Adv Syst Sci Appl 2021; 03:22–30 Published online at https://ijassa.ipu.ru. On Spectral Decomposition of Generalized Bessel Potentials M.L. Goldman1, N.H. Alkhalil1* 1 Mathematical Institute named S.M. Nikolskii Peoples’ Friendship University of Russia (RUDN University) Moscow 117198, Miklukho-Maklaya str. 6, Russia Abstract: We establish the localization condition for the γ-means spectral decomposition by the system of fundamental functions of the Laplace operator in an arbitrary multi-dimensional domain. The result is obtained in terms of belonging of decomposing function to the spaces of the generalized Bessel potential. For conditions of localization, we apply the exact estimates for the modulus of continuity of the potential. The generalized Bessel potentials are constructed using convolutions of functions with kernels that generalize the classical Bessel–MacDonald kernels. In contrast to the classical case, non-power singularities of kernels are allowed in the vicinity of the origin. Differential properties of potentials arc described by using the k-th order modulus of continuity in the uniform norm. Keywords: spectral decomposition, the generalized Bessel potential, the modulus of continuity of the potential, Laplace operator 1. INTRODUCTION The paper is organized as follows. Section 1 contains basic definitions of the potential theory. The main properties of kernels are considered and basic spaces for potentials are described. Sections 3.1, 3.2 contain some auxiliary results. In Section 3.1 we give the condition of the embedding of the space HG E (Rn) of generalized Bessel potentials into the space C(Rn) of bounded uniformly continuous functions, and obtain the estimates for modulus of continuity of potentials, see Theorem 3.1. In Section 3.2 we give the conditions for localization of γ-means of spectral decomposition in terms of properties of modulus of continuity for decomposing function, see Theorem 3.2. In Section 4.1 we prove Theorem 4.1 giving the conditions for including of the space HG E (Rn) into the scheme of spectral decomposition. Finally, Theorem 4.2 in Section 4.2 gives the conditions for localization of spectral decomposition for generalized Bessel potentials constructed over the basic weighted Lorentz spaces. ∗Corresponding author: khaleel.almahamad1985@gmail.com ON SPECTRAL DECOMPOSITION OF GENERALIZED BESSEL POTENTIALS 23 2. BASIC DEFINITIONS Let v > 0 be measurable function on R+. The Lorentz space Λp(v) is the space of measurable functions on Rn with finite (quasi) norms (see [1]) ‖f‖Λp(v) =  (∫∞ 0 f ∗(t)pv(t) dt ) 1 p ; 0 < p <∞; ess sup t∈R+ { f ∗(t)v(t) } ; p =∞. (2.1) Here f ∗ : R+ → [0,∞] is the decreasing rearrangement of function f : Rn → R, i.e f ∗ is a nonnegative decreasing right-continuous function on R+ = (0,∞) which is equimeasurable with f : µn {x ∈ Rn : |f(x)| > y} = µ1 {t ∈ R+ : |f ∗(t)| > y} , y ∈ R+, (2.2) where µn is the n-dimensional Lebesgue measure. We assume that 0 < V (t) := ∫ t 0 v(τ)d(τ) <∞, t ∈ R+, and sup t∈R+ [ V (2t) V (t) ] <∞, (2.3) is the so called ∆2-condition. Under these assumptions E(Rn) = Λp(v) is a (quasi) Banach space which gives an important example of a rearrangement invariant space (shortly:RIS), because of property: g∗ ≤ f ∗, f ∈ E(Rn) ⇒ g ∈ E(Rn), ‖g‖E ≤ ‖f‖E. (see C. Bennett and R. Sharpley [1]). Moreover, E ′ = E ′(Rn) is the associated RIS for E(Rn), i.e. E ′ is RIS with the norm: ‖g‖E′ = sup {∫ Rn |fg| dµn : f ∈ E, ‖f‖E ≤ 1 } . (2.4) For 1 < p <∞ the description of the associated space for E(Rn) = Λp(v) was obtained by E. Sawyer [11]. Namely, ‖g‖E′ = sup 0≤h↓ ∫∞ 0 g ∗ (τ)h(τ) dτ(∫∞ 0 h(τ)pv(τ) dτ ) 1 p ≈ ( ∞∫ 0 ( ξ∫ 0 g∗(τ)dτ )p′ v(ξ)dξ V (ξ)p′ ) 1 p′ , (2.5) Here the symbol ≈ means that the ratio of left and right hand sides is bounded between positive constants depending only on p (and not on v or g). The potential space HG E ≡ HG E (Rn) for E(Rn) = Λp(v) is defined as the set of convolutions of potential kernel G with all functions belonging to the basic RIS E(Rn): HG E (Rn) = { u = G ∗ f : f ∈ E(Rn) } . (2.6) We define ‖u‖HG E = inf { ‖f‖E : f ∈ E(Rn), G ∗ f = u } . (2.7) We assume that the kernel G of a representation (2.7) is admissible, i.e G ∈ L1(Rn) + E ′(Rn). Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 24 M.L. GOLDMAN, N.H. ALKHALIL Here the convolution G ∗ f is defined as the integral (G ∗ f)(x) = ∫ Rn G(x− y)f(y) d(y). For function Φ: R+ → [0,∞) we define ϕ(τ) = Φ (( τ Vn ) 1 n ) , τ ∈ R+. (2.8) Definition 2.1: Let k, n ∈ N; R ∈ R+. We say that function Φ belongs to the class Ik,n(R) if it satisfies the following conditions: 1. 0 < Φ ↓ on (0, R); ∃ c ∈ R+ such that r∫ 0 Φ(ρ)ρn−1 dρ ≤ cΦ(r)rn, r ∈ (0, R); ∞∫ R Φ(ρ)ρn−1 dρ <∞. (2.9) 2. G(x) := Φ(|x|) ∈ Ck(Rn\0), and for Gk(x) := ∑ |α|=k |DαG(x)|, x ∈ Rn\0, the estimate holds: for some c1 ∈ R+ |Gk(x)| ≤ c1Ψk(|x|), x ∈ Rn\0; (2.10) where Ψk ∈ C(R+), and for T = VnR n ϕk(τ) := Ψk (( τ Vn ) 1 n ) ≤ τ−k/nϕ(τ), τ ∈ (0, T ]; (2.11) ∞∫ T ϕk(τ) dτ <∞. (2.12) Definition 2.2: For RISE(Rn) the spaceHG E (Rn) (2.6)-(2.7) with kernelG(x) = Φ(|x|), where Φ ∈ Ik,n(R), is called the space of generalized Bessel potentials. Remark 2.1: Note that the classical Bessel–McDonald kernels have the form Gα(x) = c(α, n)ρ−βKβ(ρ), ρ = |x| ∈ R+, α ∈ (0, n), β = n− α 2 , where Kβ is the McDonald function, see [10]. The well-known properties of these kernels show that Φ(ρ) = ρ−βKβ(ρ) ∈ Ik,n(R) for fixed R ∈ R+, moreover Φ(ρ) ∼= ρα−n, ρ ∈ (0, R); Φ(ρ) ∼= ρ−β− 1 2 e−ρ, ρ > R, and our scheme includes classical Bessel potentials. Remark 2.2: Note that functions Φ ∈ Ik,n(R) may have the property Φ(ρ) = 0, ρ ∈ [2R,∞). (2.13) Thus, the case of kernels with compact support is included in our scheme. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) ON SPECTRAL DECOMPOSITION OF GENERALIZED BESSEL POTENTIALS 25 Definition 2.3: Let C(Rn) be the space of bounded and uniformly continuous functions with the norm ‖u‖C = sup x∈Rn |u(x)|. For u ∈ C(Rn) modulus of continuity of order k ∈ N is defined as ωkC(u; t) = sup { ‖∆k hu‖C : |h| ≤ t } , t ∈ R+, ∆k h(u;x) = k∑ m=0 (−1)k−mCm k u(x+mh), is the k-th difference of function u with the step h ∈ Rn at the point x ∈ Rn. For u ∈ L2(Ω), where Ω is domain in Rn, we define ωk2,Ω(u; t) = sup |h|≤t ‖∆k hu‖L2(Ωkh); where Ωkh = { x ∈ Ω : [x, x+ kh] ⊂ Ω } . Definition 2.4: Let ω ∈ C(0, 1], 0 ≤ ω(t) ↑; t−kω(t) ↓ on (0, 1]. The Nikolskii-type space with generalized smoothness Hω(·) 2 (Ω) is defined as H ω(·) 2 (Ω) = { u ∈ L2(Ω) : ‖u‖ H ω(·) 2 (Ω) <∞ } , (2.14) where ‖u‖ H ω(·) 2 (Ω) = ‖u‖L2(Ω) + sup 0 0, x where f̂y = m∑ i=1 f̂iyi. Let s > 0 and ψ be a function on (0, 1] with properties 0 < ψ ↑ on (0, 1], and ψ(t) ∼= ψ(τ) if t ∼= τ . Moreover, for s > 0 and s0 = s if s ≤ 1; s0 = 1 if s > 1 we require that 1) ψs0(t) = t∫ 0 τ s0−1ψ(τ) dτ <∞, t ∈ (0, 1], (3.22) 2) ψ ∈ C2(0, 1); |ψ′(τ)| ≤ cψ(τ)τ−1, |ψ′′(τ)| ≤ cψ(τ)τ−2, τ ∈ (0, 1], (3.23) 3) 1∫ 0 (1− τ)s−1ψ(τ) dτ = Γ(s). (3.24) Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) ON SPECTRAL DECOMPOSITION OF GENERALIZED BESSEL POTENTIALS 27 We define γ as s-th Riemann–Liouville integral γ(t) = 1 Γ(s) t∫ 0 (t− τ)s−1ψ(τ) dτ, t ∈ (0, 1]. (3.25) Let us introduce the γ-means of spectral decomposition as σγµ(f ;x) = µ∫ 0 f̂(t)y(x, t)γ ( 1− t2 µ2 ) dρ(t), µ > 0, (3.26) for f ∈ L2(F ). We see that γ(1) = 1 and if ψ(τ) ≡ Γ(s+ 1), τ ∈ (0, 1], then γ(t) = ts, and γ-means reduce to the classical Riesz means of orders, see [2, 3]. We define ω0(t) = t n−1 2 + s0 − s ψs0(t) , t ∈ (0, 1]. (3.27) Let α, β ≥ 0 be such that n− 2 2 − s < α ≤ β < min { α + 3 2 , n 2 + 1 } . (3.28) Let function ω satisfy the conditions ω(t)t−α ↑, ω(t)t−β0 ↓ on (0, 1]; β0 = min{β, k}, (3.29) and lim t→+0 ω(t) ω0(t) = 0. (3.30) Let Ω ⊂⊂ F , that is Ω is bounded domain, and Ω̄ ⊂ F , f ∈ Hω(·) 2 (Ω) ∩ L2(F ). (3.31) Theorem 3.2: [4, 5]. In the notations and assumptions of Section 3.2 (3.22)–(3.31), let D ⊂ Ω and function f satisfies the condition f(x) ≡ 0, x ∈ D. Then, for each compactK ⊂ D uniformly in x ∈ K the relation holds: lim µ→∞ σγµ(f ;x) = 0. (3.32) Remark 3.1: Theorem 3.2 gives sharp conditions for localization of γ-means of spectral decomposition. In typical situations we have s < n−1 2 , ψs0(t) ∼= ts0ψ(t), ω0(t) ∼= t n−1 2 −s ψ(t) , t ∈ (0, 1]. In particular, for Riesz means, ψ(t) ≡ Γ(s+ 1) and (3.30) gives the condition ω(t) = ¯̄0 ( t n−1 2 −s ) . Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 28 M.L. GOLDMAN, N.H. ALKHALIL 4. LOCALIZATION FOR γ-MEANS OF SPECTRAL DECOMPOSITION FOR GENERALISED BESSEL POTENTIALS 4.1. Theorem 4.1: Let the notation and assumptions (3.16)–(3.18) and (3.20) be satisfied, V (+∞) =∞, and moreover Bp := sup [ t 1 2V (t)− 1 p : t ∈ R+ ] <∞, 0 < p ≤ 2; (4.33) Bp := (∫ ∞ 0 [ t 1 2V (t)− 1 p ]s v(t) dt V (t) ) 1 s <∞, 2 < p <∞, s = 2p p− 2 . (4.34) Then, forE(Rn) = Λp(v) there is the embedding of the space of generalized Bessel potentials HG E (Rn) ⊂ L2(Rn). (4.35) Proof Note that the conditions (4.33), (4.34) give the criterion of embedding E = Λp(v) ⊂ L2(Rn). (4.36) It follows from the assertions (see [7, 8]): let Cp := sup f∈Λp(v) [(∫ ∞ 0 f ∗(τ)2 dτ ) 1 2 (∫ ∞ 0 f ∗(τ)p v(τ) dτ )− 1 p ] . Then, Cp = Bp, 0 < p ≤ 2; Cp ∼= Bp, 2 < p <∞. The embedding (4.36) implies the embedding HG E (Rn) ⊂ HG L2 (Rn). But, in the assumptions of Section 3.1 the kernel G(x) = Φ(|x|) ∈ L1(Rn), and by the generalized Minkovkii inequality for convolutions f ∈ L2(Rn)⇒ u = G ∗ f ∈ L2(Rn), ‖u‖L2(Rn) ≤ ‖G‖L1(Rn)‖f‖L2(Rn). This gives embedding (4.35). 4.2. Now, let us recall the notations and assumptions of Section 3.2. Let the conditions of Theorem 4.1 be satisfied, and HG E (Rn) be the space of generalized Bessel potentials with E(Rn) = Λp(v). We assume that the notations and assumptions of Section 3.2 hold, see (3.22)–(3.28). Under assertions (3.16)–(3.18) and (3.20), we define ω(t) := Ap(t) in (3.29), (3.30). Theorem 4.2: Under assumptions of this Section, let D, Ω, F be domains in Rn, and D ⊂ Ω ⊂⊂ F . If u ∈ HG E (Rn); u(x) ≡ 0, x ∈ D, (4.37) then for each compact K ⊂ D uniformly in x ∈ K the relation holds: lim µ→∞ σγµ(u;x) = 0. (4.38) Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) ON SPECTRAL DECOMPOSITION OF GENERALIZED BESSEL POTENTIALS 29 Proof By Theorem 4.1 we have the embedding (4.35). It means that u ∈ HG E (Rn)⇒ u ∈ L2(Rn)⇒ u ∈ L2(F ), and γ-means of spectral decomposition σγµ(u;x) are correctly defined by formula σγµ(u;x) = µ∫ 0 û(t)y(x, t)γ ( 1− t2 µ2 ) dρ(t), (4.39) where y(x, t) is the system of fundamental functions ûy = m∑ i=1 ûiyi; ûi(t) = ∫ F u(x)yi(x, t) dx, see notations in Section 3.2. Moreover, for u ∈ HG E (Rn) we have ω2,Ω(u; t) ≤ ωC(u; t)(mes Ω) 1 2 . (4.40) Indeed, ‖∆k hu‖L2(Ωkh) = (∫ Ωkh |∆k hu|2 dx ) 1 2 ≤ sup x∈Ωkh |∆k hu(x)|(mes Ωkh) 1 2 , and for ωk2,Ω(u; t) = sup |h|≤t ‖∆k hu‖L2(Ωkh) we have the estimates ωk2,Ω(u; t) ≤ sup |h|≤t sup x∈Rn |∆k hu(x)|(mes Ω) 1 2 = = sup |h|≤t ‖∆k hu‖C(Rn)(mes Ω) 1 2 = ωkC(u; t)(mes Ω) 1 2 , and (4.40) follows. Thus, by Theorem 3.1, see (3.21), we have ωk2,Ω(u; t) ≤ c2(mes Ω) 1 2‖u‖HG E (Rn)Ap(t), t ∈ (0, T ]. (4.41) Now, we apply Theorem 3.2. This estimates shows that u ∈ HG E (Rn)⇒ u ∈ Hω(·) 2 (Ω), ω(t) = Ap(t). (4.42) Finally, u ∈ Hω(·) 2 (Ω) ∩ L2(F ), and we obtain the assertion (4.38) by applying of the Theorem 3.2. Remark 4.1: Note that if kernel G of generalized Bessel potentials is compactly supported on B2R = {x ∈ Rn : |x| < 2R}, see the condition (2.13), we have for u ∈ HG E (Rn) u(x) = (G ∗ f)(x) ≡ 0, x ∈ D, Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 30 M.L. GOLDMAN, N.H. ALKHALIL function f ∈ E(Rn) satisfies the condition f(x) ≡ 0, x ∈ D2R = { x ∈ Rn : ρ(x,D) < 2R } , where ρ(x,D) = inf { |x− y| : y ∈ D } is the distance from x to D. Remark 4.2: The posing of the problem and the results of Sections 3.2 and 4.1 belong to M.L. Goldman. The other results of paper obtained by N.H. Alkhalil. Remark 4.3: In this paper we consider the differential properties of generalized Bessel potentials. Their integral properties are connected with estimates of Hardy-type operators on the cones of monotone functions considered in [12]. ACKNOWLEDGEMENTS The research of M.L. Goldman is supported by the Ministry of Science and Higher Education of the Russian Federation: agreement No. 075-03-2020-223/3 (FSSF-2020-0018). REFERENCES 1. Bennett, C., & Sharpley, R. (1988) Interpolation of operators. Boston: Acad. Press, 1988. (Pure and Appl. Math.; V. 129). 2. Il’in, V.A., & Alimov, Sh.A. (1971) Conditions for the convergence of spectral decompositions that correspond to self-adjoint extensions of elliptic operators. I, Differential Equations 1971, 7(4), 670–710. 3. Il’in, V.A., & Alimov, Sh.A. (1971) Conditions for the convergence of spectral decompositions that correspond to self-adjoint extensions of elliptic operators. II, Differential Equations 1971, 7(5), 851–882. 4. Ayele, T.G., & Goldman, M.L. (2014) Spaces of generalized smoothness in summability problems for Φ-means of spectral decomposition. Eurasian Mathem. Journal 2014, 5(1), 61–81. 5. Goldman, M.L., & Tsegaye, G.A. (2015) Spaces with Generalized Smoothness in Summability Problems for Φ-means of Spectral Decompositions. Springer International Publishing, Switzerland 2015, Current Trends in Mathematics, 163–169. 6. Alkhalil, N.Kh. Modulus of continuity for Bessel type potentials over Lorentz space. Eurasian Mathem. Journal, (to appear). 7. Burenkov, V.I., & Goldman, M.L. (1995) Calculation of the norm of a positive operator on the cone of monotone functions. Proc. of the Steklov Inst. Math., 1995, 210, 65–89. 8. Goldman, M.L. (1998) Hardy-type inequalities on the cone of quasimonotone functions. Russian Academy of Sciences, Far-Eastern Branch, Research Report 98/31, Khabarovsk, 1998, 1–70. 9. Goldman, M.L., & Malysheva, A. (2013) Estimates of the uniform modulus of continuity for Bessel potentials. Proc. of the Steklov Inst. Math. 2013, 283, 1–12. 10. Nikol’skii, S.M. Approximation of Functions of Several Variables and Imbedding Theorems. Springer, Berlin-Heidelberg-New York, 1975. 11. Sawyer, E. (1990) Boundedness of classical operators on classical Lorentz spaces. Studia Math., 96, 145–158. 12. Almohammad, Kh. (2021). The Modular Inequalities for Hardy-type Operators on Monotone Functions in Orlicz Space. Advances in Systems Science and Applications, 21(2), 133–141. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) Introduction Basic definitions Auxiliary theorems Localization for -means of spectral decomposition for generalised Bessel potentials