The Symmetric Derivative and Related Operators for Symmetric Forms with Polynomial Coefficients in ${\mathbb{R}}^{\mathit{n}}$

Abstract

The symmetric derivative and related operators for symmetric forms with polynomial coefficients in ${\mathbb{R}}^n$ are investigated. The adjoints of these operators, with respect to the scalar product for symmetric forms with polynomial coefficients, are found and proved to be of a pure algebraic character.

1. Introduction

Nowadays, differential operators constitute a highly evolving part of physics, mathematics and even of such branches of science as biology and medicine. There are many scientific papers that involve this area of investigations, e.g., [1,2,3]. Operators on a Riemannian manifold, especially in the bundle of symmetric tensors, are the subject of interest not only of contemporary geometry but also of modern medicine. Such significant concepts as Killing or conformal Killing tensors have many applications in differential geometry and also in tomography. They are also an object of a scientific study of applied mathematics, cf., e.g., [4,5,6,7,8] and earlier [9]. It is worth distinguishing here that also in the past, many scientists were interested in the X-ray theme. We can name here such great inventors as Wilhelm Röntgen and Nikola Tesla (e.g., [10]). Therefore, symmetric tensors and related topics are very important parts of mathematics, science and our everyday life.

The target of this paper is to investigate some differential operators for symmetric forms with polynomial coefficients in ${\mathbb{R}}^n$. The polynomials related to other fields are evolving subjects of modern mathematics and interesting on their own (e.g., [11]). Moreover, consideration to the objects with coefficients in subspace polynomials plays an important role in cryptography (e.g., [12]). Polynomials as solutions to first-order difference equations can also be very useful in many investigations (e.g., [13]).

The fundamental operators in this paper are symmetric derivative ${\mathrm{d}}^{\mathrm{s}}$ and its adjoint ${{\mathrm{d}}^{\mathrm{s}}}^{\ast }$ with respect to the scalar product (8), symmetric multiplication ${\epsilon }_{{\nu }^{\ast }}$ and the operator of substitution ${\iota }_{\nu }$. These operators are introduced in Section 2. In this section, the Laplace-type operators: ${\Delta }^s$ (see [14]) and $\mathrm{div}\mathrm{grad}$ (see [2,15,16]) are also defined. In the general case, they are both second-order strongly elliptic differential operators for which there is a system (for ${\Delta }^{\mathrm{s}}$ and $\mathrm{div}\mathrm{grad}$, respectively) of natural boundary conditions on a Riemannian manifold with the boundary. All the $2^{k+1}$ conditions of the system are elliptic. As a result, for each condition, there is a basis in $L^2$ composed of smooth eigenvectors for ${\Delta }^{\mathrm{s}}$ (for $\mathrm{div}\mathrm{grad}$, respectively), satisfying the boundary condition. These results for the $\mathrm{div}\mathrm{grad}$ operator were proven in [15] and then published in [16], and for the ${\Delta }^{\mathrm{s}}$ operator, these results were proven and published in [3]. The weighted Laplace-type operators ${{\Delta }_{\mathrm{ab}}}^{\mathrm{s}}$ acting on symmetric $k$-forms in ${\mathbb{R}}^n$ are also investigated here. The weighted form Laplacians and similar operators have been analysed by many famous mathematicians as L. V. Ahlfors (e.g., [17]) and H. Weyl (e.g., [18]). These kinds of operators are a subject of many scientific papers, e.g., [19,20,21].

Additionally, we investigate here an operator

$$\left(\mathfrak{C}\right)P\left(r^2\phi \right)-r^{2}P\left(\phi \right),$$

that can be treated as the commutator of an operator P and the polynomial $r^2$, where $\phi$ is a symmetric k form (see also [2]). The most important conclusions concerning this commutator are Theorems 2 and 3 and Propositions 1 and 2, which reflect relations between $r^2$ and operators: ${\Delta }^{\mathrm{s}}$, $\mathrm{div}\mathrm{grad}$, and ${{\Delta }_{\mathrm{ab}}}^{\mathrm{s}}$ along with ${\mathrm{d}}^{\mathrm{s}}$ and ${{\mathrm{d}}^{\mathrm{s}}}^{\ast }$ with their compositions, respectively.

In Section 3, we first introduce the scalar product (29) for symmetric forms with polynomial coefficients in ${\mathbb{R}}^n$. Next, we formulate and prove Theorems 4 and 5 to obtain, with respect to the scalar product (29), the shape of operators adjoint to ${\mathrm{d}}^{\mathrm{s}}$ and ${{\mathrm{d}}^{\mathrm{s}}}^{\ast }$, respectively. To conclude this section, we obtain the shape of adjoint operators to the considered Laplacians: ${\Delta }^{\mathrm{s}}$ (Theorem 6), $\mathrm{div}\mathrm{grad}$ (Theorem 7) and ${{\Delta }_{\mathrm{ab}}}^{\mathrm{s}}$ (Theorem 8), with respect to the scalar product (29). It is very curious that these adjoints are of a pure algebraic character. This fact may have many advantages and consequences, e.g., in solving some boundary problems and providing some algebraic algorithms in geometry and in algebraic investigations. For example, in the skew-symmetric case, some polynomial and algebraic motivations had significant influence on solving boundary problems for the weighted Laplacians (see [19]).

Finally, some interesting properties in the language of the mentioned operators and the operator $\Theta$ adjoint to ${\Delta }^{\mathrm{s}}$ (with respect to the scalar product (29)) are given. The operator $\Theta$ is related here to ${\mathrm{d}}^{\mathrm{s}}$ and ${{\mathrm{d}}^{\mathrm{s}}}^{\ast }$ (Propositions 3 and 4) and to the considered Laplacians: ${\Delta }^{\mathrm{s}}$, $\mathrm{div}\mathrm{grad}$, and ${{\Delta }_{\mathrm{ab}}}^{\mathrm{s}}$ (Theorems 9 and 10, respectively).

2. The Differential Operators in ${\mathbb{R}}^{\mathit{n}}$

Let us assume all the objects and morphisms to be smooth, i.e., of class $C^{\infty }$.

Let $M={\mathbb{R}}^n$ and g be the standard, flat metric. Denote by $C^{\infty }\left(M\right)$, $T=TM$ and $T^{\ast }=T^{\ast }M$ the ring of smooth functions on M, the tangent and cotangent bundles, respectively. Next, let ${T^{\ast }}^k={T^{\ast }}^{k}M$ be the bundle of k-tensors on M, and $S^k=S^{k}M$ be its sub-bundle of k-symmetric tensors (k-forms). Denote by $C^{\infty }\left(E\right)$ the $C^{\infty }\left(M\right)$ module of sections of E for any bundle E over M. Then, ${\partial }_j={\displaystyle \frac{\partial }{\partial x^j}}$ and $\mathrm{d}{x}^j$ for $j\in \{1,\dots ,n\}$ form dual bases of T and $T^{\ast }$, respectively. Consider the polynomial $r^2\left(x\right)={\sum }_{i=1}^n{\left(x^i\right)}^2$, the vector field ${\nu }_x={\sum }_{i=1}^n{x}^i{\partial }_i$ and the 1-form ${\nu }_x^{\ast }={\sum }_{i=1}^n{x}^i\mathrm{d}{x}^i$.

For any $p\in M$, we have

$$\langle ·,·\rangle =g_p(·,·):T_p\times T_p\to \mathbb{R}.$$

Extend now $\langle ·,·\rangle$ naturally to the fibers of the cotangent bundle and then to the fibers of any tensor bundle on M. Notice that the bundle of symmetric tensors $S^k$ as a sub-bundle of ${T^{\ast }}^k$ inherits this scalar product.

In $S^k$ and $S^k\otimes T$, another scalar product is also considered:

$$\langle ·|·\rangle ={\displaystyle \frac{1}{k!}}\langle ·,·\rangle .$$

(1)

Let ${\mathrm{d}}^{\mathrm{s}}:C^{\infty }\left(S^k\right)\to C^{\infty }\left(S^{k+1}\right)$ be the operator of symmetric derivative given by

$${\mathrm{d}}^{\mathrm{s}}=(k+1)\mathrm{Sym}\nabla .$$

(2)

Here, ∇ is the Levi-Civita covariant derivative of the metric g on M. It is transmitted naturally from the tangent bundle to the dual bundle, then to any tensor bundle (using the Leibniz rule) and in particular to the $S^k$ bundle.

A local expression, in our bases, for the symmetric derivative is

$${\mathrm{d}}^{\mathrm{s}}\phi =\sum _{j=1}^n\mathrm{d}{x}^j\odot {\nabla }_{{\partial }_j}\phi ,$$

(3)

for $\phi \in C^{\infty }\left(S^k\right)$. This result, in a general case, was proven in [15] and then published in [16]. See also [2].

As usual, define two trace operators as follows.

The trace operator $\mathrm{tr}:{T^{\ast }}^k\otimes T\to {T^{\ast }}^{k-1}$, that is acting on vector forms, is defined by the formula

$$\mathrm{tr}(\phi \otimes X)={\iota }_{\mathrm{X}}\phi ,\mathrm{for}k>0$$

(4)

and

$$\mathrm{tr}(\phi \otimes X)=0,\mathrm{for}k=0,$$

where $\iota$ denotes the operator of substitution.

Next, define the trace operator $\widetilde{\mathrm{tr}}:{T^{\ast }}^k\to {T^{\ast }}^{k-2}$ by

$$\widetilde{\mathrm{tr}}\phi =\sum _{i=1}^n{\iota }_{{\partial }_i}{\iota }_{{\partial }_i}\phi ,\mathrm{for}k=2,3,\dots$$

(5)

and

$$\widetilde{\mathrm{tr}}\phi =0,\mathrm{for}k=0,1.$$

(6)

Note that, for the restrictions of operators $\mathrm{tr}$ and $\widetilde{\mathrm{tr}}$ to sub-bundles of the bundles ${T^{\ast }}^k\otimes T$ and ${T^{\ast }}^k$, respectively, the same symbols will be used (in particular, to the sub-bundles $S^k\otimes T$ and $S^k$, respectively).

For any vector bundle E over M and a scalar product $\langle ·,·\rangle$ in E, define the global scalar product$\left(·,·\right)$ in the space of sections of E by

$$\left(·,·\right)={\int }_M\langle ·,·\rangle {\Omega }_M,$$

(7)

where ${\Omega }_M$ is the volume form on M defined by the orientation and the metric g (cf. [22]).

Observe that the global scalar product is defined only for such pairs of sections that the integral exists and is finite, e.g., when at least one of the section is of compact support.

Consider two global scalar products $\left(·,·\right)$ and $\left(·|·\right)$, for the bundle $S^k$, related by

$$\left(·|·\right)={\displaystyle \frac{1}{k!}}\left(·,·\right).$$

(8)

Theorem 1.

The operator ${{\mathrm{d}}^{\mathrm{s}}}^{\ast }:C^{\infty }\left(S^{k+1}\right)\to C^{\infty }\left(S^k\right)$ formally adjoint to ${\mathrm{d}}^{\mathrm{s}}:C^{\infty }\left(S^k\right)\to C^{\infty }\left(S^{k+1}\right)$ is of the form

$${{\mathrm{d}}^{\mathrm{s}}}^{\ast }=-\widetilde{\mathrm{tr}}{\nabla }_{|C^{\infty }\left(S^{k+1}\right)},$$

i.e., for $\phi \in C^{\infty }\left(S^k\right)$ and $\psi \in C^{\infty }\left(S^{k+1}\right)$, we have

$$\left({\mathrm{d}}^{\mathrm{s}}\phi \right|\psi )=\left(\phi \right|-\widetilde{\mathrm{tr}}\nabla \psi ),$$

if only φ or ψ is of compact support.

Proof. 

This result was proven, in a general case, in [15] and then published in [16]. □

Examine now the composition $\mathrm{div}\mathrm{grad}$ of two first-order differential operators’ gradient, $\mathrm{grad}=\mathfrak{a}{\mathrm{d}}^{\mathrm{s}}-{\mathrm{d}}^{\mathrm{s}}\mathfrak{a}$, and divergence, $\mathrm{div}=\mathrm{tr}{\mathrm{d}}^{\mathrm{s}}-{\mathrm{d}}^{\mathrm{s}}\mathrm{tr}$. Here, for $k=1,2,\dots$, the operator $\mathfrak{a}$ is defined by the equation

$$\mathfrak{a}\phi =\sum _{i=1}^n{\iota }_{{\partial }_i}\phi \otimes {\partial }_i$$

(9)

and for $k=0$,

$$\mathfrak{a}\phi =0,$$

where $\phi \in C^{\infty }\left(S^k\right)$.

The gradient and the divergence have good properties. For example, they serve as some form of differentiations of the algebra of symmetric tensors (see [2]). They are adjoint—each to the other—with respect to the global scalar product on M (this result, in a general case, was proven in [15] and then published in [16]).

The operator $\mathrm{div}\mathrm{grad}$ is a second-order strongly elliptic differential operator (this result, in a general case, was proven in [15] and then published in [16]).

Define also another second-order strongly elliptic operator ${\Delta }^{\mathrm{s}}$ (this result, in a general case, was proven in [3]). The Laplace operator ${\Delta }^{\mathrm{s}}:C^{\infty }\left(S^k\right)\to C^{\infty }\left(S^k\right)$ is given by the formula

$${\Delta }^{\mathrm{s}}={{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\mathrm{d}}^{\mathrm{s}}-{\mathrm{d}}^{\mathrm{s}}{{\mathrm{d}}^{\mathrm{s}}}^{\ast }.$$

(10)

Note that the Sampson Laplacian ${\Delta }^{\mathrm{s}}$ was introduced first in [14]. In the context of spectral properties, for $k=1$, the Yano rough Laplacian has been analysed in [23]. In addition, elliptic operators in the bundle of symmetric tensors have also been investigated in [5] in regard to conformal Killing tensors.

Let ${\epsilon }_{{\nu }^{\ast }}$ denote the operator of symmetric multiplication by ${\nu }^{\ast }$ and, as usual, define by

$${\epsilon }_{{\nu }^{\ast }}\phi ={\nu }^{\ast }\odot \phi ,$$

(11)

for $\phi \in C^{\infty }\left(S^k\right)$.

Recall now several facts involving the polynomial $r^2$, the vector field $\nu$, the 1-form ${\nu }^{\ast }$ and the necessary operators.

Proposition 1.

Let $\phi \in C^{\infty }\left(S^k\right)$. Then, the following properties hold:

$${\iota }_{\nu }{\epsilon }_{{\nu }^{\ast }}\phi -{\epsilon }_{{\nu }^{\ast }}{\iota }_{\nu }\phi =r^2\phi ;$$

(12)

$$\widetilde{\mathrm{tr}}\left(r^2\phi \right)=r^2\widetilde{\mathrm{tr}}\phi ;$$

(13)

$${\mathrm{d}}^{\mathrm{s}}{\epsilon }_{{\nu }^{\ast }}\phi =2g\odot \phi +{\epsilon }_{{\nu }^{\ast }}{\mathrm{d}}^{\mathrm{s}}\phi ,$$

(14)

where g is a Riemannian metric tensor;

$${{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\epsilon }_{{\nu }^{\ast }}\phi -{\epsilon }_{{\nu }^{\ast }}{{\mathrm{d}}^{\mathrm{s}}}^{\ast }\phi =-(n+k)\phi -{\nabla }_{\nu }\phi ;$$

(15)

$${{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\iota }_{\nu }\phi -{\iota }_{\nu }{{\mathrm{d}}^{\mathrm{s}}}^{\ast }\phi =-\widetilde{\mathrm{tr}}\phi ;$$

(16)

$${\mathrm{d}}^{\mathrm{s}}{\iota }_{\nu }\phi -{\iota }_{\nu }{\mathrm{d}}^{\mathrm{s}}\phi =k\phi -{\nabla }_{\nu }\phi ;$$

(17)

$${\mathrm{d}}^{\mathrm{s}}\left(r^2\phi \right)-r^2{\mathrm{d}}^{\mathrm{s}}\phi =2{\epsilon }_{{\nu }^{\ast }}\phi ;$$

(18)

$${{\mathrm{d}}^{\mathrm{s}}}^{\ast }\left(r^2\phi \right)-r^2{{\mathrm{d}}^{\mathrm{s}}}^{\ast }\phi =-2{\iota }_{\nu }\phi .$$

(19)

Proof. 

See [2]. □

It is worth mentioning here that properties (13), (18) and (19) of Proposition 1 can be investigated in the language of the operator:

$$\left(\mathfrak{C}\right)P\left(r^2\phi \right)-r^{2}P\left(\phi \right),$$

treated as the commutator of an operator P and the polynomial $r^2$, where $\phi \in C^{\infty }\left(S^k\right)$.

Let us consider more operators in the language of the above mentioned commutator.

Theorem 2.

Let $\phi \in C^{\infty }\left(S^k\right)$. Then, the following properties hold:

$${\Delta }^{\mathrm{s}}\left(r^2\phi \right)-r^2{\Delta }^{\mathrm{s}}\phi =-2n\phi -4{\nabla }_{\nu }\phi ;$$

(20)

$${\Delta }^{\mathrm{s}}\phi =-\mathrm{div}\mathrm{grad}\phi ;$$

(21)

$$\mathrm{div}\mathrm{grad}\left(r^2\phi \right)-r^2\mathrm{div}\mathrm{grad}\phi =4{\nabla }_{\nu }\phi +2n\phi .$$

(22)

Proof. 

See [2]. □

Proposition 2.

Let $\phi \in C^{\infty }\left(S^k\right)$. Then, the following properties hold:

$${{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\mathrm{d}}^{\mathrm{s}}\left(r^2\phi \right)-r^2{{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\mathrm{d}}^{\mathrm{s}}\phi =2({{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\epsilon }_{{\nu }^{\ast }}-{\iota }_{\nu }{\mathrm{d}}^{\mathrm{s}})\phi ;$$

(23)

$${\mathrm{d}}^{\mathrm{s}}{{\mathrm{d}}^{\mathrm{s}}}^{\ast }\left(r^2\phi \right)-r^2{\mathrm{d}}^{\mathrm{s}}{{\mathrm{d}}^{\mathrm{s}}}^{\ast }\phi =-2({\mathrm{d}}^{\mathrm{s}}{\iota }_{\nu }-{\epsilon }_{{\nu }^{\ast }}{{\mathrm{d}}^{\mathrm{s}}}^{\ast })\phi .$$

(24)

Proof. 

For $\phi \in C^{\infty }\left(S^k\right)$, using (19) and (18) of Proposition 1, we get

$$\begin{array}{cc}\hfill {{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\mathrm{d}}^{\mathrm{s}}\left(r^2\phi \right)& ={{\mathrm{d}}^{\mathrm{s}}}^{\ast }(2{\epsilon }_{{\nu }^{\ast }}\phi +r^2{\mathrm{d}}^{\mathrm{s}}\phi )\hfill \\ \hfill & =2{{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\epsilon }_{{\nu }^{\ast }}\phi +r^2{{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\mathrm{d}}^{\mathrm{s}}\phi -2{\iota }_{\nu }{\mathrm{d}}^{\mathrm{s}}\phi .\hfill \end{array}$$

Similarly, we have

$$\begin{array}{cc}\hfill {\mathrm{d}}^{\mathrm{s}}{{\mathrm{d}}^{\mathrm{s}}}^{\ast }\left(r^2\phi \right)& ={\mathrm{d}}^{\mathrm{s}}(-2{\iota }_{\nu }\phi +r^2{{\mathrm{d}}^{\mathrm{s}}}^{\ast }\phi )\hfill \\ \hfill & =-2{\mathrm{d}}^{\mathrm{s}}{\iota }_{\nu }\phi +r^2{\mathrm{d}}^{\mathrm{s}}{{\mathrm{d}}^{\mathrm{s}}}^{\ast }\phi +2{\epsilon }_{{\nu }^{\ast }}{{\mathrm{d}}^{\mathrm{s}}}^{\ast }\phi .\hfill \end{array}$$

□

Define now the weighted Laplace-type operators ${{\Delta }_{\mathrm{ab}}}^{\mathrm{s}}:C^{\infty }\left(S^k\right)\to C^{\infty }\left(S^k\right)$ by the formula

$${{\Delta }_{\mathrm{ab}}}^{\mathrm{s}}=a{{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\mathrm{d}}^{\mathrm{s}}-b{\mathrm{d}}^{\mathrm{s}}{{\mathrm{d}}^{\mathrm{s}}}^{\ast },$$

(25)

where $a,b\in \mathbb{R}$.

Notice that for $a=b$, the operator ${{\Delta }_{\mathrm{ab}}}^{\mathrm{s}}:C^{\infty }\left(S^k\right)\to C^{\infty }\left(S^k\right)$ coincide with the Laplace operator ${\Delta }^{\mathrm{s}}:C^{\infty }\left(S^k\right)\to C^{\infty }\left(S^k\right)$.

The weighted-form Laplacians, with $a,b>0$, were investigated, e.g., in [19], for skew-symmetric polynomial forms in the $n$-dimensional Euclidean ball. What is more, for $1$-forms, these operators are a subclass of the so-called class of non-minimal operators (see [19,20,21]).

Theorem 3.

Let $\phi \in C^{\infty }\left(S^k\right)$. Then,

$$\begin{array}{cc}\hfill {{\Delta }_{\mathrm{ab}}}^{\mathrm{s}}\left(r^2\phi \right)-r^2{{\Delta }_{\mathrm{ab}}}^{\mathrm{s}}\phi & =2\left(a{{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\epsilon }_{{\nu }^{\ast }}-b{\epsilon }_{{\nu }^{\ast }}{{\mathrm{d}}^{\mathrm{s}}}^{\ast }\right)\phi \hfill \\ \hfill & +2\left(b{\mathrm{d}}^{\mathrm{s}}{\iota }_{\nu }-a{\iota }_{\nu }{\mathrm{d}}^{\mathrm{s}}\right)\phi .\hfill \end{array}$$

(26)

Proof. 

For $\phi \in C^{\infty }\left(S^k\right)$, by Proposition, 2 we simply get

$$\begin{array}{cc}\hfill {{\Delta }_{\mathrm{ab}}}^{\mathrm{s}}\left(r^2\phi \right)-r^2{{\Delta }_{\mathrm{ab}}}^{\mathrm{s}}\phi & =a{{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\mathrm{d}}^{\mathrm{s}}\left(r^2\phi \right)-b{\mathrm{d}}^{\mathrm{s}}{{\mathrm{d}}^{\mathrm{s}}}^{\ast }\left(r^2\phi \right)+\hfill \\ \hfill & -a{r}^2{{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\mathrm{d}}^{\mathrm{s}}\phi +b{r}^2{\mathrm{d}}^{\mathrm{s}}{{\mathrm{d}}^{\mathrm{s}}}^{\ast }\phi \hfill \\ \hfill & =a({{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\mathrm{d}}^{\mathrm{s}}\left(r^2\phi \right)-r^2{{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\mathrm{d}}^{\mathrm{s}}\phi )+\hfill \\ \hfill & -b({\mathrm{d}}^{\mathrm{s}}{{\mathrm{d}}^{\mathrm{s}}}^{\ast }\left(r^2\phi \right)-r^2{\mathrm{d}}^{\mathrm{s}}{{\mathrm{d}}^{\mathrm{s}}}^{\ast }\phi )\hfill \\ \hfill & =2a({{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\epsilon }_{{\nu }^{\ast }}-{\iota }_{\nu }{\mathrm{d}}^{\mathrm{s}})\phi \hfill \\ \hfill & +2b({\mathrm{d}}^{\mathrm{s}}{\iota }_{\nu }-{\epsilon }_{{\nu }^{\ast }}{{\mathrm{d}}^{\mathrm{s}}}^{\ast })\phi .\hfill \end{array}$$

□

3. Symmetric Forms with Polynomial Coefficients

In this section, let us apply the results of Section 2 to show that some differential operators which act on forms with polynomial coefficients are of a simple algebraic character.

Let $f={\sum }_{\left|\alpha \right|=q}{a}_{\alpha }{x}^{\alpha }$ be a homogeneous polynomial of degree q in ${\mathbb{R}}^n$, where $\alpha =({\alpha }_1,\dots ,{\alpha }_n)$, $x^{\alpha }={\left(x^1\right)}^{{\alpha }_1}\cdots {\left(x^n\right)}^{{\alpha }_n}$ and $\left|\alpha \right|={\alpha }_1+\cdots +{\alpha }_n$.

With every such f associate the differential operator $f\left(D\right)$ of order q that is defined by

$$f\left(D\right)=\sum _{\left|\alpha \right|=q}{a}_{\alpha }{D}^{\alpha }:{\mathfrak{P}}_s\to {\mathfrak{P}}_{s-q},$$

(27)

where ${\mathfrak{P}}_s$ denotes the space of all homogeneous polynomials of degree s in ${\mathbb{R}}^n$, $D^{\alpha }={\left({\partial }_1\right)}^{{\alpha }_1}\circ \cdots \circ {\left({\partial }_n\right)}^{{\alpha }_n}$.

Let f, g be homogeneous polynomials of degree q in ${\mathbb{R}}^n$. Define now an inner product ${(·,·)}_q$ in ${\mathfrak{P}}_q$ by

$${(f,g)}_q=f\left(D\right)g.$$

(28)

Denote by $S_q^k$ the space of all symmetric k forms with coefficients in ${\mathfrak{P}}_q$. Let $\phi ={\sum }_{\left|\beta \right|=k}{f}_{\beta }{dx}^{\beta }$ and $\psi ={\sum }_{\left|\beta \right|=k}{g}_{\beta }{dx}^{\beta }$, where $f_{\beta },g_{\beta }\in {\mathfrak{P}}_q$ and ${dx}^{\beta }={\left({dx}^1\right)}^{{\beta }_1}\odot \cdots \odot {\left({dx}^n\right)}^{{\beta }_n}$.

Define the scalar product ${\langle ·|·\rangle }_{k,q}$ in $S_q^k$ by

$${\langle \phi |\psi \rangle }_{k,q}=\sum _{\left|\beta \right|=k}\beta !{(f_{\beta },g_{\beta })}_q,$$

(29)

where $\beta !=\left({\beta }_1\right)!\cdots \left({\beta }_n\right)!$.

Now, prove two theorems on the shape of the adjoint operators to the operators ${\mathrm{d}}^{\mathrm{s}}$ and ${{\mathrm{d}}^{\mathrm{s}}}^{\ast }$ with respect to the scalar product ${\langle \phi |\psi \rangle }_{k,q}$. It is strange and curious that these two adjoints are of a pure algebraic character. Moreover, the definition of the scalar product (29) given in the language of polynomial coefficients in common with Theorems 4 and 5 shows the connection between polynomial coefficients and algebraic adjoints.

Theorem 4.

The operator $\rho :C^{\infty }\left(S_{q-1}^{k+1}\right)\to C^{\infty }\left(S_q^k\right)$ adjoint to ${\mathrm{d}}^{\mathrm{s}}:C^{\infty }\left(S_q^k\right)\to C^{\infty }\left(S_{q-1}^{k+1}\right)$, with respect to the scalar product (29), is of the form

$$\rho \left(\psi \right)={\iota }_{\nu }\psi ,$$

(30)

for all $\psi \in C^{\infty }\left(S_{q-1}^{k+1}\right)$.

Proof. 

Let $\psi \in C^{\infty }\left(S_{q-1}^{k+1}\right)$, $\phi \in C^{\infty }\left(S_q^k\right)$. Since the scalar product is bilinear, it is enough to consider the case that

$$\phi =x^{\alpha }{dx}^{\beta },\mathrm{where}\alpha =({\alpha }_1,\dots ,{\alpha }_n),\beta =({\beta }_1,\dots ,{\beta }_n)$$

and

$$\psi ={\left(x^1\right)}^{{\alpha }_1}\cdots {\left(x^s\right)}^{{\alpha }_s-1}\cdots {\left(x^n\right)}^{{\alpha }_n}{\left(d{x}^1\right)}^{{\beta }_1}\odot \cdots {\left(d{x}^s\right)}^{{\beta }_s+1}\cdots \odot {\left(d{x}^n\right)}^{{\beta }_n}.$$

Then, using (3) and by ${\nabla }_{{\partial }_j}\left(d{x}^i\right)=0$, we have

$$\begin{array}{cc}\hfill {\mathrm{d}}^{\mathrm{s}}\phi & =\sum _{j=1}^{n}d{x}^j\odot {\nabla }_{{\partial }_j}\left(x^{\alpha }{dx}^{\beta }\right)=\sum _{j=1}^{n}d{x}^j\odot \left({\nabla }_{{\partial }_j}{x}^{\alpha }\right){dx}^{\beta }\hfill \\ \hfill & =\sum _{j=1}^n{\alpha }_j{\left(x^1\right)}^{{\alpha }_1}\cdots {\left(x^j\right)}^{{\alpha }_j-1}\cdots {\left(x^n\right)}^{{\alpha }_n}{\left(d{x}^1\right)}^{{\beta }_1}\odot \cdots {d{x}^j}^{{\beta }_j+1}\cdots \odot {\left(d{x}^n\right)}^{{\beta }_n}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {({\left(x^1\right)}^{{\alpha }_1}\cdots {\left(x^j\right)}^{{\alpha }_j-1}\cdots {\left(x^n\right)}^{{\alpha }_n},{\left(x^1\right)}^{{\alpha }_1}\cdots {\left(x^s\right)}^{{\alpha }_s-1}\cdots {\left(x^n\right)}^{{\alpha }_n})}_q\hfill \\ \hfill & =\left\{\begin{array}{c}\left({\alpha }_1\right)!\cdots ({\alpha }_s-1)!\cdots \left({\alpha }_n\right)!\mathrm{if}j=s\hfill \\ 0\mathrm{if}j\ne s,\hfill \end{array}\right.\hfill \end{array}$$

(31)

so, we have

$$\begin{array}{cc}\hfill {\langle {\mathrm{d}}^{\mathrm{s}}\phi |\psi \rangle }_{k,q}& ={\alpha }_s\left({\alpha }_1\right)!\cdots ({\alpha }_s-1)!\cdots \left({\alpha }_n\right)!\left({\beta }_1\right)!\cdots ({\beta }_s+1)!\cdots \left({\beta }_n\right)!\hfill \\ \hfill & =({\beta }_s+1)\alpha !\beta !.\hfill \end{array}$$

(32)

On the other hand,

$$\begin{array}{cc}\hfill & {\iota }_{\nu }\psi ={\iota }_{\left({\sum }_{i=1}^n{x}^i{\partial }_i\right)}\psi \hfill \\ \hfill & =\sum _{i=1}^n{x}^i{\left(x^1\right)}^{{\alpha }_1}\cdots {\left(x^s\right)}^{{\alpha }_s-1}\cdots {\left(x^n\right)}^{{\alpha }_n}{\iota }_{{\partial }_i}({\left(d{x}^1\right)}^{{\beta }_1}\odot \cdots {\left(d{x}^s\right)}^{{\beta }_s+1}\cdots \odot {\left(d{x}^n\right)}^{{\beta }_n}),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\iota }_{{\partial }_i}({\left(d{x}^1\right)}^{{\beta }_1}\odot \cdots {\left(d{x}^s\right)}^{{\beta }_s+1}\cdots \odot {\left(d{x}^n\right)}^{{\beta }_n})\hfill \\ \hfill & =\left\{\begin{array}{c}{\beta }_i{\left(d{x}^1\right)}^{{\beta }_1}\odot \cdots {\left(d{x}^i\right)}^{{\beta }_i-1}\odot \cdots {\left(d{x}^s\right)}^{{\beta }_s+1}\cdots \odot {\left(d{x}^n\right)}^{{\beta }_n}\mathrm{if}i<s\hfill \\ ({\beta }_s+1){\left(d{x}^1\right)}^{{\beta }_1}\odot \cdots {\left(d{x}^s\right)}^{{\beta }_s}\cdots \odot {\left(d{x}^n\right)}^{{\beta }_n}\mathrm{if}i=s\hfill \\ {\beta }_i{\left(d{x}^1\right)}^{{\beta }_1}\odot \cdots {\left(d{x}^s\right)}^{{\beta }_s+1}\odot \cdots {\left(d{x}^i\right)}^{{\beta }_i-1}\cdots \odot {\left(d{x}^n\right)}^{{\beta }_n}\mathrm{if}i>s\hfill \end{array}\right.\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & x^i({\left(x^1\right)}^{{\alpha }_1}\cdots {\left(x^s\right)}^{{\alpha }_s-1}\cdots {\left(x^n\right)}^{{\alpha }_n})\hfill \\ \hfill & =\left\{\begin{array}{c}{\left(x^1\right)}^{{\alpha }_1}\cdots {\left(x^i\right)}^{{\alpha }_i+1}\cdots {\left(x^s\right)}^{{\alpha }_s-1}\cdots {\left(x^n\right)}^{{\alpha }_n}\mathrm{if}i<s\hfill \\ {\left(x^1\right)}^{{\alpha }_1}\cdots {\left(x^s\right)}^{{\alpha }_s}\cdots {\left(x^n\right)}^{{\alpha }_n}\mathrm{if}i=s\hfill \\ {\left(x^1\right)}^{{\alpha }_1}\cdots {\left(x^s\right)}^{{\alpha }_s-1}\cdots {\left(x^i\right)}^{{\alpha }_i+1}\cdots {\left(x^n\right)}^{{\alpha }_n}\mathrm{if}i>s,\hfill \end{array}\right.\hfill \end{array}$$

so,

$$\begin{array}{cc}\hfill & {(x^{\alpha },x^i({\left(x^1\right)}^{{\alpha }_1}\cdots {\left(x^s\right)}^{{\alpha }_s-1}\cdots {\left(x^n\right)}^{{\alpha }_n}))}_q=\left\{\begin{array}{c}\alpha !\mathrm{if}i=s\hfill \\ 0\mathrm{if}i\ne s,\hfill \end{array}\right.\hfill \end{array}$$

(33)

and then,

$$\begin{array}{c}\hfill {\langle \phi |{\iota }_{\nu }\psi \rangle }_{k,p}=({\beta }_s+1)\beta !\alpha !.\end{array}$$

(34)

Finally, by (32) and (34), we obtain that ${\langle {\mathrm{d}}^{\mathrm{s}}\phi |\psi \rangle }_{k,q}={\langle \phi |{\iota }_{\nu }\psi \rangle }_{k,p}$, which means that $\rho ={\iota }_{\nu }$ is the adjoint operator to ${\mathrm{d}}^{\mathrm{s}}$. □

Theorem 5.

The operator $\varrho :C^{\infty }\left(S_{q-1}^{k-1}\right)\to C^{\infty }\left(S_q^k\right)$ adjoint to ${{\mathrm{d}}^{\mathrm{s}}}^{\ast }:C^{\infty }\left(S_q^k\right)\to C^{\infty }\left(S_{q-1}^{k-1}\right)$, with respect to the scalar product (29), is of the form

$$\varrho \left(\psi \right)=-{\epsilon }_{{\nu }^{\ast }}\psi ,$$

(35)

for all $\psi \in C^{\infty }\left(S_{q-1}^{k-1}\right)$.

Proof. 

Let $\phi \in C^{\infty }\left(S_q^k\right)$, $\psi \in C^{\infty }\left(S_{q-1}^{k-1}\right)$. Similarly as in Theorem 4, it is enough to consider the case that

$$\phi =x^{\alpha }{dx}^{\beta },\mathrm{where}\alpha =({\alpha }_1,\dots ,{\alpha }_n),\beta =({\beta }_1,\dots ,{\beta }_n),$$

$$\psi ={\left(x^1\right)}^{{\alpha }_1}\cdots {\left(x^s\right)}^{{\alpha }_s-1}\cdots {\left(x^n\right)}^{{\alpha }_n}{\left(d{x}^1\right)}^{{\beta }_1}\odot \cdots {\left(d{x}^s\right)}^{{\beta }_s-1}\cdots \odot {\left(d{x}^n\right)}^{{\beta }_n}.$$

Then, by Theorem 1 and the fact that ${\nabla }_{{\partial }_j}\left(d{x}^i\right)=0$, we have

$$\begin{array}{cc}\hfill & {{\mathrm{d}}^{\mathrm{s}}}^{\ast }\phi =-\widetilde{\mathrm{tr}}\nabla \phi =-\sum _{j=1}^n{\iota }_{{\partial }_j}{\nabla }_{{\partial }_j}\left(x^{\alpha }{dx}^{\beta }\right)=-\sum _{j=1}^n{\iota }_{{\partial }_j}\left(\left({\nabla }_{{\partial }_j}{x}^{\alpha }\right){dx}^{\beta }\right)\hfill \\ \hfill & =-\sum _{j=1}^n{\alpha }_j{\left(x^1\right)}^{{\alpha }_1}\cdots {\left(x^j\right)}^{{\alpha }_j-1}\cdots {\left(x^n\right)}^{{\alpha }_n}{\iota }_{{\partial }_j}\left({dx}^{\beta }\right)\hfill \\ \hfill & =-\sum _{j=1}^n{\alpha }_j{\beta }_j{\left(x^1\right)}^{{\alpha }_1}\cdots {\left(x^j\right)}^{{\alpha }_j-1}\cdots {\left(x^n\right)}^{{\alpha }_n}{\left(d{x}^1\right)}^{{\beta }_1}\odot \cdots {\left(d{x}^j\right)}^{{\beta }_j-1}\cdots \odot {\left(d{x}^n\right)}^{{\beta }_n}.\hfill \end{array}$$

So, by (31), we get

$$\begin{array}{cc}\hfill {\langle {{\mathrm{d}}^{\mathrm{s}}}^{\ast }\phi |\psi \rangle }_{k,q}& =-{\alpha }_s{\beta }_s\left({\alpha }_1\right)!\cdots ({\alpha }_s-1)!\cdots \left({\alpha }_n\right)!\left({\beta }_1\right)!\cdots ({\beta }_s-1)!\cdots \left({\beta }_n\right)!\hfill \\ \hfill & =-\alpha !\beta !.\hfill \end{array}$$

(36)

On the other hand,

$$\begin{array}{cc}\hfill & {\epsilon }_{{\nu }^{\ast }}\psi =\hfill \\ \hfill & {\nu }^{\ast }\odot {\left(x^1\right)}^{{\alpha }_1}\cdots {\left(x^s\right)}^{{\alpha }_s-1}\cdots {\left(x^n\right)}^{{\alpha }_n}{\left(d{x}^1\right)}^{{\beta }_1}\odot \cdots {\left(d{x}^s\right)}^{{\beta }_s-1}\cdots \odot {\left(d{x}^n\right)}^{{\beta }_n}=\hfill \\ \hfill & \sum _{j=1}^n{x}^j{\left(x^1\right)}^{{\alpha }_1}\cdots {\left(x^s\right)}^{{\alpha }_s-1}\cdots {\left(x^n\right)}^{{\alpha }_n}d{x}^j\odot {\left(d{x}^1\right)}^{{\beta }_1}\odot \cdots {\left(d{x}^s\right)}^{{\beta }_s-1}\cdots \odot {\left(d{x}^n\right)}^{{\beta }_n}.\hfill \end{array}$$

Then,

$$\begin{array}{cc}\hfill & d{x}^j\odot {\left(d{x}^1\right)}^{{\beta }_1}\odot \cdots {\left(d{x}^s\right)}^{{\beta }_s-1}\cdots \odot {\left(d{x}^n\right)}^{{\beta }_n}\hfill \\ \hfill & =\left\{\begin{array}{c}{\left(d{x}^1\right)}^{{\beta }_1}\odot \cdots {\left(d{x}^j\right)}^{{\beta }_j+1}\cdots \odot {\left(d{x}^s\right)}^{{\beta }_s-1}\cdots \odot {\left(d{x}^n\right)}^{{\beta }_n}\mathrm{if}j<s\hfill \\ {\left(d{x}^1\right)}^{{\beta }_1}\odot \cdots {\left(d{x}^s\right)}^{{\beta }_s}\cdots \odot {\left(d{x}^n\right)}^{{\beta }_n}\mathrm{if}j=s\hfill \\ {\left(d{x}^1\right)}^{{\beta }_1}\odot \cdots {\left(d{x}^s\right)}^{{\beta }_s-1}\odot \cdots {\left(d{x}^j\right)}^{{\beta }_j+1}\cdots \odot {\left(d{x}^n\right)}^{{\beta }_n}\mathrm{if}j>s.\hfill \end{array}\right.\hfill \end{array}$$

So, by (33), we get

$$\begin{array}{c}\hfill {\langle \phi |-{\epsilon }_{{\nu }^{\ast }}\psi \rangle }_{k,q}=-\alpha !\beta !.\end{array}$$

(37)

By (36) and (37), we thus obtain the assertion. □

Finally, to conclude this section, we will formulate and prove another three theorems about the shape of adjoint operators to the considered Laplacians—${\Delta }^{\mathrm{s}}$, ${{\Delta }_{\mathrm{ab}}}^{\mathrm{s}}$ and $\mathrm{div}\mathrm{grad}$—with respect to the scalar product (29). Also, some other properties will be given.

Denote now by * the operation of taking the adjoint with respect to the scalar product (29).

Theorem 6.

The operator $\Theta :C^{\infty }\left(S_{q-2}^k\right)\to C^{\infty }\left(S_q^k\right)$ adjoint to ${\Delta }^{\mathrm{s}}:C^{\infty }\left(S_q^k\right)\to C^{\infty }\left(S_{q-2}^k\right)$, with respect to the scalar product (29), is of the form

$$\Theta \left(\psi \right)=-r^2\psi ,$$

(38)

for all $\psi \in C^{\infty }\left(S_{q-2}^k\right)$.

Proof. 

Using the definition of ${\Delta }^{\mathrm{s}}$ and Theorems 4 and 5, we get

$$\begin{array}{cc}\hfill {\Delta }^{\mathrm{s}\mathbf{*}}& ={({{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\mathrm{d}}^{\mathrm{s}}-{\mathrm{d}}^{\mathrm{s}}{{\mathrm{d}}^{\mathrm{s}}}^{\ast })}^{\mathbf{*}}={\left({\mathrm{d}}^{\mathrm{s}}\right)}^{\mathbf{*}}{\left({{\mathrm{d}}^{\mathrm{s}}}^{\ast }\right)}^{\mathbf{*}}-{\left({{\mathrm{d}}^{\mathrm{s}}}^{\ast }\right)}^{\mathbf{*}}{\left({\mathrm{d}}^{\mathrm{s}}\right)}^{\mathbf{*}}\hfill \\ \hfill & ={\iota }_{\nu }(-{\epsilon }_{{\nu }^{\ast }})-(-{\epsilon }_{{\nu }^{\ast }}){\iota }_{\nu }={\epsilon }_{{\nu }^{\ast }}{\iota }_{\nu }-{\iota }_{\nu }{\epsilon }_{{\nu }^{\ast }}.\hfill \end{array}$$

(39)

Now, using Proposition 1 (12), we obtain the assertion. □

Proposition 3.

Let $\phi \in C^{\infty }\left(S_q^k\right)$. Then, the following properties hold:

$${\iota }_{\nu }{\epsilon }_{{\nu }^{\ast }}\phi -{\epsilon }_{{\nu }^{\ast }}{\iota }_{\nu }\phi =-\Theta \phi ;$$

(40)

$$\widetilde{\mathrm{tr}}(\Theta \phi )=\Theta \widetilde{\mathrm{tr}}\phi ;$$

(41)

$${\mathrm{d}}^{\mathrm{s}}\varrho \phi =\varrho {\mathrm{d}}^{\mathrm{s}}\phi -2g\odot \phi ,$$

(42)

where g is a Riemannian metric tensor;

$${{\mathrm{d}}^{\mathrm{s}}}^{\ast }\varrho \phi -\varrho {{\mathrm{d}}^{\mathrm{s}}}^{\ast }\phi =(n+k)\phi +{\nabla }_{\nu }\phi ;$$

(43)

$${{\mathrm{d}}^{\mathrm{s}}}^{\ast }\rho \phi -\rho {{\mathrm{d}}^{\mathrm{s}}}^{\ast }\phi =-\widetilde{\mathrm{tr}}\phi ;$$

(44)

$${\mathrm{d}}^{\mathrm{s}}\rho \phi -\rho {\mathrm{d}}^{\mathrm{s}}\phi =k\phi -{\nabla }_{\nu }\phi ;$$

(45)

$${\mathrm{d}}^{\mathrm{s}}(\Theta \phi )-\Theta {\mathrm{d}}^{\mathrm{s}}\phi =2\varrho \phi ;$$

(46)

$${{\mathrm{d}}^{\mathrm{s}}}^{\ast }(\Theta \phi )-\Theta {{\mathrm{d}}^{\mathrm{s}}}^{\ast }\phi =2\rho \phi .$$

(47)

Proof. 

The assertion is a direct consequence of Theorem 6 and Proposition 1. □

Theorem 7.

The operator adjoint to $\mathrm{div}\mathrm{grad}:C^{\infty }\left(S_q^k\right)\to C^{\infty }\left(S_{q-2}^k\right)$, with respect to the scalar product (29), is of the form

$${\left(\mathrm{div}\mathrm{grad}\right)}^{\mathbf{*}}=-\Theta .$$

(48)

Proof. 

It is a direct consequence of Theorem 2 (21) and Theorem 6. □

Theorem 8.

The operator adjoint to ${{\Delta }_{\mathrm{ab}}}^{\mathrm{s}}:C^{\infty }\left(S_q^k\right)\to C^{\infty }\left(S_{q-2}^k\right)$, with respect to the scalar product (29), is of the form

$${\left({{\Delta }_{\mathrm{ab}}}^{\mathrm{s}}\right)}^{\mathbf{*}}=b{\epsilon }_{{\nu }^{\ast }}{\iota }_{\nu }-a{\iota }_{\nu }{\epsilon }_{{\nu }^{\ast }}.$$

(49)

Proof. 

As a simple consequence of Theorems 4 and 5, we get

$$\begin{array}{cc}\hfill {\left({{\Delta }_{\mathrm{ab}}}^{\mathrm{s}}\right)}^{\mathbf{*}}& ={(a{{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\mathrm{d}}^{\mathrm{s}}-b{\mathrm{d}}^{\mathrm{s}}{{\mathrm{d}}^{\mathrm{s}}}^{\ast })}^{\mathbf{*}}=a{\left({\mathrm{d}}^{\mathrm{s}}\right)}^{\mathbf{*}}{\left({{\mathrm{d}}^{\mathrm{s}}}^{\ast }\right)}^{\mathbf{*}}-b{\left({{\mathrm{d}}^{\mathrm{s}}}^{\ast }\right)}^{\mathbf{*}}{\left({\mathrm{d}}^{\mathrm{s}}\right)}^{\mathbf{*}}\hfill \\ \hfill & =a{\iota }_{\nu }(-{\epsilon }_{{\nu }^{\ast }})-b(-{\epsilon }_{{\nu }^{\ast }}){\iota }_{\nu }.\hfill \end{array}$$

□

Theorem 9.

Let $\phi \in C^{\infty }\left(S_q^k\right)$. Then, the following properties hold:

$${\Delta }^{\mathrm{s}}(\Theta \phi )-\Theta {\Delta }^{\mathrm{s}}\phi =4{\nabla }_{\nu }\phi +2n\phi ;$$

(50)

$$\mathrm{div}\mathrm{grad}(\Theta \phi )-\Theta \mathrm{div}\mathrm{grad}\phi =-4{\nabla }_{\nu }\phi -2n\phi .$$

(51)

Proof. 

The assertion is a direct consequence of Theorems 2 and 6. □

Proposition 4.

Let $\phi \in C^{\infty }\left(S_q^k\right)$. Then, the following properties hold:

$${{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\mathrm{d}}^{\mathrm{s}}(\Theta \phi )-\Theta {{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\mathrm{d}}^{\mathrm{s}}\phi =2({\iota }_{\nu }{\mathrm{d}}^{\mathrm{s}}-{{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\epsilon }_{{\nu }^{\ast }})\phi ;$$

(52)

$${\mathrm{d}}^{\mathrm{s}}{{\mathrm{d}}^{\mathrm{s}}}^{\ast }(\Theta \phi )-\Theta {\mathrm{d}}^{\mathrm{s}}{{\mathrm{d}}^{\mathrm{s}}}^{\ast }\phi =2({\mathrm{d}}^{\mathrm{s}}{\iota }_{\nu }-{\epsilon }_{{\nu }^{\ast }}{{\mathrm{d}}^{\mathrm{s}}}^{\ast })\phi .$$

(53)

Proof. 

Using Proposition 2 and Theorem 6, we simply obtain the assertion. □

Theorem 10.

Let $\phi \in C^{\infty }\left(S_q^k\right)$. Then, the following property holds:

$$\begin{array}{cc}\hfill {{\Delta }_{\mathrm{ab}}}^{\mathrm{s}}(\Theta \phi )-\Theta {{\Delta }_{\mathrm{ab}}}^{\mathrm{s}}\phi & =2\left(b{\epsilon }_{{\nu }^{\ast }}{{\mathrm{d}}^{\mathrm{s}}}^{\ast }-a{{\mathrm{d}}^{\mathrm{s}}}^{\ast }{\epsilon }_{{\nu }^{\ast }}\right)\phi \hfill \\ \hfill & +2\left(a{\iota }_{\nu }{\mathrm{d}}^{\mathrm{s}}-b{\mathrm{d}}^{\mathrm{s}}{\iota }_{\nu }\right)\phi .\hfill \end{array}$$

(54)

Proof. 

The assertion is a simple consequence of Theorems 3 and 6. □

The consideration of differential operators that act on forms with polynomial coefficients gives many advantages. Investigating this area of mathematics can be priceless in the process of solving certain boundary problems. For example, in [19], the four natural boundary problems for the weighted-form Laplacians $L=ad\delta +b\delta d$ were explicitly solved, where d is the exterior derivative, $\delta$ is the coderivative and $a,b>0$, for polynomial differential forms in the $n$-dimensional Euclidean ball. Furthermore, an algebraic algorithm for generating a solution from the boundary data were given in each case. Besides, the algebraic and differential nature, for example, of the Laplace transform of the stationary distribution, can be studied in enumerative combinatorics (see [24]). Polynomials can also be very useful in the context of PDE models based on differential algebra. Such investigations involve some types of inverse problems and have applications in finance and biomedicine (see [25]). Recently, an analytic extension of the Kashaev invariant and of the coloured Jones polynomial has been investigated in [26]. This fact has drawn promising expectations for differential operators that act on symmetric forms with polynomial coefficients. The simple algebraic character of their adjoints could be crucial for such considerations.