad=t^{-1}(2\mu+d){\langle}x,\nabla_{x}{\rangle}+(2\mu+d)\partial_{t}-(2\mu+d+{\gamma}+1)(t\partial_{t}+{\langle}x,\nabla{\rangle})$
$\displaystyle\qquad\quad+t(1-t)(t^{-1}{\langle}x,\nabla_{x}{\rangle}+\partial_{t})^{2}.$

The last term can be written as

$\displaystyle
t(1-t)(\partial_{t}+t^{-1}{\langle}x,\nabla_{x}{\rangle})^{2}=(1-t)(t\partial_{t}+{\langle}x,\nabla_{x}{\rangle})(\partial_{t}+t^{-1}{\langle}x,\nabla_{x}{\rangle})$
$\displaystyle\quad=t(1-t)\partial_{t}^{2}+2(1-t){\langle}x,\nabla_{x}{\rangle}\partial_{t}-{\langle}x,\nabla{\rangle}^{2}+{\langle}x,\nabla_{x}{\rangle}+t^{-1}\left({\langle}x,\nabla{\rangle}^{2}-{\langle}x,\nabla_{x}{\rangle}\right).$

Putting together shows that

$\displaystyle{\mathfrak{D}}_{\mu,{\gamma}}=$
$\displaystyle\frac{1}{t^{d}W_{\mu,{\gamma}}(x,t)}\left((t^{-1}{\langle}x,\nabla_{x}{\rangle}+\partial_{t})t^{d+1}W_{\mu,{\gamma}+1}(x,t)(t^{-1}{\langle}x,\nabla_{x}{\rangle}+\partial_{t})\right)$
$\displaystyle+t^{-1}\left(t^{2}\Delta_{x}-{\langle}x,\nabla_{x}{\rangle}^{2}-(2\mu+d-1){\langle}x,\nabla_{x}{\rangle}\right).$

To conclude the proof, we need the following identity,

$\displaystyle
t^{2}\Delta_{x}-{\langle}x,\nabla_{x}{\rangle}^{2}-(2\mu+d-1){\langle}x,\nabla_{x}{\rangle}$
$\displaystyle\quad=\frac{1}{W_{\mu,{\gamma}}(x,t)}\sum_{i=1}^{d}\partial_{x_{i}}\big{(}W_{\mu+1,{\gamma}}(x,t)\partial_{x_{i}}\big{)}+\sum_{1\leq
i<j\leq d}(D_{i,j}^{(x)})^{2},$

which we claim to hold. Indeed, if we dilate this identity by $x=ty$, and use
the relations $t^{2}\Delta_{x}=\Delta_{y}$,
${\langle}x,\nabla_{x}{\rangle}={\langle}y,\nabla_{y}{\rangle}$,
$D_{i,j}^{(x)}=D_{i,j}^{(y)}$, and

(4.21)
$\frac{1}{W_{\mu,{\gamma}}(x,t)}\partial_{x_{i}}\left(W_{\mu+1,{\gamma}}(x,t)\partial_{x_{i}}\right)=\frac{1}{\varpi_{\mu}(y)}\partial_{y_{i}}\left(\varpi_{\mu+1}(y)\partial_{y_{i}}\right),$

where $\varpi_{\mu}(y)=(1-\|y\|^{2})^{\mu-\frac{1}{2}}$, we see that the
claimed identity becomes

$\displaystyle\Delta_{y}-{\langle}y,\nabla_{y}{\rangle}^{2}-(2\mu+d-1){\langle}y,\nabla_{y}{\rangle}$
$\displaystyle\qquad=\frac{1}{\varpi_{\mu}(y)}\sum_{i=1}^{d}\partial_{y_{i}}\left(\varpi_{\mu+1}(y)\partial_{y_{i}}\right)+\sum_{1\leq
i<j\leq d}(D_{i,j}^{(y)})^{2},$

which can be easily verified and is in fact the decomposition of the spectral
differential operator for the unit ball ${\mathbb{B}}^{d}$ (cf. [13, Section
5.2]). This completes the proof. ∎

###### Proof of Theorem 2.7.

Using the decomposition (4.20) of ${\mathfrak{D}}_{\mu,{\gamma}}$ and making a
change of variable $x=yt$, we can write

$\displaystyle\int_{{\mathbb{V}}^{d+1}}{\mathfrak{D}}_{\mu,{\gamma}}\cdot
gW_{\mu,{\gamma}}\mathrm{d}x\mathrm{d}t=\int_{0}^{1}t^{d}\int_{{\mathbb{B}}^{d}}{\mathfrak{D}}_{\mu,{\gamma}}f(ty,t)\cdot
g(ty,t)W_{\mu,{\gamma}}(ty,t)\mathrm{d}y\mathrm{d}t$
$\displaystyle=\int_{{\mathbb{B}}^{d}}\int_{0}^{1}(\partial_{t}+t^{-1}{\langle}x,\nabla_{x}{\rangle})\left(t^{d+1}W_{\mu,{\gamma}+1}(x,t)(\partial_{t}+t^{-1}{\langle}x,\nabla_{x}{\rangle})\right)f(ty,t)\cdot
g(x,t)\mathrm{d}t\mathrm{d}y$
$\displaystyle\qquad\quad+\int_{0}^{1}t^{2\mu+d-2}\sum_{i=1}^{d}\int_{{\mathbb{B}}^{d}}\partial_{y_{i}}\left((1-\|y\|^{2})^{\mu+\frac{1}{2}}\partial_{y_{i}}\right)f(ty,t)\cdot
g(x,t)\mathrm{d}y(1-t)^{\gamma}\mathrm{d}t$
$\displaystyle\qquad\quad+\sum_{1\leq i<j\leq
d}\int_{0}^{1}t^{2\mu+d-2}\int_{{\mathbb{B}}^{d}}(D_{i,j}^{(y)})^{2}f(ty,t)\cdot
g(ty,t)\varpi_{\mu}(y)\mathrm{d}y\,(1-t)^{\gamma}\mathrm{d}t,$

where we have used (4.21) in the second term in the righthand side and
$D_{i,j}^{(x)}=D_{i,j}^{(y)}$ in the third term in the righthand side.
Assuming ${\gamma}>0$, we observe that

$\frac{\mathrm{d}}{\mathrm{d}t}[f(ty,t)]=\big{(}(t^{-1}{\langle}x,\nabla{\rangle}+\partial_{t})f\big{)}(ty,t),$

so that we can do integration by parts with respect to the variable $t$ in the
first integral in the right-hand side. Integration by parts also shows that
the second integral in the right-hand side is equal to

$\displaystyle-\int_{0}^{1}t^{2\mu+d-2}\sum_{i=1}^{d}\int_{{\mathbb{B}}^{d}}\partial_{y_{i}}f(ty,t)\cdot\partial_{y_{i}}g(ty,t)\varpi_{\mu+1}(y)\mathrm{d}y(1-t)^{\gamma}\mathrm{d}t$
$\displaystyle\qquad=-\int_{0}^{1}t^{d-1}\sum_{i=1}^{d}\int_{{\mathbb{B}}^{d}}\partial_{x_{i}}f(x,t)\cdot\partial_{x_{i}}g(x,t)W_{\mu+1,{\gamma}}(x,t)\mathrm{d}x\mathrm{d}t,$

which is the second integral in the right-hand side of (2.16). Next, sing the
result on the unit ball, the third integral in the right-hand side is equal to

$\displaystyle-\sum_{1\leq i<j\leq
d}\int_{0}^{1}t^{2\mu+d-2}\int_{{\mathbb{B}}^{d}}D_{i,j}^{(y)}f(ty,t)\cdot
D_{i,j}^{(y)}g(ty,t)\varpi_{\mu}(y)\mathrm{d}y\,(1-t)^{\gamma}\mathrm{d}t$
$\displaystyle\qquad\quad=-\sum_{1\leq i<j\leq
d}\int_{0}^{1}t^{d-1}\int_{{\mathbb{B}}^{d}}D_{i,j}^{(x)}f(x,t)\cdot
D_{i,j}^{(x)}g(x,t)W_{\mu,{\gamma}}(x,t)\mathrm{d}x\,\mathrm{d}t,$

which gives the third integral in the right-hand side of (2.16). Finally,
analytic continuation shows that the identity holds for ${\gamma}>-1$. ∎

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