Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 161405, 22 pages doi:10.1155/2009/161405 Research Article A Kind of Estimate of Difference Norms in Anisotropic Weighted Sobolev-Lorentz Spaces Jiecheng Chen 1 and Hongliang Li 1, 2 1 Department of Mathematics, Zhejiang University, Hangzhou 310027, China 2 Department of Mathematics, Zhejiang Education Institute, Hangzhou 310012, China Correspondence should be addressed to Hongliang Li, honglli@126.com Received 27 April 2009; Accepted 2 July 2009 Recommended by Shusen Ding We investigate the functions spaces on R n for which the g eneralized partial derivatives D r k k f exist andbelongtodifferent Lorentz spaces Λ p k ,s k w,wherep k > 1andw is nonincreasing and satisfies some special conditions. For the functions in these weighted Sobolev-Lorentz spaces, the estimates of the Besov type norms are found. The methods used in the paper are based on some estimates of nonincreasing rearrangements and the application of B p , B p,∞ weights. Copyright q 2009 J. Chen and H. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In this paper we study functions f on R n which possess the generalized partial derivatives D r k k f ≡ ∂ r k f ∂x r k k  r k ∈ N  . 1.1 Our main goal is to obtain some norm estimates for the differences Δ r k k  h  f  x  ≡ r k  j0 −1 r k −j  r k j  f  x  jhe k   h ∈ R  1.2 e k being the unit coordinate vector. 2 Journal of Inequalities and Applications The classic Sobolev embedding theorem asserts that for any function f in Sobolev space W 1 p R n 1 ≤ p<n f q ∗ ≤ C n  k1     ∂f ∂x k     p ,q ∗  np n − p . 1.3 Sobolev proved this inequality in 1938 for p>1. His method, based on integral representations, did not work in the case p  1. Only at the end of fifties Gagliardo and Nirenberg gave simple proofs of inequality 1.3 for all 1 ≤ p<n.Inequality 1.3 has been generalized in various directions see 1–6 for details. It was proved that the left hand side in 1.3 can be replaced by the stronger Lorentz norm, that is, there holds the inequality f q ∗ ,p ≤ C n  k1     ∂f ∂x k     p , 1 ≤ p<n. 1.4 For p>1 the result follows by interpolation see 7, 8. In the case p  1 some geometric inequalities were applied to prove 1.4see 9–13. The sharp estimates of the norms of differences for the functions in Sobolev spaces have firstly been proved by Besov et al. 1, Volume 2, page 72. For the space W 1 p R n 1 ≤ p< n Il’in’s result reads as follows: If n ∈ N, 1 <p<q<∞ and α ≡ 1 − n1/p − 1/q > 0, then n  k1   ∞ 0  h −α Δ 1 k  h  f q  p dh h  1/p ≤ C n  k1     ∂f ∂x k     p . 1.5 Actually, this means that there holds the continuous embedding to the Besov space W 1 p  R n  → B α p,q  R n  . 1.6 It is easy to see that inequality 1.5 fails to hold for p  n  1, but, it was proved in 14 that 1.5 is true for p  1andn ≥ 2. The generalization of the inequality 1.5 to the spaces W r 1 , ,r n p was given in 12.That is n  k1   ∞ 0  h −α k Δ r k k  h  f q,p  p dh h  1/p ≤ C n  k1 D r k k f p , 1.7 where 0 < 1/p − 1/q < r/n, r  n  n i1 r −1 i  −1 , and α k  r k 1 − r/n1/p − 1/q;the inequality is valid if p>1,n≥ 1orp  1,n≥ 2. Using 1.7, we get the following continuous embedding: W r 1 , ,r n p  R n  → B α 1 , ,α n q,p  R n  . 1.8 For p>1 this embedding was proved by Besov et al. 1, Volume 2, page 72. The main result in 12 is the proof of 1.7 for p  1,n≥ 2. Journal of Inequalities and Applications 3 In 15, there was the sharp estimates of the type 1.7 when the derivatives D r k k f belong to different Lorentz spaces L p k ,s k . Before stating the theorem, we give some notations. Let S 0 R n  be the class of all measurable and almost everywhere finite functions f on R n such that for each y>0, λ f  y      x ∈ R n :   f  x    >y    < ∞. 1.9 Let r k ∈ N and 1 ≤ p k ,s k < ∞ for k  1, ,nn ≥ 2. Denote r  n  n  k1 1 r k  −1 ,p n r  n  k1 1 p k r k  −1 , s  n r  n  k1 1 s k r k  −1 . 1.10 Now we state the main theorem in 15. Theorem 1.1. Let n ≥ 2,r k ∈ N, 1 ≤ p k ,s k < ∞, and s k  1 if p k  1.Letr, p, and s be the numbers defined by 1.10. For every p j 1 ≤ j ≤ n satisfying the condition ρ j ≡ r n  1 p j − 1 p > 0, 1.11 take arbitrary q j >p j such that 1 q j > 1 p − r n , 1.12 and denote H j  1 − 1 ρ j  1 p j − 1 q j  ,α j  H j r j , 1 θ j  1 − H j s  H j s j , 1.13 then for any function f ∈ S 0 R n  which has the weak derivatives D r k k f ∈ L p k ,s k R n k  1, ,n there holds the inequality   ∞ 0  h −α j    Δ r j j  h  f    q j ,1  θ dh h  1/θ j ≤ C n  k1 D r k k f p k ,s k , 1.14 where C is a constant that does not depend on f. In many cases, the Lorentz space should be substituted by more general space, the weighted Lorentz space. In this paper, we will generalize the above result when the weighted Lorentz spaces Λ p k ,s k w take place of L p k ,s k , where w is a weight on R  which satisfies some special conditions. 4 Journal of Inequalities and Applications 2. Auxiliary Proposition Let MX, μ be the class of all measurable and almost everywhere finite functions on X. For f ∈MX, μ, a nonincreasing rearrangement of f is a nonincreasing function f ∗ on R  ≡ 0, ∞, that is, equimeasurable with |f|. The rearrangement f ∗ can be defined by the equality f ∗  t   inf  λ : μ f  λ  ≤ t  , 0 <t<∞, 2.1 where μ f  λ   μ  x ∈ X :   f  x    >λ  ,λ≥ 0. 2.2 If X  R n ,μE|E|, then the following relation holds 16, Chapter 2: sup |E|t  E   f  x    dx   t 0 f ∗  u  du. 2.3 Set f ∗∗  t   1 t  t 0 f ∗  s  ds. 2.4 Assume that 0 <q,p<∞. A function f ∈MX, μ belongs to the Lorentz space L q,p X if f q,p    ∞ 0 t 1/q f ∗ t p dt t  1/p < ∞. 2.5 For 0 <p<∞, the space L p,∞ X is defined as the class of all f ∈MX, μ such that f p,∞  sup t>0 t 1/p f ∗  t  < ∞. 2.6 We also let L ∞,∞ XL ∞ X.Letw be a weight in R  nonnegative locally integrable functions in R  . If X, μR  ,wtdt, we replace L q,p X with L q,p w. For 0 <p,q<∞,or0<p≤∞ and q  ∞, the weighted Lorentz space Λ p,q R n wΛ p,q w is defined in 9, Chapter 2 by Λ p,q  w    f ∈M  R n  : f Λ p,q  w   f ∗  L p,q  w  < ∞  . 2.7 If p  q, denote Λ p wΛ p,p w. It is well known that Λ p,q  1   L p,q  R n  , 2.8 Journal of Inequalities and Applications 5 and if 0 <p,q<∞, then Λ p,q  w  Λ q  w  , 2.9 where w  t   W q/p−1  t  w  t  ,W  t    t 0 w  s  ds. 2.10 In following part of this paper, we will always denote Wt  t 0 wsds. The weighted Lorentz spaces have close connection with weights of B p ,B p,∞ for 0 < p<∞ see 9, C hapter 1.LetA be the Hardy operator as follows: Af  t   1 t  t 0 f  s  ds, t > 0. 2.11 The space L p dec is the cone of all nonnegative nonincreasing functions in L p . We denote w ∈ B p if A : L p dec  w  −→ L p  w  2.12 is bounded and denote w ∈ B p,∞ if A : L p dec  w  −→ L p,∞  w  2.13 is bounded. Lemma 2.1 Generalized Hardy’s inequalities. Let ψ be nonnegative, measurable on 0, ∞ and suppose −∞ <λ<1, 1 ≤ q ≤∞, and w is a weight in R  , W∞∞, then one has   ∞ 0  Wt λ 1 Wt  t 0 ψswsds  q wt Wt dt  1/q ≤ 1 1 − λ   ∞ 0  Wt λ ψt  q wt Wt dt  1/q ,   ∞ 0  Wt 1−λ  ∞ t ψs ws Ws ds  q wt Wt dt  1/q ≤ 1 1 − λ   ∞ 0  Wt 1−λ ψt  q wt Wt dt  1/q 2.14 (with the obvious modification if q  ∞). Proof. It is easy to obtain this result applying Hardy’s inequality 16. Lemma 2.2. Let ψ ∈ Λ p,s w1 ≤ p, s < ∞ be a nonnegative nonincreasing function on R  , w be a nonincreasing weight on R  and there exists A>0, such that W  ξt  ≥ ξ A W  t  , ∀ξ>1, ∀t>0, 2.15 6 Journal of Inequalities and Applications Then for δ>0 there exists a continuously differentiable φ on R  such that i ψt ≤ Cφt,t∈ R  , ii φtWt 1/p−δ decreases and φtWt 1/pδ increases on R  , iii φ Λ p,s w ≤ Cψ Λ p,s w , where C is a constant depends only on p, δ, and A. Proof. Without loss of generality, we may suppose that δ<1/p.Set φ 1  t   Wt δ−1/p  ∞ t/2 ψ  u  Wu 1/p−δ w  u  W  u  du. 2.16 Then φ 1 tWt 1/p−δ decreases and φ 1  t  ≥ Wt δ−1/p  t t/2 ψ  u  Wu 1/p−δ w  u  W  u  du ≥ Wt δ−1/p ψ  t  Wt 1/p−δ − Wt/2 1/p−δ 1/p − δ . 2.17 Using the conditions which w satisfy, it gives φ 1  t  ≥ Cψ  t  . 2.18 Furthermore, noticing w is nonincreasing and applying Lemma 2.1,wegetthat φ 1  Λ p,s w   2  ∞ 0  W2h δ  ∞ h Wu 1/p−δ ψu wu Wu du  s w2h W2h dh  1/s ≤ 2 1/sδ   ∞ 0  Wh δ  ∞ h Wu 1/p−δ ψu wu Wu du  s wh Wh dh  1/s ≤ C   ∞ 0  Wh 1/p ψh  s wh Wh dh  1/s  Cψ Λ p,s w . 2.19 now set φ  t    δ  1 p  Wt −1/p−δ  t 0 φ 1  u  Wu δ1/p w  u  W  u  du. 2.20 Then φtWt 1/pδ increases on R  ,and φ  t  ≥ φ 1  t  ≥ Cψ  t  . 2.21 Journal of Inequalities and Applications 7 Furthermore, φ  t  Wt 1/p−δ   δ  1 p  Wt −2δ  t 0 φ 1  u  Wu δ1/p w  u  W  u  du  Wt −2δ  t 0 φ 1  u  dWu δ1/p  Wt −2δ  Wt 2δ 0 φ 1  h  v  v 1/p−δ/2δ dv, 2.22 where v  Wu 2δ ,hvu, that is, hvW −1 v 1/2δ . Since φ 1 tWt 1/p−δ is decreasing function on R  ,thusφ 1 hvv 1/p−δ/2δ is decreasing and φtWt 1/p−δ is also decreasing on R  . Finally, using Lemma 2.1 and 2.19,wegetiii.TheLemma 2.2 is proved. Let r k ∈ N and 1 <p k < ∞ for k  1, ,nn ≥ 2. Denote r  n ⎛ ⎝ n  j1 1 r j ⎞ ⎠ −1 ,p n r ⎛ ⎝ n  j1 1 p j r j ⎞ ⎠ −1 , γ k  1 − 1 r k  r n  1 p k − 1 p  . 2.23 Then γ k > 0and n  k1 γ k  n − 1. 2.24 To prove our main results we use the estimates of the rearrangement of a given function in term of its derivatives D r k k f k  1, ,n. We will use the notations 2.23. Lemma 2.3. Let r k ∈ N, 1 <p k < ∞, 1 ≤ s k < ∞ for k  1, ,nn ≥ 2 and w is continuous weight on R  .Set s  n r ⎛ ⎝ n  j1 1 s j r j ⎞ ⎠ −1 . 2.25 Let 0 <δ< 1 4 min γ j <1  1 − γ j  , 2.26 8 Journal of Inequalities and Applications and suppose that φ k ∈ Λ p k ,s k wk  1, ,n are positive continuously differentiable functions with φ  k t < 0 on R  such that φ k tWt 1/p k −δ decreases and φ k tWt 1/p k δ increases on R  .Setfor u, t > 0, η k  u, t    W  t  u  r k φ k  t  , 2.27 σ  t   sup  min 1≤k≤n η k  u k ,t  : n  k1 u k  W  t  n−1 ,u k > 0  . 2.28 Then i there holds the inequality   ∞ 0 Wt s1/p−r/n−1 σt s wtdt  1/s ≤ C  n  k1 φ k  r/nr k  Λ p k ,s k  w  ; 2.29 ii there exist continuously differentiable functions u k t on R  such that n  k1 u k  t   W  t  n−1 , σ  t   η k  u k  t  ,t  t ∈ R  ,k 1, ,n  ; 2.30 iii for any k such that 1 p k > 1 p − r n 2.31 the function u k tWt δ−1 decreases on R  . Proof. The proof is similar to 15, Lemma 2.2. All the argument holds true when we substitute the weight wt in this lemma for wt1. The Lebesgue measure of a measurable set A ⊂ R k will be denoted by mes k A. For any F σ −set E ⊂ R n denote by E j the orthogonal projection of E onto the coordinate hyperplane x j  0. By the Loomis-Whitney inequality 17, Chapter 4 mes n E n−1 ≤ n  j1 mes n−1 E j . 2.32 Journal of Inequalities and Applications 9 Let f ∈ S 0 R n ,t>0, and let E t be a set of type F σ and measure t such that |fx|≥ f ∗ t for all x ∈ E t . Denote by λ j t the n − 1-dimensional measure of the projection E j t j  1, ,n. By 2.32, we have that n  j1 λ j  t  ≥ t n−1 . 2.33 Lemma 2.4. Let n ≥ 2,r k ∈ N k  1, ,n,wbe nonincreasing, and wt → a when t →∞ where a>0. Function f ∈ S 0 R n  has weak derivatives D r k k f ∈ L loc R n k  1, ,n. Then for all 0 <t<τ<∞ and k  1, ,none has f ∗  t  ≤ K  f ∗  τ    τ t  r k  Wt λ  k t  r k D r k k f ∗∗  τ   , 2.34 where  n k1 λ  k t ≥ Wt n−1 and K is a constant depending on r 1 , ,r n and a. Proof. Let λ  k t1/ n √ aWt/tλ k t, then n  k1 λ  k  t   1 a  Wt t  n n  k1 λ k  t  . 2.35 Due to the conditions of w and 2.33, we can get n  k1 λ  k  t  ≥ Wt n−1 . 2.36 In 2, 12, 15, we have f ∗  t  ≤ K  f ∗  τ    τ λ k  t   r k  D r k k f  ∗∗  τ   . 2.37 So we immediately get 2.34. Lemma 2.5. If w ∈ B 1,∞ , 1 <p 0 < ∞ and 1 ≤ s 0 < ∞,thenv ≡ Wt s 0 /p 0 −1 wt ∈ B s 0 . Proof. Let w ∈ B 1,∞ . Since B 1,∞ ⊂ B p 0 , so by 9, Chapter 1 we get  r 0 1 Wt 1/p 0 dt ≤ C r Wr 1/p 0 , ∀r>0. 2.38 Then  r 0 1 V t 1/s 0 dt ≤ C r V r 1/s 0 , ∀r>0, 2.39 10 Journal of Inequalities and Applications where V  t    t 0 v  t  dt. 2.40 So v ∈ B s 0 . Lemma 2.6. Let n ≥ 2,r k ∈ N, 1 <p k < ∞, 1 ≤ s k < ∞ for k  1, ,n.Assume that weight w on R  satisfies the following conditions: i it is nonincreasing, continuous, and lim t →∞ wta, a > 0, ii exists A>0, such that W  ξt  ≥ ξ A W  t  , ∀ξ>1, ∀t>0. 2.41 Set r  n  n  k1 1 r k  −1 ,p n r  n  k1 1 p k r k  −1 , s  n r  n  k1 1 s k r k  −1 . 2.42 Assume that a locally integrable function f ∈ S 0 R n  has weak derivatives D r k k f ∈ Λ p k ,s k wk  1, ,n. Then for any ξ>1 f ∗  t  ≤ K  f ∗  ξt   ξ r σ  t   , 2.43 where r  max r k , the constants K depends only on r 1 , ,r n ,w, and   ∞ 0 Wt s1/p−r/n−1 wtσt s dt  1/s ≤ C n  k1 D r k k f r/nr k  Λ p k ,s k w . 2.44 Proof. For every fixed k  1, ,nwe take ψ k  t    D r k k f  ∗∗  t  . 2.45 Thanks to Lemma 2.5,andw ∈ B 1,∞ for w is nonincreasing,weknow v  Wt s k /p k −1 w  t  ∈ B s k . 2.46 Thus ψ k  Λ p k ,s k w     D r k k f  ∗∗   L s k v ≤ C    D r k k f  ∗   L s k v  C   D r k k f   Λ p k ,s k w . 2.47 [...]... Mart´n and M Milman, “Symmetrization inequalities and Sobolev embeddings,” Proceedings of the ı American Mathematical Society, vol 134, no 8, pp 2335–2347, 2006 19 J Mart´n and M Milman, “Higher-order symmetrization inequalities and applications,” Journal of ı Mathematical Analysis and Applications, vol 330, no 1, pp 91–113, 2007 20 V I Kolyada, “Inequalities of Gagliardo-Nirenberg type and estimates... 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Journal of Inequalities and Applications Therefore, we get, applying 3.27 and 3.34 J 1/θ1 ≤ C ∞ W t s 1/p−r/n −1 w t σ t s dt 1−H1 /s 0 ≤C n r D11 f H1 Λp1 ,s1 w 3.46 1−H1 r Dkk f k 1 r/ nrk Λpk ,sk w r D11 f H1 Λp1 ,s1 w Since n k r nrk 1 1, 3.47 we get the inequality 3.6 The theorem is proved Let X X Rn be a rearrangement invariant space r.i space , Y be an r.i space over s s 1 s integral part of . Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 161405, 22 pages doi:10.1155/2009/161405 Research Article A Kind of Estimate of Difference Norms. class of all measurable and almost everywhere finite functions on X. For f ∈MX, μ, a nonincreasing rearrangement of f is a nonincreasing function f ∗ on R  ≡ 0, ∞, that is, equimeasurable. 2.53 12 Journal of Inequalities and Applications Proof. Let f  g  h,withg ∈ Λ 1 w and h ∈ Λ p 0 w. Applying H ¨ older’s inequality and noticing W∞∞ and w is nonincreasing, we obtain J 1 ≡  ∞ 1 f ∗θ  t  Wt θ/q−1 w  t  dt ≤  ∞ 1 g ∗θ  t 2  W  t  θ/q−1 w  t  dt