ISSN 1064 5624, Doklady Mathematics, 2011, Vol 84, No 2, pp 672–674 © Pleiades Publishing, Ltd., 2011 Original Russian Text © H.H Bang, V.N Huy, 2011, published in Doklady Akademii Nauk, 2011, Vol 440, No 4, pp 456–458 MATHEMATICS Behavior of Sequences of Norms of Primitives of Functions Depending on Their Spectrum H H Banga and V N Huyb Presented by Academician V.S Vladimirov January 20, 2011 Received May 11, 2011 DOI: 10.1134/S1064562411060263 In this paper, the behavior of the sequence of Lp(ޒn) norms of primitives of distributions (general ized functions) is studied depending on their spectrum (on the support of their Fourier transform) The following result was proved in [1]: let ≤ p ≤ ∞ and f (m) ∈ Lp()ޒ, m = 0, 1, 2, … Then the limit lim f m→∞ ( m ) 1/m p always exists and lim f m→∞ ( m ) 1/m p e where is at the jth place I j f ∈ S'(ޒn) is called the = σ f = sup { ξ : ξ ∈ suppfˆ}, e where ˆf is the Fourier transform of f This result shows that the behavior of the sequence f pose, we need the concept of the primitive of general ized functions, which was introduced by Vladimirov for generalized functions in D'(a, b), a, b ∈ [ ޒ15] Developing his idea, we define this concept for tem pered generalized functions of many variables Let S(ޒn) and S'(ޒn) be the Schwartz spaces of test func tions and generalized functions, respectively For h ∈ S'(ޒn) and ∀ϕ ∈ S(ޒn), we define h(ϕ) := 〈h, ϕ〉 Let f ∈ S'(ޒn) and ej = (0, …, 0, 1, 0, …, 0) ∈ ގn, e is completely characterized by the spectrum of f, which is the support of the generalized function ˆf lim I f m→∞ –1 = σ , n +∞ ψ ( x ) = ϕ ( x ) – θ ( xj ) This issue has been investigated by numerous authors (see, e.g., [1–12]) A natural question arises as to what happens if the derivatives are replaced by primitives For p = 2, Tuan answered this question in [13]: Let f ∈ L2( )ޒand σ := inf{|ξ|: ξ ∈ suppˆf } > Then there exists primitives Imf, Imf ∈ L2( )ޒfor all m, and 1/m e 〈 I j f, D j ϕ〉 = – 〈 f, ϕ〉 ∀ϕ ∈ S ( ) ޒ For ϕ ∈ S(ޒn), we define ( m ) 1/m p m e ejth primitive of f if D j ( I j f ) = f; i.e., ∫ ϕ ( x , x , …, x j – 1, t, x j + , –∞ x j + , …, x n )dt, xj Φ(x) = ∫ ψ ( x , x , …, x j – 1, t, x j + , x j + , …, x n )dt, –∞ where ∞ θ ∈ C ( ( – 1, ) ), where suppˆf is understood as the smallest closed set outside of which ˆf = a.e In the general case of ≤ p ≤ ∞ and σ ≥ 0, this prob lem was solved in [14] Below, the problem is studied in the case of many variables and ≤ p ≤ ∞ For this pur ∫ θ ( t ) dt = ޒ ej If I f ∈ S'(ޒn) is some ejth primitive, then e 〈 I j f, ϕ〉 = – 〈 f, Φ〉 + 〈 g j, ϕ〉 , (1) where gj ∈ S'(ޒn) is defined as a Institute of Mathematics, Vietnamese Academy of Science and Technology, Hoang Quoc Viet Road 18, Cau Giay, Hanoi, 10307 Vietnam e mail: hhbang@math.ac.vn b Department of Mathematics, College of Science, Hanoi National University, Hanoi, Vietnam e mail: nhat_huy85@yahoo.com 〈 g j, ϕ〉 = ∫ f j ( x 1, x 2, …, x j – 1, x j + 1, x j + 2, …, x n ), × ϕ ( x 1, x 2, …, x j – 1, t, x j + 1, x j + 2, …, x n ) dt ޒ 672 ∫ BEHAVIOR OF SEQUENCES OF NORMS OF PRIMITIVES OF FUNCTIONS Here, fj ∈ S'(ޒn – 1) and n ∏ sin x 〈 f j, h〉 e = 〈 I j f ( x ), θ ( x j )h ( x 1, x 2, …, x j – 1, x j + 1, x j + 2, …, x n )〉 for every h ∈ S(ޒn – 1) Then gj is the jth constant tem pered generalized function Note that, if n = 1, then gj is a constant Conversely, for any jth constant tem e pered generalized function gj, the functional I j f over S(ޒn) defined by formula (1) is the ejth primitive of f Thus, we have proved the following result: each func ej tion f ∈ S'( ޒhas its own primitive I f in S'( ޒand every ejth primitive of f is given by (1), where gj is any jth constant tempered generalized function Let I 0f = f In what follows, for j = 1, 2, …, n and a n) n) ej function f ∈ S'(ޒn), the symbol I f stands for some of e e its ejth primitives; i.e., I j f ∈ ᏼ j ( f ) For any multi ej α – ej ej α – ej index α ∈ ގn with |α| ≥ 1, we define Iαf = I α ( I α f), where jα := max{j: αj ≥ 1}; i.e., Iαf ∈ ᏼ α ( I α f ) Therefore, Dα(Iαf ) = f for all α ∈ ގn Let ≤ p ≤ ∞ and f ∈ Lp(ޒn) We write (Iαf ) α ∈ ގn ⊂ Lp(ޒn) if for any α ∈ ގn there exists an e jα th primitive of I I α – ej 673 α – ej α f (denoted by f) that belongs to Lp(ޒn) For δ > 0, let α n n ( ޒ, δ ) = { ξ = ( ξ 1, ξ 2, …, ξ n ) ∈ ޒ: { ξ , ξ , …, ξ n } > δ } It was shown above that ∀f ∈ S'(ޒn) and ∀α ∈ ގn there always exists a primitive Iαf ∈ S'(ޒn) The ques tion arises as to what happens if S'(ޒn) is replaced by Lp(ޒn) A function f ∈ S'(ޒn) is said to satisfy condition (O) if there exists δ > such that suppˆf ⊂ (ޒn, δ) It turns out that condition (O) ensures the existence of some primitive of any order of f ∈ Lp(ޒn) Theorem Let ≤ p ≤ ∞, f ∈ Lp(ޒn), and condition (O) be satisfied Then, for any j = 1, 2, …, n, there exists ej precisely one ejth primitive of f, which is denoted by I f, e such that I j f belongs to Lp(ޒn) and satisfies condition (O) Moreover, e suppI j f = suppfˆ Assume that f ∈ Lp(ޒn) but condition (O) does not j (then suppˆf = {ξ ∈ ޒn: ξj ∈ {0, –2, 2}, j = 1, j=1 n 2, …, n}); and if ≤ p < ∞ and f(x) = j=1 DOKLADY MATHEMATICS Vol 84 No 2011 (then xj suppˆf = [–2, 2]n) Note also that, if p < ∞, f ∈ Lp(ޒn), and j = 1, 2, …, n, e then there exists at most one I j f ∈ Lp(ޒn) Moreover, e if p = ∞ and f, I j f ∈ L∞(ޒn) for some j ∈ {1, 2, …, n}, then f(x) + c = g(x) ∈ L∞(ޒn) for all c ∈ ރ, c ≠ 0, while e there is no I j g ∈ L∞(ޒn) n Let p = ∞ and f(x) = ∏ cos x Then suppˆf j = j=1 {x ∈ ޒn: xj ∈ {–1, 1}, j = 1, 2, …, n} and there exists any αth primitive of f in L∞(ޒn) Theorem Let ≤ p ≤ ∞ and f ∈ Lp(ޒn) Then there exists at most one sequence (Iαf ) α ∈ ގn ⊂ Lp(ޒn) Theorem Let ≤ p ≤ ∞, f ∈ Lp(ޒn), and condition (O) be satisfied Then there exists precisely one sequence α of primitives (Iαf ) α ∈ ގn ⊂ Lp(ޒn) Moreover, supp I f = supp ˆf ∀α ∈ ގn The following results hold for the sequence of norms of generalized primitives Theorem Let ≤ p ≤ ∞, (Iαf ) α ∈ ގn ⊂ Lp(ޒn), f ≡ 0, and condition (O) be satisfied Then there always exists the limit α α lim ( ( inf ξ ) I f α →∞ ξ ∈ suppfˆ p) 1/ α = Note that, if condition (O) is not satisfied, then α Theorem does not hold, since inf ξ = for all α ∈ | ގαj| ≥ 1, j = 1, 2, …, n ξ ∈ suppfˆ n: Theorem Let ≤ p ≤ ∞, (Iαf ) α ∈ ގn ⊂ Lp(ޒn), n n and σ = (σ1, σ2, …, σn) ∈ ޒ+ Then suppˆf ⊂ α α lim ( σ I f p ) α →∞ 1/ α ∏ ( –∞, k=1 –σk] ∪ [σk, +∞) if and only if ≤ (2) Theorem Let ≤ p < ∞, (Iαf ) α ∈ ގn ⊂ Lp(ޒn), and n e hold Then it is possible that there is no primitive I j f ∈ Lp(ޒn), j = 1, 2, …, n This occurs, for example, if p = ∞ and f ≡ (then suppˆf = {0}); if p = ∞ and f(x) = ∏ sin x j σ = (σ1, σ2, …,σn) ∈ n ޒ+ Assume that suppˆf –σk] ∪ [σk, +∞) Then ⊂ ∏ ( –∞, k=1 674 BANG, HUY α α lim σ I f p α →∞ = (3) Remark Theorem does not hold if p = ∞ Theorem Let ≤ p ≤ ∞, P(x) be a polynomial in n ∞ variables, f ≡ 0, and (ᏼmf ) m = ⊂ Lp(ޒn) Assume that ˆf has a compact support Then there always exist the limits n Indeed, let σ ∈ ޒ+ and m lim ᏼ f m→∞ n f(x) = and ∏ sin σ x j j m lim ᏼ f j=1 m→∞ 1/m p ∏ ( –∞, –σj] ∪ [σj, +∞) and j=1 n σ ||I f ||∞ = 1, α ∈ ގ Let f ∈ S'(ޒn) and P(x) be a polynomial in n vari ables Define α α –1 = ( inf P ( ξ ) ) n Then suppˆf ⊂ 1/m p , ξ ∈ suppfˆ ACKNOWLEDGMENTS This work was supported by the Vietnamese National Foundation for Science and Technology Development, project no 101.01.50.09 n ( ސf ) = { h ∈ S' ( ) ޒ: P ( D )h = f }, where the differential operator P(D) is obtained from P(x) by making the substitution xj → – i ∂ Each ele ∂x j ment h ∈ (ސf ) is called a Pth primitive of f Let ᏼ0f = f and ᏼf be some Pth primitive of f; i.e., ᏼf ∈ (ސf ), and ᏼm + 1f ∈ (ސᏼmf ), m = 0, 1, 2, … Then Pk(D)ᏼm + kf = ᏼmf for k, m = 0, 1, … Let ≤ p ≤ ∞ and f ∈ Lp(ޒn) If for any m = 0, 1, … there is a Pth primitive of ᏼmf (denoted by ᏼm + 1f ) that belongs to Lp(ޒn), then we ∞ write (ᏼmf ) m = ⊂ Lp(ޒn) Theorem Let ≤ p ≤ ∞, P(x) be a polynomial in n ∞ variables, f ≡ 0, and (ᏼmf ) m = ⊂ Lp(ޒn) Then, for all m ∈ ގ, m suppfˆ = suppᏼ f Theorem Let ≤ p ≤ ∞, P(x) be a polynomial in n ∞ variables, f ≡ 0, and (ᏼmf ) m = ⊂ Lp(ޒn) Then m lim ᏼ f m→∞ 1/m p –1 ≥ ( inf P ( ξ ) ) ξ ∈ suppfˆ REFERENCES H H Bang, Proc Am Math Soc 108, 73–76 (1990) H H Bang and M Morimoto, Tokyo J Math 14 (1), 231–238 (1991) H H Bang, Bull Polish Akad Sci 40, 197–206 (1993) H H Bang, Trans Am Math Soc 347, 1067–1080 (1995) H H Bang, Tokyo J Math 18 (1), 123–131 (1990) H H Bang, J Math Sci Univ Tokyo 2, 611–620 (1995) H H Bang, Dokl Math 53, 420–422 (1996) H H Bang, Izv Math 61, 399–434 (1997) H H Bang, Dokl Math 55, 377–380 (1997) 10 N B Andersen, Pacific J Math 213 (1), 1–13 (2004) 11 J J Betancor, J D Betancor, and J M R Méndez, Publ Math Debrecen 60, 347–358 (2002) 12 V K Tuan and A Zayed, J Math Anal Appl 266, 200–226 (2002) 13 V K Tuan, J Fourier Anal Appl 7, 319–323 (2001) 14 H H Bang and V N Huy, J Approx Theory 162, 1178–1186 (2010) 15 V S Vladimirov, Methods of the Theory of Generalized Functions (Taylor & Francis, New York, 2002) DOKLADY MATHEMATICS Vol 84 No 2011 .. .BEHAVIOR OF SEQUENCES OF NORMS OF PRIMITIVES OF FUNCTIONS Here, fj ∈ S'(ޒn – 1) and n ∏ sin x 〈 f j, h〉 e = 〈 I j f ( x ),... precisely one ejth primitive of f, which is denoted by I f, e such that I j f belongs to Lp(ޒn) and satisfies condition (O) Moreover, e suppI j f = suppfˆ Assume that f ∈ Lp(ޒn) but condition (O)... tion arises as to what happens if S'(ޒn) is replaced by Lp(ޒn) A function f ∈ S'(ޒn) is said to satisfy condition (O) if there exists δ > such that suppˆf ⊂ (ޒn, δ) It turns out that condition