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

Trong bài báo này, về cơ bản phát triển các phương pháp [1] và [10], chúng tôi cung cấp chomột phần mở rộng của sự bất bình đẳng Gagliardo-Nirenberg không gian Lorentz.
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Báo cáo toán học: "A Gagliardo-Nirenberg Inequality for Lorentz Spaces" 9LHWQDP -RXUQDOVietnam 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, 2004Abstract. In this paper, essentially developing the method of [1] and [10], we givean extension of the Gagliardo-Nirenberg inequality to Lorentz spaces.Let Φ : [0, ∞) → [0, ∞) be a non-zero concave function, which is non-decreasingand 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 0where λf (y ) = mes{x ∈ R : |f (x)| > y } , (y ≥ 0). The space NΦ (Rn ) consisting nof measurable functions f such that f NΦ < ∞ is a Banach space. Denote byMΦ (Rn ) the space of measurable functions g such that 1 |g (x)|dx : Δ ⊂ Rn , 0 < mes Δ < ∞ < ∞. g = sup MΦ Φ(mes Δ) ΔThen 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 ThuLemma 1. If f ∈ NΦ (Rn ), g ∈ MΦ (Rn ) then f g ∈ L1 (Rn ) and |f (x)g (x)|dx ≤ f g MΦ . NΦ RnLemma 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 suchthat f and its generalized derivatives Dβ f , 0 < |β | ≤ , belong to L∞ (Rn ). Thefollowing 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 hasapplications to partial differential equations and interpolation theory.Theorem 1. Let ≥ 2, f and its generalized derivatives Dβ f , |β | = be inNΦ (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 vLet ...