DSpace at VNU: Behavior of Sequences of Norms of Primitives of Functions Depending on Their Spectrum

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