The study of differential operators in functional spaces plays a vital role in modern analysis, particularly in the theory of Partial Differential Equations (PDEs) and mathematical physics. Among the various types of differential operators, those of higher order, such as fourth-order operators, arise naturally in a broad spectrum of physical models, including elasticity theory, plate and beam problems, quantum mechanics, and other applied disciplines. Understanding the analytic and spectral properties of these operators, especially in weighted spaces, remains a subject of considerable theoretical and practical interest. One key concept in this area is separability. This structural property ensures that a differential operator behaves well under decomposition and allows solutions to be obtained via expansion in terms of suitable basis functions. Separability has direct implications for solvability, uniqueness, and numerical approximation of differential equations. The term “separability” in the context of differential operators was introduced and developed by Everitt and Giertz [1–4], who studied the Sturm–Liouville operator and established foundational results. Later developments, particularly those by Boymatov [2] and Otelbaev [5], and their followers, extended these ideas to broader classes of elliptic and higher-order operators. Closely related to separability is the concept of coercivity, which is a type of a priori estimate that provides lower bounds for operator norms in terms of function norms. Coercive estimates are essential tools in proving well-posedness of boundary value problems and in establishing energy estimates for PDEs. The work of Otelbaev [5] and others has shown that coercivity conditions often serve as sufficient criteria for separability. However, the exact relationship between these two concepts becomes more intricate in weighted function spaces and for higher-order operators. While much of the classical theory focused on second-order elliptic operators, recent research efforts have turned toward the more challenging fourth-order operators. These operators, due to their increased complexity and the presence of higher derivatives, demand more refined analytical techniques, especially when considered in weighted spaces where the geometry and growth conditions of the weight functions significantly influence the analysis. In this paper, we examine a fourth-order differential operator acting in a weighted Hilbert space. We aim to derive new coercive inequalities that serve as sufficient conditions for separability. By precisely characterizing the classes of admissible weight functions and functions involved in the operator, we formulate a theorem that guarantees separability under specific analytic assumptions. The main novelty of our work lies in extending existing results for second-order and biharmonic operators to more general fourth-order operators, and doing so within the broader framework of weighted spaces, which are often encountered in realistic models where non-uniformities or singularities are present.

The remainder of the article is structured as follows. In Section 2, we provide the precise formulation of the problem and state the main theorem. Section 3 develops the auxiliary lemmas and technical estimates needed for the proof. Section 4 is devoted to proving the main result, including detailed coercive inequalities and the role they play in establishing separability. We conclude by summarizing the implications of our findings and suggesting directions for further research in the analysis of nonlinear and higher-order differential operators in weighted functional frameworks. They examined the separability of the Sturm–Liouville operator and its degrees. A significant contribution to the future development of the theory of separability of differential expressions was introduced by Boimatov [2], Otelbaev [5], and their disciples. The separability of differential expressions and the corresponding coercivity inequalities have been studied in many papers (See [1], [2], [5], [6] and references therein). Coercivity estimates and separability for second-order elliptic differential equations were studied in [3]. In [4] and [7] investigated the separability of a biharmonic operator with a matrix potential. The paper [7] is devoted to the study of nonlinear elliptic differential operators of nondivergent form in a weighted space. In this paper, we consider an operator of the fourth order in the weight space L2,ρ(R ), establish the corresponding coercivity inequalities and, on the basis of these inequalities, obtain new sufficient conditions of separability for this operator.

Let's introduce a weight L2,ρ(R ) with finite

∥ u, L2,ρ(R ) ∥= {∫ ρ(x)|u(x)| dx} ,

Rn

where ρ is a positive function.

The space L2,ρ(R ) is a Hilbert space, and in it the scalar product is defined by the equality 〈u, υ〉ρ = ∫ ρ(x)u(x)υ(x)dx.

Rn

In the space L2,ρ(R ) consider the differential equation:

∂ u(x) L[u] = ∑ + V(x) ⋅ u(x) = f(x), (1)

where V(x)- is a positive function.

Definition 1. Eq. (1) and the corresponding differential-operator are called separable in the L2,ρ(R ), если

∂ u(x) n n 4 n n

4 , V(x) ⋅ u(x) ∈ L2,ρ(R ) для всех u(x) ∈ L2,ρ(R ) ∩ W2,loc(R ) such that f(x) ∈ L2,ρ(R ).

Definition 2. We say that the function belongs to a class if the following conditions are satisfied for all:

− ∂ V(x)

2 −1

∑ ‖V (x) V (x)‖ ≤ σ1. (2)

− ∂V(x) ∂u ∂ u ‖V 2(x) ‖ ≤ σ ‖V2(x) ‖.

2 2 (3) ∂xi ∂xi ∂x

Let us formulate the main result of the paper. The following is true.

δ1,δ2

Theorem 1. Let the function V(x) belong to the class Tσ ,σ . Let the weight function ρ(x) belong to the class

2 n n

C (R ), and for all x ∈ R inequalities hold:

∂ρ − ∂u ∂ u

−1 2 2

‖ρ V ‖ ≤ σ3 ∥ V |. (4) ∂xi ∂xi ∂x

∂ρ 1 ∂V(x)

−1 − −1

∑ ‖ρ V 2 V ‖ ≤ σ .

4 (5)

∂ ρ −

−1 2

∑ ‖ρ V ‖ ≤ δ1. (6)

∂ρ ∂u ∂ u ‖ρ ‖ ≤ δ2 | |. (7) ∂xi ∂xi ∂x

Then, if the conditions specified in Eq. (8) are satisfied.

δ1 < 4(1 − 2δ2), δ1 + σ1 + 2σ4 < 2√1 − 2σ2 − 2σ3. (8)

n n 4 n

Eq. (1) is separable in the space L2,ρ(R ), and for all solutions of u(x) ∈ L2,ρ(R ) ∩ W2,loc(R ) equations

n ∂ u(x)

∑ + V(x)u(x) = f(x).

With the right part f(x) ∈ L2,ρ(R ) the following coercive inequality is satisfied:

∂ u(x) ∂u (x)

n n 2 n

∥ ∑ , L2,ρ(R ) ∥ +∥ V(x)u(x), L2,ρ(R ) + ∑ ∥ V (x) , L2,ρ(R ) ∥≤ M ∂xi ∂xi (9)

i=1 i=1

∥ f(x), L2,ρ(R ),

where the positive number M is independent of u(x), f(x).

We will frequently use the following lemma, which is proved by integration by parts.

Lemma 1. Then, for any two functions φ and ψ, the following equality holds:

′ ′ ′ 1 ∞ n

〈ω1, ψω2〉 = −〈ω1, ψ ω2〉 − 〈ω1, ψω2〉, (ω1, ω2 ∈ W2,loc(R), ψ ∈ C0 (R )). (10)

Lemma 2. Let the function f(x) in Eq. (1) belong to the space L2,ρ(R ), the function u(x) belong to the space

n 4 n ∂ u(x) n

L (R ) ∩ W (R ). Then functions V2(x)u(x), , i = 1, n belong to the space L (R ).

2,ρ 2,loc 2 2,ρ

Proof: Let φ be a test function such that a fixed non-negative function is a function that converges to one at. For any positive number let.

There is equality:

∂ u(x) 〈f(x), ρφε(x)u(x)〉 = ∑ 〈 , ρφεu(x)〉 + 〈V(x)u(x), ρφεu(x)〉, (11)

where 〈, 〉 - are the scalar product in space L2,ρ(R ).

Then, applying Lemma 1 several times, we obtain from Equality (11):

〈f(x), ρφε(x)u(x)〉 ∂ u(x) ∂ ρ ∂ u(x) ∂ φε = ∑ 〈 , φεu(x)〉 + ∑ 〈 , ρ u(x)〉 ∂xi ∂xi ∂xi ∂xi

i=1 i=1

∂ u(x) ∂ u(x) ∂ u(x) ∂ρ ∂φε + ∑ 〈 , ρφε 〉 + 2 ∑ 〈 , u(x)〉 2 2 2 (12) ∂x ∂x ∂x ∂xi ∂xi

i i i i=1 i=1

∂ u(x) ∂ρ ∂u(x) ∂ u(x) ∂φε ∂u(x) + 2 ∑ 〈 , φε 〉 + 2 ∑ 〈 , ρ 〉 ∂x ∂xi ∂xi ∂x ∂xi ∂xi

i=1 i=1

+ 〈 V(x, u(x))u(x), φε(x)ρu(x)〉.

Equating the real parts of this equality, we have

Re〈f(x), ρφε(x)u(x)〉 ∂ u(x) ∂ u(x) = ∑ 〈 , ρφε 〉 + 〈V(x, u(x))u(x), φε(x)ρu(x)〉 2 2 (13)

+ A1,ε(u) + A2,ε(u) + 2A3,ε(u) + 2A4,ε(u) + 2A5,ε(u),

∂ u(x) ∂ ρ A1,ε(u) = Re ∑ 〈 , φεu(x)〉,

∂ u(x) ∂ φε A2,ε(u) = Re ∑ 〈 , ρ u(x)〉,

n ∂ u(x) ∂ρ ∂φε

A3,ε(u) = Re ∑ 〈 , u(x)〉, n ∂ u(x) ∂ρ ∂u(x)

A4,ε(u) = Re ∑ 〈 , φ 〉,

i=1 2 ε

n ∂ u(x) ∂φε ∂u(x)

A5,ε(u) = Re ∑ 〈 , ρ 〉.

We now evaluate the absolute values of the functionals Aj(u), j = 1,5,one by one.

Since the function φε(x) is real-valued, and

∂φε ∂ φε | | ≤ M0ε, | | ≤ M1ε , for all x ∈ R , ∂xi ∂x

∂φε ∂ φε M0 = sup | | , M1 = sup | |.

∂xi ∂x

Then A2,ε(u), A3,ε(u) and A5,ε(u) tend to zero at ε → 0. To evaluate A1,ε(u), A4,ε(u), applying the inequality:

y1y2 ≤ |y1| + | y2| , (14) 2 2α

is valid for any α > 0 and all y1 и y2, in view of Inequalitie (6) and Inequalitie (7), we find the following estimates:

n 2 2

αδ1 ∂ u δ1

2 n 2 2 n

|A1,ε(u)| ≥ − ∑ ‖φ , L2,ρ(R )‖ − ‖φ V u;L2,ρ(R )‖ , (15) 2 ∂x 2α

|A4,ε(u)| ≥ −δ2 ∑ ‖φ , L2,ρ(R )‖ . (16)

Passing to the limit in Equality (13) at, using the Cauchy-Bunyakovsky inequality, at an arbitrary positive number, and constants from Condition (8) and Condition (9), we find

∥ f(x); L2,ρ(R ) ∥∥ u(x); L2,ρ(R ) ∥≥ Re 〈f(x), u(x)〉 ≥

δ1α ∂ u δ1

n 2 n

(1 − − 2δ2)∑ ‖ ; L2,ρ(R )‖ + (1 − ) ‖V u;L2,ρ(R )‖ .

2 ∂x 2α

This completes the proof of Lemma 2.

For any ν > 0, the following equality holds

∂ u ∂ u 〈f(x), φερf(x)〉 = 〈Vu(x), φερVu(x)〉 + ν 〈∑ , φερ 〉 + A1(u) − A2(u), (17)

A1(u) = −(1 + ν)Re ∑ 〈 , φερVu〉.

A2(u) = (1 − ν)Re ∑ 〈 , φερf(x)〉.

For any α1 > 0, the following inequality holds.

α1|1 − ν| ∂ u |1 − ν|

ε 2 2 2 2

|A2(u)| ≤ ∑ ∥ φ (x) ∥ + ∥ φ (x)f(x) ∥ , 2 ∂x 2α1

where ∥, ∥ - are norms in the space L2,ρ(R ).

Therefore, we obtain

∂ u ∂φε ∂ u ∂ρ A1(u) = −(1 + ν)Re ∑ 〈 , ρVu(x)〉 − (1 + ν)Re ∑ 〈 , φε(x) Vu(x)〉 ∂x ∂xi ∂x ∂xi

i=1 i i=1 i

∂ u ∂V ∂ u ∂u(x) −(1 + ν)Re ∑ 〈 , φε(x)ρ u(x)〉 − (1 + ν)Re ∑ 〈 , φε(x)ρV 〉.

∂x ∂xi ∂x ∂xi

i=1 i i=1 i

Using Lemma 1, after simple transformations, we find

ε n ∂ u ∂ u ε ε ε

A1(u) = (1 + ν) ∑i=1 〈 , φ (x)ρV(x) (x)〉 + B1(u) + B2(u) + B3(u) +

2 ε 2

(18)

2B4(u) + 2B5(u) + 2 6( ) + 2 7( ) + 2 8( ) + 2 9( ),

∂ u ∂ φε B1(u) = (1 + ν)Re ∑ 〈 , ρV(x)u(x)〉,

ε n ∂ u ∂ ρ

B (u) = (1 + ν)Re ∑ 〈 , φ (x) V(x)u(x)〉,

2 i=1 2 ε 2 ∂x ∂x

ε n ∂ u ∂ V(x)

B (u) = (1 + ν)Re ∑ 〈 , φ (x)ρ u(x)〉,

3 i=1 2 ε 2 ∂x ∂x

ε n ∂ u ∂φε ∂ρ

B (u) = (1 + ν)Re ∑ 〈 , V(x)u(x)〉,

4 i=1 2

ε n ∂ u ∂φε ∂V

B (u) = (1 + ν)Re ∑ 〈 , ρ (x)u(x)〉,

5 i=1 2

n ∂ u ∂φε ∂u(x)

B (u) = (1 + ν)Re ∑ 〈 , ρV(x) 〉,

6 i=1 2

ε n ∂ u ∂V(x) ∂u(x)

B (u) = (1 + ν)Re ∑ 〈 , φ (x)ρ 〉,

7 i=1 2 ε

ε n ∂ u ∂ρ ∂u(x)

B (u) = (1 + ν)Re ∑ 〈 , φ (x) V(x) 〉,

8 i=1 2 ε

ε n ∂ u ∂ρ ∂V(x)

B (u) = (1 + ν)Re ∑ 〈 , φ (x) u(x)〉.

9 i=1 2 ε

ε ε ε ε ε

We evaluate alternately Bj (u), j = 1,9. Functionals B1(u), B4(u), B5(u) and B6(u) tend to zero at ε → 0 by

ε ε ε ε ε

virtue of Lemma 1. When evaluating the functionals B2(u), B3(u), B7(u), B8(u) and B9(u) for any u(x) ∈ L2,ρ(R ), applying inequalities (14), we obtain the following estimates:

βδ1 ∂ u ∂ u δ1

ε 2 2

|B2(u)| ≤ ∑ 〈V , φερV 〉 + (Vu, φερVu), (19) 2 ∂x ∂x 2β

βσ1 ∂ u ∂ u σ1

ε 2 2

|B3(u)| ≤ ∑ 〈V , φερV 〉 + (Vu, φερVu), (20) 2 ∂x ∂x 2β

∂ u ∂ u

ε 2 2

|B7(u)| ≤ σ2 ∑ 〈V , φερV 〉. (21)

∂ u ∂ u

ε 2 2

|B8(u)| ≤ σ3 ∑ 〈V , φερV 〉, (22)

βσ4 ∂ u ∂ u σ4

ε 2 2

|B9(u)| ≤ ∑ 〈V , φερV 〉 + (Vu, φερVu). (23) 2 ∂x ∂x 2β

Here β is an arbitrary positive number, δ1, σ1, σ2, σ3 and σ4 - are constants from Conditions (2)–(6) respectively. Based on the obtained estimates from Equality (17), taking into account inequality:

−|Z| ≤ ReZ ≤ |Z|.

Go to the inequality:

2 2 2 2 2

‖φ f(x))‖ ≥∥ φ Vu(x) ∥ + ν ∑ ∥ φ ∥

ε ε ε 4

α1|1 − ν| ∂ u |1 − ν| − ∑ ∥ φ ∥ − ∥ φ f(x) ∥ 2 ∂x 2α1

+(1 + ν) {∑ ∥ φ V ∥ − |B2(ε)| − |B3(ε)| − 2|B7(ε) − 2|B8(ε)| − 2|B9(ε)|} ,

where ν, β > 0, α1 - are any positive numbers.

Further, having in mind Enequalities (20)–(23) for any ν, α1, α > 0, we obtain the inequality

2 2 2 2 2

‖φ f(x))‖ ≥∥ φ Vu(x) ∥ + ν ∑ ∥ φ ∥ −

ε ε ε 4

α1|1 − ν| ∂ u |1 − ν| − ∑ ∥ φ ∥ − ∥ φ f(x) ∥ 2 ∂x 2α1

+(1 + ν) {∑ ∥ φ V ∥ −

β(δ1 + σ1 + 2σ4) + 4σ2 + 4σ3 ∂ u δ1 + σ1 + 2σ4

2 2 2 2 2

−( ) ∑ ∥ φ V ∥ − ∥ φ Vu(x) ∥ } .

2 ∂x 2β

Passing to the limit at ε → 0, we obtain |1 − ν| (1 + ν)(δ1 + σ1 + 2σ4) (1 + )‖f(x))‖ ≥ (1 − ) ∥ Vu(x) ∥ 2α1 2β α1|1 − ν| ∂ u +(ν − ) ∑ ∥ ∥ 2 ∂x

β(δ1 + σ1 + 2σ4) + 4σ2 + 4σ3 ∂ u +(1 + ν) (1 − ) ∑ ∥ V ∥ .

2 ∂x

Hence, the following is obvious: if |1 − ν| < 2α1, α1|1 − ν| < 2ν, (1 + ν)(δ1 + σ1 + 2σ4) < 2β and β(δ1 + σ1 + 2σ4) + 4σ2 + 4σ3 < 2, then the Coercive Estimate (6). The existence of the numbers α1, β, ν > 0 is deduced from Inequality (8).

The separability of the Nonlinear Operator (1) in the space L2,ρ(R ) follows from the coercive Inequality (12).

This completes the proof of the theorem.

In this work, we investigated the coercive properties and separability of a fourth-order differential operator in a weighted function space. Through the establishment of coercive inequalities, we derived sufficient conditions that ensure the separability of the operator under consideration. These results build upon and extend existing research on second-order and biharmonic operators, providing a broader theoretical foundation for the analysis of higher-order differential equations. The developed criteria contribute significantly to the theory of elliptic differential operators, particularly in weighted Hilbert spaces, where traditional techniques may not be directly applicable. The findings also highlight the importance of weight functions and function classes in determining operator behavior. This study opens the door for future research on nonlinear operators, matrix potentials, and boundary value problems in complex domains, thereby deepening our understanding of the interplay between operator theory and functional analysis.

Acknowledgements enable you to thank all those who have helped in carrying out the research. Careful thought needs to be given concerning those whose help should be acknowledged and in what order. The general advice is to express your appreciation in a concise manner and to avoid strong emotive language.

Conceptualization, O.K. and J.N.; Methodology, O.K.; Software, O.K.; Validation, O.K. and J.N.; formal analysis, O.K.; investigation, O.K.; resources, O.K.; data maintenance, J.N.; writing-creating the initial design, J.N.; writing-reviewing and editing. All authors have read and agreed to the published version of the manuscript.

This section indicates any support that is not included in the Author Contribution or Funding sections. This may include administrative and technical support or in-kind donations. Data availability: all data supporting the reported findings in this research paper are provided within the manuscript.

The authors declare that there is no conflict of interest concerning the reported research findings. Funders played no role in the study’s design, in the collection, analysis, or interpretation of the data, in the writing of the manuscript, or in the decision to publish the results.