in this paper we study the general uncertainty principal , we obtain the best constant as application of Sobolev inequalities for higher order fractional derivatives 

Abbaker Abd alla, Ali Hassan

English

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Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.6, 2017  182 Application of Sobolev inequalities for higher order fractional derivatives  Dr Ali Hassan Abbaker Abd alla 1- Alzaim alazhary university - sudan 2- Faculty of arts & science - Albaha University - PO box 1988-KSA Email :dralihassan1973@gmail.com Abstract in this paper we study the general uncertainty principal , we obtain the best constant as application of Sobolev inequalities for higher order fractional derivatives .  1. Introduction  Sobolev inequality has many applications in mathematics , and it is  important to estimates constants in these inequalities. 𝑘 ∈ ℕ  is  an integer of Sobolev space which can be defined  in  𝐻𝑘(ℝ𝑛)  as a function 𝑓 ∈ 𝑙2(ℝ𝑛)  satisfying |𝛻ℓ𝑓 | ∈  𝐿2(ℝ𝑛), 1 ≤ ℓ ≤ 𝑘. The sobolev imbedding theorem asserts that 𝐻𝑘(ℝ𝑛)  ⊆  𝐿𝑞(ℝ𝑛) for  𝑞 =  2𝑛/(𝑛 −  2𝑘). For example, let 𝑘 =  1, 𝑛 ≥  3 and 𝑞 =  2𝑛/(𝑛 −  2),  Then we have inequality                 ‖𝑓‖ 2𝑛𝑛−22 ≤ 𝐶𝑛‖∇𝑓‖22,    𝑓 ∈ 𝐶0∞(ℝ𝑛)                                   ( 1 ) (𝐶𝑛 is constant)  The best value for the constant  𝐶𝑛  in inequality (1) has been estimated to be               𝐶𝑛 = 𝜋−1 𝑛−1(𝑛 − 2)−1 [Γ(𝑛)Γ(𝑛 2⁄ )]2𝑛⁄                                         ( 2 ) If we take the formula  𝛤 (𝑛)Γ(𝑛 2⁄ )=2𝑛−1𝜋1 2⁄𝛤((𝑛 + 1) 2⁄ ) Then  𝐶𝑛  = 4𝑛(𝑛 − 2)|𝕊𝑛|−2 𝑛⁄ = 2−2 𝑛⁄ 𝜋−(𝑛+1) 𝑛⁄4𝑛(𝑛 − 2)[Γ (𝑛 + 12)]2 𝑛⁄ Since 𝑆𝑛 is the n- dimential unit sphere and |𝑆𝑛| is the surface area. Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.6, 2017  183 Proposition 1: for every 𝑓 ∈ 𝐻𝑠(ℝ𝑛)  we have ‖𝑓 − 𝑒−𝑡(−Δ)𝑠𝑓‖2= √𝑡 ‖(−Δ)𝑠2⁄ 𝑓‖2 Proof: let 𝑓 ∈ 𝐻𝑠(ℝ𝑛) then ‖𝑓 − 𝑒−𝑡(−Δ)𝑠𝑓‖22= ∫|𝑓(𝑘)|2(1 − 𝑒−𝑡(2𝜋|𝑘|)2𝑠)2𝑑𝑘 Let 1 − 𝑒−𝑥 ≤ 𝑥       for   𝑥 ≥ 0 Hence   ‖𝑓 − 𝑒−𝑡(−Δ)𝑠𝑓‖22≤ 𝑡 ∫(2𝜋|𝑘|)2𝑠|𝑓(𝑘)|2𝑑𝑘 = 𝑡‖(−∆)𝑠 2⁄ 𝑓‖22 By taking square root of both sides we have  ‖𝑓 − 𝑒−𝑡(−Δ)𝑠𝑓‖2= √𝑡 ‖(−Δ)𝑠2⁄ 𝑓‖2  Theorem 1: let 𝑛 > 2𝑠 and 𝑞 = 2𝑛 (𝑛 − 2𝑠)⁄  then ‖𝑓‖𝑞2 ≤ 𝑆(𝑛, 𝑠)‖(−∆)𝑠 2⁄ ‖22                          , 𝑓 ∈ 𝐻𝑠(ℝ𝑛).          (3) where  𝑆(𝑛, 𝑠) = 2−2𝑠𝜋−𝑠Γ(𝑛−2𝑠2)Γ(𝑛+2𝑠2)[Γ(𝑛)Γ(𝑛 2⁄ )]2𝑠 𝑛⁄                            (4) We have equality in (3) if and only if 𝑓(𝑥) = 𝐶(𝜇2 + (𝑥 − 𝑥0)2)−𝑛−2𝑠2                  , 𝑥 ∈ ℝ𝑛. where  𝐶 ∈ ℝ , 𝜇 > 0  and  𝑥0 ∈ ℝ𝑛  are fixed constants. Also we have  𝑆(𝑛, 𝑠) =Γ (𝑛 − 2𝑠2 )Γ (𝑛 + 2𝑠2 )|𝑆𝑛|−2𝑠𝑛  Proof: First :if 𝐶0∞(ℝ𝑛)  is dense in 𝐻𝑠(ℝ𝑛)  and 𝑓 ∈ 𝐶0∞(ℝ𝑛) then the relation (3)  is true. Now , let 𝑓, 𝑔 ∈ 𝐶0∞(ℝ𝑛)  then (𝑓, 𝑔) = (𝑓, ?̂?) = ∫|𝐾|𝑠𝑓(𝑘)̅̅ ̅̅ ̅̅ |𝑘|−𝑠?̂?(𝑘)𝑑𝑘 = ∫(−∆)𝑠 2⁄̂ 𝑓(𝑘)(−∆)−𝑠 2⁄̂ 𝑔(𝑘)𝑑𝑘                                                = ((−∆)𝑠 2⁄ 𝑓 ∙ (−∆)−𝑠 2⁄ 𝑔).   (5) Hence |(𝑓, 𝑔)| ≤ ‖(−∆)𝑠 2⁄ 𝑓‖2‖(−∆)−𝑠 2⁄ 𝑔‖2                         (6) Hardy-Littlewood-Sobolev inequality defined as : Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.6, 2017  184 ‖(−∆)−𝑠 2⁄ (𝑔)‖2≤ 2−𝑠𝜋−𝑠 2⁄ (Γ(𝑛−2𝑠2)Γ(𝑛+2𝑠2))12[Γ(𝑛)Γ(𝑛 2⁄ )]𝑠 𝑛⁄‖𝑔‖𝑝         (7) where 1𝑝+1𝑞= 1 , i.e 𝑝 =2𝑛(𝑛+2𝑠) combining (6) and (7) we have  |(𝑓, 𝑔)| ≤ (𝑆(𝑛, 𝑠))12‖(−∆)𝑠 2⁄ (𝑓)‖2‖𝑔‖𝑝.                        (8) Let us take 𝑔 = 𝑓𝑞−1 , there for  we have  |(𝑓, 𝑔)| = |(𝑓, 𝑓𝑞−1)| = ‖𝑓‖𝑞𝑞 ‖𝑔‖𝑝 = ‖𝑓𝑞−1‖𝑝 = ‖𝑓‖𝑞𝑞−1 Hence (8) be comes ‖𝑓‖𝑞2 ≤ 𝑆(𝑛, 𝑠)‖(−∆)𝑠 2⁄ ‖22 Notation : If 𝑥 =  (𝑥1, . . . , 𝑥𝑛), 𝑘 =  (𝑘1, . . . , 𝑘𝑛)  ∈  ℝ𝑛 , then we denote (𝑘, 𝑥) =  𝑘1𝑥1  + · · ·+ 𝑥𝑛𝑘𝑛   and |𝑥|  =  (𝑥, 𝑥)1 2⁄ . If  𝑓, 𝑔 ∈  𝐿2(ℝ𝑛), then we denote (𝑓, 𝑔) = ∫ 𝑓(𝑥)𝑔(𝑥 )𝑑𝑥 .  Theorem 2: Sobolev’s inequality For 𝑛 ≥  3  let 𝑓 is a function in 𝐶1(ℝ𝑛) with compact support. A constant 𝐶𝑛 exists depending on the dimension rather than  𝑓  so that  ‖𝑓‖𝑝 ≤ 𝑆𝑛‖∇𝑓‖ 2 Where  𝑝 =2𝑛𝑛 − 2  Which is denoted as  Sobolev index. Remark 1: if  𝑛 ≥ 3, does not make a statement in 2 and 3 dimensions. Remark 2: The Sobolev index can be understood as follows. assuming the inequality holds, pick any function f and consider its  as a scaled verion 𝑓(𝜆𝑥) with 𝜆  >  0 arbitrary. Then, by changing variables (∫ |𝑓(𝜆𝑥)|𝑝ℝ𝑛𝑑𝑥)1 𝑝⁄= 𝜆−𝑛 𝑝⁄ (∫ |𝑓(𝑥)|𝑝ℝ𝑛𝑑𝑥)1 𝑝⁄ Which is  ≤ 𝐶𝑛 (∫ |∇𝑓(𝜆𝑥)|2ℝ𝑛𝑑𝑥)1 2⁄= 𝜆1−𝑛 2⁄ 𝐶𝑛 (∫ |∇𝑓(𝑥)|2ℝ𝑛𝑑𝑥)1 2⁄ Thus, the   𝜆   exponents must necessarily be the same, i.e., 𝑛/𝑝 =  𝑛/2 −  1. Remark 3:The best possible constant in Sobolev’s inequality is defined  and it has the value. Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.6, 2017  185 𝑛(𝑛 −  2)4|𝑆𝑛|𝑛 2⁄  where |𝑆𝑛| is the surface area of the unit n-sphere in   ℝ𝑛+1, i.e., |𝑆𝑛| =2𝜋(𝑛+1) 2⁄Γ (𝑛 + 12 ). The functions which yield equality are of the form 𝑘(𝜇2 + |𝑥 − 𝑎|2)(𝑛−2) 2⁄.              𝑘 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡.  See [1] , [3] and [4] for other proofs.  Theorem 3 : The values of  𝑝  is a possible for the inequality                  ‖𝑓‖𝑝 ≤ 𝐶𝑛,𝑞‖∇𝑓‖𝑞.                                                         ( 9 )  to hold is                                            𝑝 =𝑞𝑛𝑛−𝑞.  In particular for     𝑞 =  1,    𝑝 =  𝑛/(𝑛 −  1). Proof: the proof of inequality (9)  is due to Gagliardo and Nirenberg, we prove  it in 3-space  .  Using the fundamental theorem of calculus  𝑓(𝑥, 𝑦, 𝑧)  = ∫ 𝜕𝑥𝑓𝑥−∞(𝑟, 𝑦, 𝑧)𝑑𝑟  and in particular  |𝑓(𝑥, 𝑦, 𝑧)| ≤ ∫ |𝜕𝑥𝑓(𝑟, 𝑦, 𝑧)|∞−∞𝑑𝑟 = 𝑔1(𝑦, 𝑧).   Similarly, repeating the same argument in the other variables  |𝑓(𝑥, 𝑦, 𝑧)|3  ≤  𝑔1(𝑦, 𝑧)𝑔2(𝑥, 𝑧)𝑔3(𝑥, 𝑦) , and hence ‖𝑓‖3 2⁄ ≤ (∫ √𝑔1(𝑦, 𝑧)√𝑔2(𝑥, 𝑧)√𝑔3(𝑥, 𝑦)𝑑𝑥𝑑𝑦𝑑𝑧)2 3⁄, Using Schwarz’ inequality on the 𝑥- variable yields the upper bound (∫ √𝑔1(𝑦, 𝑧)√∫ 𝑔2(𝑥, 𝑧)𝑑𝑥 √∫ 𝑔3(𝑥, 𝑦)𝑑𝑥 𝑑𝑦𝑑𝑧)2 3⁄ Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.6, 2017  186  Applying Schwarz’ inequality once more in the 𝑦-variable yields (∫ √∫ 𝑔1(𝑦, 𝑧)𝑑𝑦 √∫ 𝑔2(𝑥, 𝑧)𝑑𝑥 √∫ 𝑔3(𝑥, 𝑦)𝑑𝑥𝑑𝑦 𝑑𝑧)2 3⁄,  (√𝑔1(𝑦, 𝑧)𝑑𝑦𝑑𝑧√𝑔2(𝑥, 𝑧)𝑑𝑥𝑑𝑧√𝑔3(𝑥, 𝑦)𝑑𝑥𝑑𝑦)2 3⁄,                       =  (∫ 𝑔1(𝑦, 𝑧)𝑑𝑦𝑑𝑧 ∫ 𝑔2(𝑥, 𝑧)𝑑𝑥𝑑𝑧 ∫ 𝑔3(𝑥, 𝑦)𝑑𝑥𝑑𝑦)1 3⁄ ,                                = (‖𝜕𝑥𝑓‖1‖𝜕𝑦𝑓‖1‖𝜕𝑧𝑓‖1)1 3⁄,                                                ≤ ‖∇𝑓‖1.                          Thus it is established that                         ‖𝑓‖3 2⁄ ≤ ‖∇𝑓‖1 .                                            (10)  To arrive at the general inequality, replace f by  |𝑓|𝑠 for a number 𝑠 >  0 to be chosen later and calculate  ‖𝑓𝑠‖3 2⁄ ≤ 𝑠‖|∇𝑓||𝑓|𝑠−1‖1  Using Hölder’s inequality on the right side yields the estimate                   ‖𝑓𝑠‖3 2⁄ ≤ 𝑠‖|∇𝑓|‖𝑞‖|𝑓|𝑠−1‖𝑞ꞌ                                   (11)   where 1/𝑞 +  1/𝑞ꞌ =  1 or 𝑞ꞌ =  𝑞/(𝑞 −  1). Now if we choose 𝑠 =  2𝑞/(3 −  𝑞) so that 3𝑠2=(𝑠 −  1)𝑞𝑞 −  1=3𝑞3 − 𝑞= 𝑝, we get from (11)  ‖𝑓‖𝑝2𝑝 3⁄ ≤ 2𝑞 (3 − 𝑞)⁄ ‖||∇𝑓||‖𝑞‖𝑓‖𝑝𝑝(𝑞−1) 𝑞⁄  and upon dividing both sides by  ‖𝑓‖𝑝𝑝(𝑞−1) 𝑞⁄ we obtain  ‖𝑓‖𝑝2𝑝 3−𝑝⁄ (𝑞−1) 𝑞⁄ ≤ 2𝑞 (3 − 𝑞)⁄ ‖|∇𝑓|‖𝑞,  which is our desired inequality. Note, as a check, that Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.6, 2017  187  𝑝[2/3 − (𝑞 −  1)/𝑞]  =  1 .  Remark 4: The sharp constant in (10) is strongly related to the isoperimetric inequality. This is a substantial subject all by itself and we just touch it with a few remarks. The inequality (10) on ℝ𝑛 in its sharp form reads as ‖𝑓‖ 𝑛𝑛−1≤ 𝑛−(𝑛−1)𝑛 |𝑆𝑛−1|−1 𝑛⁄ ‖∇𝑓‖1. In other words, we claim that 𝑠𝑢𝑝𝑓≠0‖𝑓‖ 𝑛𝑛−1‖∇𝑓‖1= 𝑛−(𝑛−1)𝑛 |𝑠𝑛−1|−1 𝑛⁄ . The constant is exactly the surface area of a ball divided by the (𝑛 −  1)/𝑛 – th power of its volume. The constant is not attained by any function whose gradient is integrable but can be obtained arbitrarily close as the following calculation shows.  Consider  Sobolev inequality (10), in ℝ𝑛, A among all functions the spherically symmetric functions delivers the worst constant. Thus, it is assumed that all the level sets are rearranged into balls with radius [𝑛|𝑆𝑛−1|]1𝑛|{𝑥: |𝑓(𝑥)| > 𝛼}|1 𝑛⁄  and hence this inequality reads 𝐶𝑛 ≥ [1𝑛 − 1]𝑛−1𝑛|𝑆𝑛−1|−1 𝑛⁄ 𝑠𝑢𝑝𝑓≠0[∫ 𝛼1 (𝑛−1)⁄ (𝛼)𝑛𝑛−1𝑑𝛼∞0]𝑛−1𝑛∫ 𝜆(𝛼)𝑑𝛼∞0               (12) where 𝜆(𝛼)  =  |{𝑥 ∶  |𝑓(𝑥)|  >  𝛼}|𝑛−1𝑛 . Two observations about the function 𝜆(𝛼): it is a non increasing function and can be assumed that ∫ 𝜆(𝛼)𝑑𝛼∞0= 0 as well as 𝜆(0)  =  1, since the scaling 𝜆(𝛼)  →  𝐶𝜆(𝐷𝛼) leaves the ratio in (12) fixed. To maximize [∫ 𝛼1 (𝑛−1)⁄ (𝛼)𝑛𝑛−1𝑑𝛼∞0]𝑛−1𝑛  over all such functions 𝜆(𝛼) we proceed as follows. The functional 𝜆(𝛼) ⟼ ℱ(𝜆) = [∫ 𝛼1 (𝑛−1)⁄ (𝛼)𝑛𝑛−1𝑑𝛼∞0]𝑛−1𝑛 is convex. Now restrict the set over which to maximize in order to consist of non-increasing functions haveing the value 1 at 𝛼 =  0, whose integral equals 1 and are 0 outside the interval [0, 𝑁] for some 𝑁 larger values. This set is called  𝑇𝑁 and that 𝑇𝑁 is a convex set and Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.6, 2017  188  𝐹(𝑁)  =  𝑠𝑢𝑝𝜆∈𝑇𝑁ℱ(𝜆)  is a non- decreasing function of   𝑁.           Suppose that our  functional is convex it attains the maximum on the set  𝑇𝑁 at the extreme points which consists of functions which have only 0 and 1 values. Since the function is  a non-increasing, has the value 1 at 𝛼 =  0 and integrates to 1. It  must be  𝜆𝑜𝑝𝑡(𝛼)  =  𝜆[0,1](𝛼) .                           (13)  which does not depend on the value of  𝑁  as long as 𝑁 >  1, and hence inserting  the inequality (12),  the result is:  𝐶𝑛  ≥  |𝑆𝑛−1|−1 𝑛⁄ 𝑛−(𝑛−1) 𝑛⁄  ,  which demonstrates our claim.   References: [1] Talenti, G., Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976),353-372.6 [2] Tavoularis, N ,A., Best constant for sobolev inequalities for higher order fractional derivatives,  [3] Lieb, E.h., sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math. 118 (1983), 349-374. [4] Carlen, E.A. and Loss, M. Extremals of functionals with competing symmetries, J. Funct. Anal. 88 (1990), 194-211 [5] Aubin, T., Meilleures constantes dans le th´eor`eme d’inclusion de Sobolev et un th e´or`eme de Fredholm non-lineaire pour la transformation conforme de la courbure scalaire, J. Funct. Anal. 57 (1979), 148-174. [6] Brothers, J. and Ziemer, W. P., Minimal rearrangements of Sobolev functions, J. Reine Angew. Math. 384 (1988), 153-179. 

Application of Sobolev inequalities for higher order fractional derivatives

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Application of Sobolev inequalities for higher order fractional derivatives

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