Báo cáo toán học: "A Gagliardo-Nirenberg Inequality for Lorentz Spaces"

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Trong bài báo này, về cơ bản phát triển các phương pháp [1] và [10], chúng tôi cung cấp cho một phần mở rộng của sự bất bình đẳng Gagliardo-Nirenberg không gian Lorentz.

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9LHWQDP -RXUQDO Vietnam Journal of Mathematics 33:2 (2005) 207–213 RI 0$7+(0$7,&6 9$67 A Gagliardo-Nirenberg Inequality for Lorentz Spaces* Mai Thi Thu Ca Mau Pedagogical College, Nguyen Tat Thanh Road, Ca Mau City, Vietnam Received July 23, 2004 Revised November 26, 2004 Abstract. In this paper, essentially developing the method of [1] and [10], we give an extension of the Gagliardo-Nirenberg inequality to Lorentz spaces. Let Φ : [0, ∞) → [0, ∞) be a non-zero concave function, which is non-decreasing and Φ(0+) = Φ(0) = 0. We put Φ(∞) = limt→∞ Φ(t). For an arbitrary mea- surable function f we deﬁne ∞ f = Φ λf (y ) dy, NΦ 0 where λf (y ) = mes{x ∈ R : |f (x)| > y } , (y ≥ 0). The space NΦ (Rn ) consisting n of measurable functions f such that f NΦ < ∞ is a Banach space. Denote by MΦ (Rn ) the space of measurable functions g such that 1 |g (x)|dx : Δ ⊂ Rn , 0 < mes Δ < ∞ < ∞. g = sup MΦ Φ(mes Δ) Δ Then MΦ (Rn ) is a Banach space, see [7 - 9]. We have the following results [8 - 9]: ∗ This work was supported by the Natural Science Council of Vietnam.
208 Mai Thi Thu Lemma 1. If f ∈ NΦ (Rn ), g ∈ MΦ (Rn ) then f g ∈ L1 (Rn ) and |f (x)g (x)|dx ≤ f g MΦ . NΦ Rn Lemma 2. If f ∈ NΦ (Rn ) then f = sup f (x)g (x)dx . NΦ g MΦ ≤1 Rn Let ≥ 2. Denote by W ,∞ (Rn ) the set of all measurable functions f such that f and its generalized derivatives Dβ f , 0 < |β | ≤ , belong to L∞ (Rn ). The following is the well-known Gagliardo-Nirenberg inequality: Lemma 3. [6] For ﬁxed α, 0 < |α| < , there is the best constant Cα, such that |α| |α| 1− Dα f Dβ f ≤ Cα, f , ∞ ∞ ∞ |β |= ,∞ (Rn ). for any f ∈ W The following result is an extension of the Gagliardo-Nirenberg inequality ([2 - 6]) to Lorentz spaces. Note that the Gagliardo-Nirenberg inequality has applications to partial diﬀerential equations and interpolation theory. Theorem 1. Let ≥ 2, f and its generalized derivatives Dβ f , |β | = be in NΦ (Rn ). Then Dα f ∈ NΦ (Rn ) for all α, 0 < |α| < and 1− |α| |α| Dα f Dβ f ≤ Cα, f ( NΦ ) , (1) NΦ NΦ |β |= where the constant Cα, is deﬁned in Lemma 3. Proof. We begin to prove (1) with the assumption that Dα f ∈ NΦ (Rn ), 0 ≤ |α| ≤ . Fix 0 < |α| < . By Lemma 2 we have Dα f Dα f (x)v (x)dx . = sup NΦ v MΦ ≤1 Rn > 0. We choose a function v ∈ MΦ (Rn ) such that v Let = 1 and MΦ f (x)v (x)dx ≥ f − /2. NΦ Rn By Lemma 1, there is H := [−H, H ]n such that f (x)v (x)dx ≥ f −, (2) NΦ Rn where v = v (H, ) := χH v and χH is the characteristic function of H. Put
A Gagliardo-Nirenberg Inequality for Lorentz Spaces 209 F (x) = f (x + y )v (y )dy. Rn Then F ∈ L∞ (Rn ) by virtue of Lemma 1, and it is easy to check that Dβ Fε (x) = Dβ f (x + y )v (y )dy, 0 ≤ |β | ≤ (3) Rn in the distribution sense. For all x ∈ Rn , clearly, |Dβ Fε (x)| ≤ Dβ f (x + ·) ≤ Dβ f v NΦ . (4) NΦ MΦ Now we prove the continuity of Dβ Fε on Rn (0 ≤ |β | ≤ ). We show this for β = 0. Clearly, it suﬃces to prove that for any x ∈ Rn , lim χH (·) f (x + t + ·) − f (x + ·) = 0. NΦ t→0 Assume the contrary that for some δ > 0, point x0 and sequence tm → 0, χH (·) f (x0 + tm + ·) − f (x0 + ·) ≥ δ, m ≥ 1. (5) NΦ 0 n n For simplicity of notation we suppose x = 0. Since f ∈ NΦ (R ), f ∈ L1, oc (R ). So, it is known that |f (x + tm ) − f (x)|dx → 0 as m → ∞. H Therefore, there exists a subsequence {tmj }, which we still denote by {tm }, such that f (· + tm ) → f a.e. on H. Deﬁne gn (x) = inf |f (x + tm )|, x ∈ H; m≥ n then {gn } is a non-decreasing sequence and gn → |f | a.e. on H. It is easy to see that λχH gn (t) → λχH |f | (t) as n → ∞, for every t > 0. We have Φ(λχH |f | (t)) = lim Φ(λχH |gm | (t)) ≤ lim Φ(λχH |f (·+tm )| (t)), t > 0. (6) m→∞ m→∞ It follows from the deﬁnition of Φ that Φ(a + b) ≤ Φ(a) + Φ(b) for a, b ≥ 0. Observing that, for any f, g ∈ NΦ (Rn ) and t > 0 we have λχH (f +g) (2t) ≤ λχH f (t) + λχH g (t), then Φ(λχH |f (·+tm )−f | (2t)) ≤ Φ(λχH |f (·+tm )| (t)) + Φ(λχH |f | (t)). Hence for all m ≥ 1, 0 ≤ Φ(λχH |f (·+tm )| (t)) + Φ(λχH |f | (t)) − Φ(λχH |f (·+tm )−f | (2t)), ∀t > 0.
210 Mai Thi Thu It is easy to check that χH f (· + tm ) ∀m ≥ 1. = χH f NΦ , NΦ Applying Fatou’s lemma to the sequence {Φ(λχH |f (·+tm )| (t)) + Φ(λχH |f | (t)) − Φ(λχH |f (·+tm )−f | (2t))}, we obtain ∞ lim Φ(λχH |f (·+tm )| (t)) + Φ(λχH |f | (t)) − Φ(λχH |f (·+tm )−f | (2t)) dt m→∞ 0 ∞ ≤ lim Φ(λχH |f (·+tm )| (t)) + Φ(λχH |f | (t)) − Φ(λχH |f (·+tm )−f | (2t) dt m→∞ 0 ∞ ∞ 1 Φ(λχH |f | (t))dt − lim =2 Φ(λχH |f (·+tm )−f | (t))dt. (7) 2 m→∞ 0 0 On the other hand, λχH |f (·+tm )−f | (t) = mes{x ∈ H : |f (x + tm ) − f (x)| > t}. Therefore, since f (· + tm ) → f a.e. on H, we have lim λχH |f (·+tm )−f | (t) = 0 m→∞ and then lim Φ(λχH |f (·+tm )−f | (t)) = 0. m→∞ So, by (6) we get for any t > 0 2Φ(λχH |f | (t)) = lim Φ(λχH |gm | (t)) + Φ(λχH |f | (t)) − lim Φ(λχH |f (·+tm )−f | (2t)) m→∞ m→∞ ≤ lim Φ(λχH |f (·+tm )| (t)) + Φ(λχH |f | (t)) − Φ(λχH |f (·+tm )−f | (2t)) . m→∞ (8) From (7) and (8), we have ∞ ∞ ∞ 1 Φ(λχH |f | (t))dt ≤ 2 Φ(λχH |f | (t))dt − lim 2 Φ(λχH |f (·+tm )−f | (t))dt. 2 m→∞ 0 0 0 Hence ∞ Φ(λχH |f (·+tm )−f | (t))dt → 0 as m → ∞, 0 i.e., χH f (· + tm ) − f lim = 0, NΦ m→∞
A Gagliardo-Nirenberg Inequality for Lorentz Spaces 211 which contradicts (5). The cases 1 ≤ |β | ≤ are proved similarly. The continuity of Dα Fε , 0 ≤ |β | ≤ has been proved. The functions Dβ Fε , 0 ≤ |β | ≤ are continuous and bounded on Rn . There- fore, it follows from Lemma 3 and (2) - (3) that Dα f ||NΦ − ≤ |Dα Fε (0)| ≤ Dα Fε ≤ ∞ |α| |α| 1− Dβ Fε ≤ Cα, Fε , ∞ ∞ |β |= which together with (4) implies |α| 1− |α| ||Dα f ||NΦ − ≤ Cα, f Dβ f . NΦ NΦ |β |= → 0 we have (1). By letting Step 2. To complete the proof, it remains to show that Dα f ∈ NΦ (Rn ), ∀α : 0 < |α| < if f, Dα f ∈ NΦ (Rn ), |α| = . Since f ∈ L1, oc (Rn ) and Dα f ∈ L1, oc (Rn ), |α| = , we get Dα f ∈ L1, oc (Rn ), 0 < |α| < (see [5], p. 7). Let ψ (x) ∈ C0 (Rn ), ψ (x) ≥ 0, ψ (x) = 0 for |x| ≥ 1 and ∞ ψ (x)dx = 1. We put Rn ψλ (x) = λ1 ψ ( λ ), λ > 0 and fλ = f ∗ ψλ . Then Dα fλ = f ∗ Dα ψλ , |α| ≥ 0 x n and D fλ = Dα f ∗ ψλ , 0 ≤ |α| ≤ in the D (Rn ) sense. Actually, for every α ϕ ∈ C0 (Rn ), ∞ Dα fλ (x), ϕ(x) = (−1)|α| fλ (x), Dα ϕ(x) = (−1)|α| f (x − y )ψλ (y )dy Dα ϕ(x)dx Rn Rn = (−1)|α| f (x − y )Dα ϕ(x)dx dy ψλ (y ) Rn Rn Dα f (x − y )ϕ(x)dx dy = ψλ (y ) Rn Rn Dα f (x − y )ψλ (y )dy ϕ(x)dx = Rn Rn Dα f (x − y )ψλ (y )dy , ϕ(x) = . Rn Taking Lemma 1 into account, we get easily Dβ fλ = f ∗ Dβ ψλ ∈ NΦ (Rn ), 0 ≤ |β | ≤ and = f ∗ ψλ ≤ f (x − ·) fλ ψλ =f NΦ , (9) NΦ NΦ NΦ 1 D α fλ = Dα f ∗ ψλ ≤ Dα f (x − ·) = Dα f ψλ NΦ . (10) NΦ NΦ NΦ 1
212 Mai Thi Thu Applying (1) for fλ , we have from (9) - (10) for every α (0 < |α| < ) |α| |α| 1− ||Dα fλ ||NΦ ≤ Cα, fλ D β fλ NΦ NΦ |β |= |α| |α| 1− Dβ f ≤ Cα, f . (11) NΦ NΦ |β |= Fix α (0 < |α| < ). Since Dα f ∈ L1, oc (Rn ), for each j = 1, 2, . . . , we have Dα f ∗ ψλ → Dα f in L1 ([−j, j ]n ) as λ → 0. Therefore, there is a sequence ∞ {λj }k=1 , λj 0 such that Dα fλj → Dα f a.e. in [−j, j ]n as k → ∞. So, by k k k the diagonal process, there exists a subsequence denoted by {λm }∞ : λm → 0 1 such that lim Dα fλm (x) = Dα f (x) (12) m→∞ a.e. in Rn . For each function v ∈ MΦ (Rn ), v MΦ ≤ 1 and m ≥ 1, by (11) - (12) and the deﬁnition of the Lorentz norm we get |α| |α| 1− (Dα fλm )(x)v (x) dx ≤ Cα, f Dβ f . (13) NΦ NΦ |β |= Rn Therefore, using Fatou’s lemma, (12) and (13), we obtain Dα f (x)v (x)dx ≤ lim inf Dα fλm (x)v (x) dx m→∞ Rn Rn (Dα fλm )(x)v (x) dx ≤ lim inf m→∞ Rn |α| |α| 1− Dβ f ≤ Cα, f . (14) NΦ NΦ |β |= Because (14) is true for all v ∈ MΦ (Rn ), v ≤ 1, by the deﬁnition of · MΦ NΦ we have |α| |α| 1− ||Dα f ||NΦ ≤ Cα, f Dβ f < ∞, 0 < |α| < . NΦ NΦ |β |= The proof is complete. By Theorem 1, we have Theorem 2. Let ≥ 2, f and its generalized derivatives Dβ f, |β | = be in NΦ (Rn ). Then Dα f ∈ NΦ (Rn ) for all α, 0 < |α| = r < and r 1− r Dα f Dβ f ≤ Cr, f . NΦ NΦ NΦ |α|=r |β |=
A Gagliardo-Nirenberg Inequality for Lorentz Spaces 213 Corollary 1. Let ≥ 2, f and its generalized derivatives Dβ f, |β | = be in NΦ (Rn ). Then Dα f ∈ NΦ (Rn ) for all α, 0 < |α| = r < and r ≤ Ch− Dα f Dβ f f + Ch NΦ , −r NΦ NΦ |α|=r |β |= for all h > 0 and C does not depend on f . Remark 1. Note that the Gagliardo-Nirenberg inequality for Orlicz spaces LΦ (Rn ) was proved in [1] but the techniques used there cannot be applied to Lorentz spaces NΦ (Rn ). In conclusion the author would like to thank professor Ha Huy Bang for the helpful suggestions. References 1. H. H.Bang and M.T.Thu, A Gagliardo-Nirenberg inequality for Orlicz spaces, East J. on Approximations 10 (2004) 371–377. 2. O. V. Besov, V. P. Ilin, and S. M. Nikol’skii, Integral Representation of Functions and Imbedding Theorems, V. H. Winston and Sons, Washington, D. C. , 1978, 1979, Vol 1, 2. 3. T. Donaldson and N. S. Trudinger, Orlicz-Sobolev spaces and imbedding theo- rems, J. Funct. Anal. 8 (1971) 52–75. 4. E. Gagliardo, Ulterori propriet` di alcune classi di funzioni in pi` variabili, a u Ricerche di Math. 8 (1959) 24–51. 5. V. G. Maz’ja, Sobolev Spaces, Springer-Verlag, Berlin-Heidelberg, New York, Tokyo, 1980. 6. N. Nirenberg, On elliptic partial diﬀerential equations, Ann. Scuola Norm. Sup. di Pisa, ser. III, 13 (1959) 115–162. 7. M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, Inc., New York, 1991. 8. M. S. Steigerwalt and A. J. White, Some function spaces related to Lp , Proc. London. Math. Soc. 22 (1971) 137–163. 9. M. S. Steigerwalt, Some Banach Function Spaces Related to Orlicz Spaces, Uni- versity of Aberdeen Thesis, 1967. 10. E. M. Stein, Functions of exponential type, Ann. Math. 65 (1957) 582–592.