Optimal non-homogeneous improvements for the series expansion of Hardy's inequality
aa r X i v : . [ m a t h . A P ] M a y Optimal non-homogeneous improvementsfor the series expansion of Hardy’s inequality
K. T. Gkikas ∗ G. Psaradakis † Abstract
We consider the series expansion of the L p -Hardy inequality of [BFT2], in the particular casewhere the distance is taken from an interior point of a bounded domain in R n and 1 < p = n . For p < n we improve it by adding as a remainder term an optimally weighted critical Sobolev norm,generalizing the p = p > n weimprove it by adding as a remainder term the optimally weighted H¨older seminorm, extending theHardy-Morrey inequality of [Ps] to the series case. Keywords: improved Hardy inequality · critical Sobolev norm · modulus of continuity MSC: · · Let Ω be any domain in R n , n ≥
3, containing the origin. Hardy’s inequality asserts that for all u ∈ H ( Ω ) Z Ω | ∇ u | d x ≥ (cid:16) n − (cid:17) Z Ω | u | | x | d x , (1.1)with the best possible constant. As in [S, pg 262], an integration by parts shows that ( n − ) Z Ω | u | | x | d x = − Z Ω | u || x | − x · ∇ | u | d x , and then applying the Cauchy-Schwarz inequality on the right gives (1.1). The best constant in this easilyobtained functional inequality has applications in various branches of analysis. For instance, it is usedin [S, Appendix B] to prove the non-existence of stable minimal cones in R n + , 2 ≤ n ≤ § / | x | (see for example [BrV], [GP],[Gk1, Gk2] [CM], [VzZ], , [FMT], [FT], [VnZ] & [Erv]). ∗ Konstantinos T. Gkikas: Centro de Modelamiento Matem´atico (UMI 2807 CNRS), Universidad de Chile, Casilla 170Correo 3, Santiago, Chile; [email protected] † Georgios Psaradakis: Institut f¨ur Mathematik (Lehrst¨uhl f¨ur Mathematik IV), Universit¨at Mannheim, A5, Mannheim68131, Deutschland; [email protected] Ω has finite Lebesgue measure L n ( Ω ) ,by adding the term C ( n , q ) (cid:0) L n ( Ω ) (cid:1) / ⋆ − / q (cid:18) Z Ω | u | q d x (cid:19) / q , ≤ q < ⋆ : = nn − , (1.2)on its right hand side. Such a non-homogeneous improvement fails in the case of the critical Sobolevexponent q = ⋆ . This can be seen for example from the improvement given for bounded Ω in [FT,Theorem A]. There, the term C ( n ) (cid:18) Z Ω | u | ⋆ X + ⋆ / ( | x | / D ) d x (cid:19) / ⋆ , X ( t ) : = ( − log t ) − , t ∈ ( , ] , (1.3)where D : = sup x ∈ Ω | x | , was added on the right hand side of (1.1). It is then shown that the exponent1 + ⋆ / X cannot be decreased, stating thus the failure of adding the term (1.2) for q = ⋆ .It was also established in [FT, Theorem D] in case of bounded Ω , that the homogeneous term14 Z Ω | u | | x | X ( | x | / D ) d x , (1.4)can be added on the right hand side of (1.1). Here, the constant 1 / X cannot be decreased. This type of optimal logarithmic homogeneous improvements to Hardyinequalities originated in [BrM]. Compared to (1.3), one has that (1.4) is not a weaker remainder term.In particular, one cannot deduce (1.1) with remainder term (1.4) from (1.1) with remainder term (1.3) byapplying H ¨older’s inequality (not even with some other positive constant instead of 1 / X in (1.4) implies via H ¨older’s inequality the optimality of the exponent1 + ⋆ / X in (1.3) (see [PsSp] and [FPs] for similar arguments).In the question what is an optimal non-homogeneous term one can add on the right hand side of (1.1)with remainder term (1.4), the answer is C ( n ) (cid:18) Z Ω | u | ⋆ X + ⋆ / ( | x | / D ) X + ⋆ / ( | x | / D ) d x (cid:19) / ⋆ , X ( t ) : = X ( X ( t )) , t ∈ ( , ] . Furthermore, it is proved in [FT, Theorem A ′ ] that for any k ∈ N ∪ { } and all u ∈ H ( Ω ) Z Ω | ∇ u | d x − (cid:16) n − (cid:17) Z Ω | u | | x | d x − Z Ω | u | | x | k ∑ i = i ∏ j = X j ( | x | / D ) d x ≥ C ( n ) (cid:18) Z Ω | u | ⋆ k + ∏ i = X + ⋆ / i ( | x | / D ) d x (cid:19) / ⋆ , X j + ( t ) : = X ( X j ( t )) , t ∈ ( , ] , (1.5)where the exponent 1 + ⋆ / X k + cannot be decreased. That the left hand side in this inequality isnonnegative and each term appears with best constant 1 / ∏ kj = i X j was firstestablished in [FT, Theorem D]. For a second proof of (1.5) with the best constant C ( n ) see [AdFT].The purpose of this paper is to extend inequality (1.5) to the case of the k -improved p -Hardy inequal-ity for any 1 < p < ∞ , p = n . More precisely, the p -Hardy inequality in a domain Ω of R n , n ≥
2, con-taining the origin, asserts that if p ≥ p = n , then for all u ∈ W , p ( Ω ) if p < n , or all u ∈ W , p ( Ω \ { } ) if p > n , we have Z Ω | ∇ u | p d x ≥ (cid:12)(cid:12)(cid:12) n − pp (cid:12)(cid:12)(cid:12) p Z Ω | u | p | x | p d x , (1.6)with the best possible constant. A proof of the same simplicity as in the case p = Ω containing the origin there exists b = b ( n , p ) ≥ k ∈ N theterms p − p (cid:12)(cid:12)(cid:12) n − pp (cid:12)(cid:12)(cid:12) p − Z Ω | u | p | x | p k ∑ i = i ∏ j = X j ( | x | / D ) d x , D = bD , (1.7)can be added on its right hand side. Moreover, for every k ∈ N , each one of these terms appears with thebest possible constant and the exponent 2 on ∏ kj = X j cannot be decreased. In Theorem 2.7 we providean alternative proof of (1.7) using a suitable ground state transformation (as was done in the p = p ∈ ( , n ) of (1.5). We denote below by p ⋆ the criticalSobolev exponent in this case; that is p ⋆ : = np / ( n − p ) . Theorem A
Let Ω be a bounded domain in R n , n ≥ , containing the origin and let < p < n. Thereexist constants B = B ( n , p ) ≥ and C = C ( n , p ) > such that for any k ∈ N ∪ { } and all u ∈ W , p ( Ω ) Z Ω | ∇ u | p d x − (cid:16) n − pp (cid:17) p Z Ω | u | p | x | p d x − p − p (cid:16) n − pp (cid:17) p − Z Ω | u | p | x | p k ∑ i = i ∏ j = X j ( | x | / D ) d x ≥ C (cid:18) Z Ω | u | p ⋆ k + ∏ i = X + p ⋆ / pi ( | x | / D ) d x (cid:19) p / p ⋆ , D = B sup x ∈ Ω | x | . (1.8) Moreover, for each k ∈ N ∪ { } , the weight function ∏ k + i = X + p ⋆ / pi is optimal in the sense that the power + p ⋆ / p on X k + cannot be decreased. Inequality (1.8) for p = k =
0. In fact, for k = L q -weighted remainder term with q < p ⋆ ) inequality with an optimal power on the logarithmicweight, that tends to the one appearing in (1.8) as q → p ⋆ , is in [BFT1, Theorem C (2)]. It is referredthere as an open question whether (1.8) for k = k = p < p > < p < ( ii ) which is the extension of Proposition 3.4 of [BFT1] toall k ∈ N . The reason is that for k ∈ N , the naturally choice for a function to be used in the groundstate transformation fails to be a supersolution of the corresponding Euler-Langrange equation (see alsoRemark 3.1). To solve this problem we invent a new supersolution (see (2.4)).For p >
2, by applying the natural ground state transform, two lower bounds for the left hand sideof (1.8) may be produced (for k = p = p = p > n , we address here the question of what is an optimal nonhomogeneous termone can add on the right hand side of (1.6) with remainder term (1.7). It is known that the Dirichletintegral in R n exceeds a constant multiple of the C , − n / p -seminorm. More precisely, there exists apositive constant C = C ( n , p ) such that for all u ∈ W , p ( R n ) (cid:18) Z R n | ∇ u | p dx (cid:19) / p ≥ C sup x , y ∈ R n x = y | u ( x ) − u ( y ) || x − y | − n / p , and the modulus of continuity 1 − n / p is optimal. This is Morrey’s inequality. In Hardy’s inequality(1.6), an optimally weighted C , − n / p -seminorm was added in [Ps] in case of a bounded Ω . The precisestatement asserts the existence of constants C = C ( n , p ) > B = B ( n , p ) ≥ u ∈ W , p ( Ω \ { } ) (cid:18) Z Ω | ∇ u | p dx − (cid:16) p − np (cid:17) p Z Ω | u | p | x | p dx (cid:19) / p ≥ C sup x , y ∈ Ω x = y | u ( x ) − u ( y ) || x − y | − n / p X / p (cid:16) | x − y | D (cid:17) , D = B diam ( Ω ) . The correction X / p on the modulus of continuity was shown to be optimal in the sense that the power1 / p on X cannot be decreased. The following inequality is reduced to the above one when k =
0, andgives the complete picture for the series improvement of Hardy’s inequality
Theorem B
Let Ω be a bounded domain in R n , n ≥ , containing the origin and let p > n. There existconstants B = B ( n , p ) ≥ and C = C ( n , p ) > such that for any k ∈ N ∪ { } and all u ∈ W , p ( Ω \ { } ) (cid:18) Z Ω | ∇ u | p d x − (cid:16) p − np (cid:17) p Z Ω | u | p | x | p d x − p − p (cid:16) p − np (cid:17) p − Z Ω | u | p | x | p k ∑ i = i ∏ j = X j ( | x | / D ) d x (cid:19) / p ≥ C sup x , y ∈ Ω x = y | u ( x ) − u ( y ) || x − y | − n / p k + ∏ i = X / pi ( | x − y | / D ) , D = B diam ( Ω ) . (1.9) Moreover, for each k ∈ N ∪ { } , the weight function ∏ k + i = X / pi is optimal in the sense that the power / pon X k + cannot be decreased. The paper is organised as follows: In §
2, after setting the notation and a couple of technical calculusfacts, we use the ground state transform to produce lower estimates for the series expansion of Hardy’sinequality. These are used in § § § < p <
2, Theorem A for p > p > n , respectively. These are also used in § In this paper we assume • < p = n , where n ∈ N \ { } , • Ω is a bounded domain in R n containing the origin, • D : = sup x ∈ Ω | x | .Furhermore, L n stands for the Lebesgue measure in R n and H n − for the n − R n . B r ( x ) is the open ball in R n having radius r > x ∈ R n ; ∂ B r ( x ) is its boundary. Whenthe centre is of no importance we simply write B r . When the center is the origin and r = S n − instead of ∂ B ( ) . Also, ω n : = L n ( B ) and so H n − ( ∂ B ) = n ω n . Throughout, an expression of theform b ( n , p , ... ) , B ( n , p , ... ) , c ( n , p , ... ) or C ( n , p , ... ) stands for a positive constant that may change valuefrom line to line but always depending only on its arguments n , p ... . The particular constant dependingonly on p that appears in (2.15) or (2.17), is denoted by c p . All functions having compact support areextended by zero outside it. Definition 2.1.
For any t ∈ ( , ] we define the function X ( t ) : = ( − log t ) − and then X k ( t ) : = X ( X k − ( t )) , k = , , ..., Y k ( t ) : = k ∏ i = X i ( t ) , k ∈ N , Z k ( t ) : = k ∑ i = Y i ( t ) , k ∈ N . The following computational lemma gives a formula for the derivative of X k , Y k and Z k . Lemma 2.2.
For any k ∈ N and t ∈ ( , ] there holdsddt (cid:0) X k ( t ) (cid:1) = t Y k ( t ) X k ( t ) , ddt (cid:0) Y k ( t ) (cid:1) = t Y k ( t ) Z k ( t ) , ddt (cid:0) Z k ( t ) (cid:1) = t (cid:16) Z k + k ∑ i = Y i ( t ) (cid:17) . Proof.
The first one follows easily by induction. The proof of the second one is ddt (cid:0) Y k ( t ) (cid:1) = k ∑ j = ddt (cid:0) X j ( t ) (cid:1) k ∏ i = i = j X i ( t ) = t k ∑ j = Y j ( t ) X j ( t ) k ∏ i = i = j X i ( t ) = t Y k ( t ) Z k ( t ) , where the first one is used in the middle equality. For the third one, notice that one has the elementaryidentity k ∑ i = Y i Z i = (cid:16) Z k + k ∑ i = Y i (cid:17) , for which we include its proof for clarity2 k ∑ i = Y i Z i = k ∑ i , j = j ≤ i Y i Y j = k ∑ i , j = j < i Y i Y j + k ∑ i = Y i = (cid:16) k ∑ i = Y i (cid:17) + k ∑ i = Y i = Z k + k ∑ i = Y i . Now we may easily conclude ddt (cid:0) Z k ( t ) (cid:1) = k ∑ i = ddt (cid:0) Y i ( t ) (cid:1) = t k ∑ i = Y i ( t ) Z i ( t ) = t (cid:16) Z k ( t ) + k ∑ i = Y i ( t ) (cid:17) , where the second one is used in the middle equality. Remark 2.3.
The infinite series Z ∞ ( t ) : = ∑ ∞ k = Y k ( t ) , t ∈ ( , ] , converges if and only if t ∈ ( , ) . Aproof of this fact can be extracted from [FT, §
6] (see [D, Appendix] for the details).A technical lemma follows
Lemma 2.4.
Let α , β , R > . For all r ∈ ( , R ] , all c > / α and any D ≥ η R, where η = η ( α , β , c ) > ,there holds Z r t α − Y − β k ( t / D ) d t ≤ cr α Y − β k ( r / D ) . Proof.
For c > D ≥ R set f ( r ) : = Z r t α − Y − β k ( t / D ) d t − cr α Y − β k ( r / D ) , r ∈ ( , R ] . It suffices to show for suitable values of the parameters c and D , that f ( r ) ≤ r ∈ ( , R ) . Since f ( +) =
0, it is enough to choose c and D so that f is decreasing in ( , R ) . To this end, with the aid ofLemma 2.2 we compute f ′ ( r ) = cr α − Y − β k ( r / D ) (cid:2) / c − α + β Z k ( r / D ) (cid:3) , r ∈ ( , R ] . By Remark 2.3 the series Z ∞ ( R / D ) is convergent if R < D , hence for c > / α we can find large enough η > α , β and c , such that for all D ≥ η R to have f ′ ( r ) ≤ r ∈ ( , R ) . Definition 2.5.
Given u ∈ C ∞ c ( Ω \ { } ) and D ≥ D we set I [ u ; D ] ≡ I [ u ] : = Z Ω | ∇ u | p d x − (cid:12)(cid:12)(cid:12) n − pp (cid:12)(cid:12)(cid:12) p Z Ω | u | p | x | p d x , I k [ u ; D ] : = I k − [ u ; D ] − p − p (cid:12)(cid:12)(cid:12) n − pp (cid:12)(cid:12)(cid:12) p − Z Ω | u | p | x | p Y k ( | x | / D ) d x = I [ u ] − p − p (cid:12)(cid:12)(cid:12) n − pp (cid:12)(cid:12)(cid:12) p − Z Ω | u | p | x | p k ∑ i = Y i ( | x | / D ) d x , k ∈ N . In [BFT2] the following successive homogeneous improvements to Hardy’s inequality were obtained
Theorem 2.6 ([BFT2]-Theorem A) . There exists a constant b = b ( n , p ) ≥ such that for any k ∈ N I k [ u ; D ] ≥ for all u ∈ C ∞ c ( Ω \ { } ) , (2.1) where D = bD . Moreover, for each k ∈ N : ( i ) the weight function Y k is optimal, in the sense that the power 2 cannot be decreased, and ( ii ) the constant appearing on the term with this weight function is sharp. In [BFT1] the authors obtained various auxiliary improvements for Hardy’s inequality (1.6). Inparticular, given u ∈ C ∞ c ( Ω \ { } ) , the ground state transformation u ( x ) = | x | − n / p v ( x ) , (2.2)plus elementary vectorial inequalities lead to the following lower bounds on I [ u ] in terms of the function v (see [BFT1, Lemma 3.3 & Proposition 3.4]) I [ u ] ≥ c ( p ) Z Ω | x | p − n | ∇ v | p d x , I [ u ] ≥ c ( p ) Z Ω | x | − n | v | p − | ∇ v | d x , both in case p ≥
2, and I [ u ] ≥ c ( n , p ) Z Ω | x | p − n | ∇ v | p X − p ( | x | / D ) d x , D ≥ D , (2.3)in case p <
2. Our aim here is to extend these estimates to arbitrary k ∈ N . More precisely, we have thefollowing theorem which readily implies (2.1) Theorem 2.7.
For a ≥ , D ≥ D and k ∈ N ∪ { } setf a , k , D ( x ) : = sgn ( n − p ) | x | − n / p Y − / pk ( | x | / D ) (cid:0) − aX ( | x | / D ) (cid:1) , x ∈ Ω \ { } . (2.4) For simplicity we write f in place of f , , D and f k , D instead of f , k , D . Then, ( i ) for p ≥ , there exists a constant b ′ = b ′ ( n , p ) ≥ such that for any k ∈ N ∪ { } , all u ∈ C ∞ c ( Ω \{ } ) and any D ≥ b ′ D I k [ u ; D ] ≥ c p Z Ω | x | p − n | ∇ v | p Y − k ( | x | / D ) d x , (2.5) I k [ u ; D ] ≥ c p Z Ω | x | − n | v | p − | ∇ v | (cid:12)(cid:12)(cid:12) p − np − p Z k ( | x | / D ) (cid:12)(cid:12)(cid:12) p − Y − k ( | x | / D ) d x , (2.6) where v is defined through the ground state transformation u = f k , D v. ( ii ) If p < , there exist constants a = a ( n , p ) > and b ′′ = b ′′ ( n , p ) ≥ such that for any k ∈ N ∪ { } ,all u ∈ C ∞ c ( Ω \ { } ) and any D ≥ b ′′ D I k [ u ; D ] ≥ c ( n , p ) Z Ω | x | p − n | ∇ v | p Y − pk + ( | x | / D ) Y − k ( | x | / D ) d x , (2.7) where v is defined through the ground state transformation u = f a , k , D v. Remark 2.8.
Clearly, p < n = ⇒ (cid:12)(cid:12)(cid:12) p − np − p Z k ( | x | / D ) (cid:12)(cid:12)(cid:12) = n − pp + p Z k ( | x | / D ) ≥ n − pp , for all x ∈ Ω and all k ∈ N . A similar estimate is true for p > n . In particular, by Remark 2.3 the series Z ∞ ( | x | / D ) is convergent if | x | < D and thus for suitable b ′′′ >
1, depending only on n , p , we may choose D ≥ b ′′′ D so that p − np − p Z k ( | x | / D ) ≥ C ( n , p ) > , for all x ∈ Ω and all k ∈ N . Consequently, from (2.6) we get Corollary 2.9.
For p ≥ , there exists a constant b ′′′ = b ′′′ ( n , p ) ≥ such that for all u ∈ C ∞ c ( Ω \ { } ) ,any D ≥ b ′′′ D and any k ∈ N ∪ { } I k [ u ; D ] ≥ C ( n , p ) Z Ω | x | − n | v | p − | ∇ v | Y − k ( | x | / D ) d x . (2.8)To prove Theorem 2.7 we need the following key lemma Lemma 2.10. ( i ) For p ≥ , there exists a constant b ′ = b ′ ( n , p ) ≥ such that for any D ≥ b ′ D and any k ∈ N , the function f k , D (defined by (2.4) with a = ) is a supersolution of the followingp-Laplace equation with k + singular potential terms − ∆ p w − (cid:16)(cid:12)(cid:12)(cid:12) p − np (cid:12)(cid:12)(cid:12) p + p − p (cid:12)(cid:12)(cid:12) p − np (cid:12)(cid:12)(cid:12) p − k ∑ i = Y i ( | x | / D ) (cid:17) | w | p − w | x | p = , in Ω \ { } . (2.9) ( ii ) For p < , there exist constants a = a ( n , p ) > and b ′′ = b ′′ ( n , p ) ≥ such that for any D ≥ b ′′ D and any k ∈ N , the function f a , k , D (defined in (2.4) ) is a supersolution of (2.9) . Proof.
Let a ≥ < ε <
1. In view of Remark 2.3 we choose δ = δ ( a , p ) ≥
1, such that with D : = D δ to have1 − aX ( | x | / D ) ≥ − p and X ( | x | / D ) ≤ ∞ ∑ i = Y i ( | x | / D ) ≤ pX ( | x | / D ) , ∀ x ∈ Ω . (2.10)We further set A a , k ( | x | / D ) : = p − np − p Z k ( | x | / D ) − a X ( | x | / D ) − aX ( | x | / D ) . (2.11)Using Lemma 2.2 we compute (from now on in this proof we write f k , A k , X k , Y k , Z k instead of f a , k , D ( x ) , A a , k ( | x | / D ) , X k ( | x | / D ) , Y k ( | x | / D ) , Z k ( | x | / D ) ) ∇ f k = f k | x | A k x | x | so that − ∆ p f k = − div n | f k | p − f k | x | p − | A k | p − A k x | x | o . Direct computations reveal the next identities which are valid for any x ∈ Ω \ { }− div n | f k | p − f k | x | p − x | x | o = (cid:16) p − n − ( p − ) A k (cid:17) | f k | p − f k | x | p , − ∇ (cid:16) | A k | p − A k (cid:17) = − ( p − ) | A k | p − ∇ A k = p − | x | | A k | p − (cid:18) p (cid:16) Z k + k ∑ i = Y i (cid:17) + a X − aX + a X ( − aX ) | {z } : = F ( X ) (cid:19) x | x | , where in the last one we used Lemma 2.2 in order to compute ∇ A k . We conclude − ∆ p f k = | A k | p − (cid:18) ( p − n ) A k − ( p − ) A k + p − p (cid:16) Z k + k ∑ i = Y i (cid:17) + ( p − ) F ( X ) (cid:19) | f k | p − f k | x | p = | A k | p − (cid:18)(cid:16) p − np (cid:17) + ( p − n )( p − ) p Z k + ( p − )( p − ) p Z k + p − p k ∑ i = Y i (cid:19) | f k | p − f k | x | p + a | A k | p − (cid:18) ( p − n )( p − ) p + ( p − ) (cid:16) X − p Z k (cid:17)(cid:19) X − aX | f k | p − f k | x | p . It turns out that given 1 < p < n it is enough to establish the following inequality for some nonnegativeconstant a = a ( n , p ) and for any x ∈ Ω \ { }| A k | p − (cid:18)(cid:16) p − np (cid:17) + ( p − n )( p − ) p Z k + ( p − )( p − ) p Z k + p − p k ∑ i = Y i (cid:19) + a | A k | p − (cid:18) ( p − n )( p − ) p + ( p − ) (cid:16) X − p Z k (cid:17)(cid:19) X − aX ≥ (cid:12)(cid:12)(cid:12) p − np (cid:12)(cid:12)(cid:12) p + p − p (cid:12)(cid:12)(cid:12) p − np (cid:12)(cid:12)(cid:12) p − k ∑ i = Y i , (2.12)and the reverse inequality if p > n (note that sgn f k = sgn ( n − p ) ). In the case p = a = p = h : = p − np , t : = Z k p , s : = a X − aX and λ : = p − p k ∑ i = Y i , so that all we need to prove is that for all sufficiently small t and some nonnegative constant a dependingpossibly only on n , p , there holds (cid:12)(cid:12)(cid:12) − t + sh (cid:12)(cid:12)(cid:12) p − (cid:16) h + λ + h ( p − )( t + s ) + ( p − )( p − ) t + ( p − )( X − t ) s (cid:17) ≥ h + λ , if 1 < p < n , and the reverse inequality if p > n . By a further rearrangement of terms, this is the same as ( h + λ ) (cid:18) − (cid:12)(cid:12)(cid:12) − t + sh (cid:12)(cid:12)(cid:12) − p (cid:19) + h ( p − )( t + s ) + ( p − )( p − ) t + ( p − )( X − t ) s ≥ , (2.13)if 1 < p < n , and the reverse inequality if p > n . The Taylor expansion of g ( x ) : = | − x | − p at x = g ( x ) = + ( p − ) x + ( p − )( p − ) x + p ( p − )( p − ) x + O ( x ) , ( p − )( X − t ) s + ( p − )( p − ) (cid:0) t − ( t + s ) (cid:1) − p − h (cid:16) p ( p − ) ( t + s ) + λ ( t + s ) (cid:17) + O (cid:0) λ ( t + s ) (cid:1) ≥ , (2.14)if 1 < p < n , and the reverse inequality if p > n .We distinguish two cases: • < p = n . In this case we take a = s =
0. The trivial fact that ∑ ki = Y i ≤ Z k , k ∈ N , istranslated to 2 λ ≤ p ( p − ) t . Therefore if p < n we have h < − p − h (cid:16) p ( p − ) t + λ t (cid:17) + O (cid:0) λ t (cid:1) ≥ , if 2 < p < n , while if p > n then h > • < p <
2. By (2.10) we obtain X − t = X − Z k / p > k ∈ N and we can choose for example a = p and hence s > Lemma 2.11.
For all w ∈ W , p ( Ω ) and all D ≥ D , there holds Z Ω | w | p | x | n Y k ( | x | / D ) X k + ( | x | / D ) d x ≤ p p Z Ω | x | p − n | ∇ w | p Y − pk ( | x | / D ) X − pk + ( | x | / D ) d x . Proof.
A direct computation using Lemma 2.2 shows thatdiv (cid:8) | x | − n X k + ( | x | / D ) x (cid:9) = | x | − n Y k ( | x | / D ) X k + ( | x | / D ) , x ∈ Ω \ { } . Hence, integrating by parts, Z Ω | w | p | x | n Y k ( | x | / D ) X k + ( | x | / D ) d x = − p Z Ω X k + ( | x | / D ) | w | p − ∇ | w | · x | x | n d x ≤ p Z Ω X k + ( | x | / D ) | w | p − | ∇ w || x | − n d x = p Z Ω n | w | p − | x | n ( − / p ) Y − / pk ( | x | / D ) X ( − / p ) k + ( | x | / D ) on | ∇ w || x | n / p − Y / p − k ( | x | / D ) X / p − k + ( | x | / D ) o d x ≤ p (cid:18) Z Ω | w | p | x | n Y k ( | x | / D ) X k + ( | x | / D ) d x (cid:19) − / p (cid:18) | ∇ w | p | x | n − p Y − pk ( | x | / D ) X − pk + ( | x | / D ) d x (cid:19) / p . The result follows by rearranging terms and taking the p -th power. Proof of Theorem 2.7 for p ≥ . Setting u ( x ) = f k , D ( x ) v ( x ) we get Z Ω | ∇ u | p d x = Z Ω | v ∇ f k , D + f k , D ∇ v | p d x ≥ Z Ω | v | p | ∇ f k , D | p d x + c p Z Ω | f k , D | p | ∇ v | p d x + Z Ω f k , D | ∇ f k , D | p − ∇ f k , D · ∇ | v | p d x , α , β ∈ R n , n ≥ p ≥ | α + β | p ≥ | α | p + c p | β | p + p | α | p − α · β . (2.15)Noting thatdiv { f k , D | ∇ f k , D | p − ∇ f k , D } = | ∇ f k , D | p + f k , D ∆ p f k , D , (2.16)we perform an integration by parts in the last term to arrive at Z Ω | ∇ u | p d x ≥ c p Z Ω | f k , D | p | ∇ v | p d x − Z Ω | v | p f k , D ∆ p f k , D d x = c p Z Ω | f k , D | p | ∇ v | p d x − Z Ω | u | p f − k , D | f k , D | − p ∆ p f k , D d x . Inequality (2.5) follows now from Lemma 2.10- ( i ) . If instead of (2.15) we use | α + β | p ≥ | α | p + c p | α | p − | β | + p | α | p − α · β , (2.17)valid for all α , β ∈ R n , n ≥ p ≥
2, we similarly obtain (2.6) from Lemma 2.10- ( i ) . Proof of Theorem 2.7 for < p < . By the fact that (see [L]) | α + β | p − | α | p ≥ p | α | p − α · β + p ( p − ) | β | ( | α | + | β | ) − p , for all α , β ∈ R n , n ≥ p ∈ ( , ) , we get setting u ( x ) = f a , k , D ( x ) v ( x ) , Z Ω | ∇ u | p d x = Z Ω | v ∇ f a , k , D + f a , k , D ∇ v | p d x ≥ Z Ω | v | p | ∇ f a , k , D | p d x + Z Ω f a , k , D | ∇ f a ,, k , D | p − ∇ f a ,, k , D · ∇ | v | p d x + c ( p ) Z Ω f a , k , D | ∇ v | (cid:0) | v || ∇ f a , k , D | + | f a , k , D || ∇ v | (cid:1) − p d x . By the same integration by parts and (2.16), but this time using Lemma 2.10- ( ii ) , we get for any D ≥ b ′ D I k [ u ; D ] ≥ c ( p ) Z Ω f a , k , D | ∇ v | (cid:0) | v || ∇ f a , k , D | + | f a , k , D || ∇ v | (cid:1) − p d x = : c ( p ) M . (2.18)Next we define M to have the same integrand as in M but with the measure ρ − p d x in place of d x , where ρ ( x ) : = − aX ( | x | / D ) , x ∈ Ω . Also, we set M : = Z Ω f pk , D | ∇ v | p Y − pk + d x , M : = Z Ω f pk , D | x | p | v | p Y k + d x . To get (2.7) from (2.18), it suffices to show M ≤ C ( n , p ) M . Noting that f k , D = f a , k , D / ρ , we use H ¨older’sand Minkowski’s inequalities as follows M = Z Ω f pa , k , D | ∇ v | p (cid:0) | v || ∇ f a , k , D | + | f a , k , D || ∇ v | (cid:1) ( − p ) p / (cid:0) | v || ∇ f a , k , D | + | f a , k , D || ∇ v | (cid:1) ( − p ) p / Y − pk + d x ρ p ≤ M p / (cid:16) Z Ω (cid:0) | v || ∇ f a , k , D | + | f a , k , D || ∇ v | (cid:1) p Y k + d x ρ p (cid:17) − p / ≤ M p / (cid:18)(cid:16) Z Ω | v | p | ∇ f a , k , D | p Y k + d x ρ p (cid:17) − p / + M − p / (cid:19) , (2.19)2where we also have used ( α + β ) q ≤ α q + β q for all α , β ≥ q = p ( − p / ) ∈ ( , ] and the simple factthat Y k + ( t ) ≤ Y − pk + ( t ) for all t ∈ ( , ] . From Remark 2.3 we know that for sufficiently large D ≥ BD , B = B ( n , p ) ≥
1, we have Z k ( | x | / D ) ≤ C ( n , p ) for all x ∈ Ω . Hence, taking into account (2.10) we get | A a , k ( | x | / D ) | ≤ C ( n , p ) for all x ∈ Ω , where A a , k is given by (2.11). Therefore | ∇ f a , k , D | = (cid:12)(cid:12)(cid:12) f a , k , D | x | A a , k x | x | (cid:12)(cid:12)(cid:12) ≤ C ( n , p ) | f a , k , D || x | , so that Z Ω | v | p | ∇ f a , k , D | p Y k + d x ρ p ≤ C ( n , p ) M . But notice that Lemma 2.11 asserts M ≤ p p M . Plugging these into (2.19) we arrive at M ≤ C ( n , p ) M . Finally, M ≤ ( − p ) − p M because of (2.10), and the proof is complete. Remark 2.12.
In the case p = D ≥ D and B ( n , p ) = < p ≤ We start with a series of reductions. First, since 0 ∈ Ω , if u ∈ C ∞ c ( Ω ) then u ∈ C ∞ c ( B D ( )) . Hence itis enough to establish (1.8) for Ω = B D ( ) . Furthermore, (1.8) being scaling invariant, it is enough toprove it for Ω = B ( ) only. Finally, given u ∈ C ∞ c ( B ( )) \ { } , the transform u = f a , k , D v implies throughTheorem 2.7- ( ii ) that it suffices to find a constant c ( n , p ) > C : = inf v ∈ C ∞ c ( B ( )) \{ } R B ( ) | x | p − n | ∇ v | p Y − pk + ( | x | / D ) Y − k ( | x | / D ) d x (cid:16) R B ( ) | x | − n | v | p ⋆ Y k ( | x | / D ) X + p ⋆ / pk + ( | x | / D ) d x (cid:17) p / p ⋆ ≥ c ( n , p ) . (3.1) Remark 3.1.
Let k =
0. Then the above sufficiency of (3.1) is straightforward from (2.3) through thetransform (2.2); that is u = f , v . It is for k ∈ N that we need the transform u = f a , k , D v for some a > ( ii ) .To carry on with the proof, consider the Emden-Fowler transformation v ( x ) = w ( τ , θ ) , where τ : = X k + ( r / D ) , θ : = xr with r : = | x | . A simple computation using Lemma 2.2 givesd τ d r = − r Y k ( r / D ) , therefore, | ∇ v | = ( ∂ r v ) + r | ∇ θ v | = r Y k ( r / D ) (cid:0) ( ∂ τ w ) + Y − k ( r / D ) | ∇ θ w | (cid:1) . F ( t ) denote the inverse function of X ( t ) and define F i + ( t ) : = F (cid:0) F i ( t ) (cid:1) , i = , ..., k . With thisnotation, from the transformation we readily get r / D = F k + ( / τ ) , X i ( r / D ) = F k + − i ( / τ ) , i = , ..., k . Hence Y k ( r / D ) = ∏ ki = F i ( / τ ) . Setting τ : = X − k + ( / D ) , we deduce C = inf w ∈ C ∞ ([ τ , ∞ ) × S n − ) w ( τ , θ )= R ∞τ R S n − τ p − (cid:16) ( ∂ τ w ) + (cid:0) ∏ ki = F i ( / τ ) (cid:1) − | ∇ θ w | (cid:17) p / d H n − ( θ ) d τ (cid:16) R ∞τ R S n − τ − − p ∗ / p | w | p ∗ d H n − ( θ ) d τ (cid:17) p / p ∗ . (3.2)Suppose next that n ≥ S : = inf u ∈ C ∞ c ( B R ) \{ } R B R | ∇ u | p d x (cid:16) R B R | u | p ∗ d x (cid:17) p / p ∗ . From [T] we know S = S ( n , p ) >
0. Consider the transformation u ( x ) = z ( t , θ ) , where t : = r n − p , θ : = xr with r : = | x | . An elementary computation gives | ∇ u | = ( ∂ r u ) + r | ∇ θ u | = ( n − p ) t ( n − p + ) / ( n − p ) (cid:16) ( ∂ t z ) + (cid:0) ( n − p ) t (cid:1) − | ∇ θ z | (cid:17) . Therefore, S ( n , p )( n − p ) p ( n − ) / n = inf z ∈ C ∞ ([ R p − n , ∞ ) × S n − ) z ( R p − n , θ )= R ∞ R p − n R S n − t p − (cid:16) ( ∂ t z ) + (cid:0) ( n − p ) t (cid:1) − | ∇ θ z | (cid:17) p / d H n − ( θ ) d t (cid:16) R ∞ R p − n R S n − t − − p ∗ / p | z | p ∗ d H n − ( θ ) d t (cid:17) p / p ∗ . (3.3)To compare the expressions on the right of (3.2) and (3.3), we first choose R such that R p − n = τ . Thenwe observe for τ ≥ τ τ − k ∏ i = F i ( / τ ) ≤ τ − k ∏ i = F i ( / τ ) = τ − Y k ( / D ) ≤ ≤ n − p , the last inequality because of n ≥
3. Thus ∏ ki = F i ( / τ ) ≤ ( n − p ) τ for any τ ≥ τ and inserting this to(3.2) we conclude with C ≥ S ( n , p )( n − p ) p ( n − ) / n . This is (3.1) for n ≥ n = S ′ : = inf u ∈ C ∞ ( B R ( )) \{ } R B R ( ) | x | α p | ∇ u | p d x (cid:16) R B R ( ) | x | α p ⋆ | u | p ⋆ d x (cid:17) p / p ⋆ . n = a = r = p ⋆ there), we know that S ′ = S ′ ( α , p ) > α > − / p . In particular, taking α = − / p and considering the transformation u ( x ) = z ( t , θ ) , where t : = r , θ : = xr with r : = | x | , we deduce by a straightforward calculation S ′ ( − / p , p ) = inf z ∈ C ∞ ([ R − , ∞ ) × S ) z ( R − , θ )= R ∞ R − R S t p − (cid:0) ( ∂ t z ) + t − | ∇ θ z | (cid:1) p / d H ( θ ) d t (cid:16) R ∞ R − R S t − − p ∗ / p | z | p ∗ d H ( θ ) d t (cid:17) p / p ∗ . (3.5)To compare the expressions on the right of (3.2) and (3.5), we choose R such that R − = τ . Then (3.4)says ∏ ki = F i ( / τ ) ≤ τ for any τ ≥ τ and inserting this to (3.2) we conclude with C ≥ S ′ ( − / p , p ) ;that is (3.1) for n = + p ⋆ / p on X k + , k ∈ N , cannot be decreased. The argumentapplies for any 1 < p < n . Suppose for the sake of contradiction, that ε ∈ [ , ) is such that the followinginequality holds for some D ≥ D and k ∈ N I k [ u ] ≥ c (cid:16) Z Ω | u | p ⋆ Y + p ⋆ / pk ( | x | / D ) X ( + p ⋆ / p ) ε k + ( | x | / D ) d x (cid:17) p / p ⋆ ∀ u ∈ C ∞ c ( Ω ) , (3.6)with c being a positive constant independent of u . Applying H ¨older’s inequality with conjugate exponents n / p and p ⋆ / p and using (3.6) we have (in the first displayed line below we write Y k , X k + instead of Y k ( | x | / D ) , X k + ( | x | / D ) ) Z Ω | u | p | x | p Y k X γ k + d x = Z Ω (cid:8) | x | − p Y − p / p ⋆ k X γ − ( + p / p ⋆ ) ε k + (cid:9)(cid:8) | u | p Y + p / p ⋆ k X ( + p / p ⋆ ) ε k + (cid:9) d x ≤ c − (cid:16) Z Ω | x | − n Y k ( | x | / D ) X β ( | x | / D ) d x (cid:17) p / n I k [ u ] , (3.7)where β : = h γ − (cid:16) + pp ⋆ (cid:17) ε i np . The integral on the right is a constant depending on n , p , ε , γ and Ω if and only if β > γ > − (cid:16) + pp ⋆ (cid:17) ( − ε ) . (3.8)Thus for values of γ determined from (3.8), we get from (3.7) that Z Ω | u | p | x | p Y k ( | x | / D ) X γ k + ( | x | / D ) d x ≤ c − C ( n , p , ε , γ , Ω ) I k [ u ] ∀ u ∈ C ∞ c ( Ω ) . However, Proposition 3.1-(i) of [BFT1] (for the case κ = n there) asserts the last inequality is possibleonly if γ ≥
2. This and (3.8) forces ε ≥
1, a contradiction.5 p > We need the following special improvement to the series expansion of the L -Hardy inequality, which isvalid only for radially symmetric functions. Lemma 4.1.
Let ≤ p < n. Then for any D ≥ , any k ∈ N ∪ { } and all radially symmetric functions ζ ∈ H (cid:0) B ( ) (cid:1) we have Z B ( ) | ∇ζ | d x − (cid:16) n − (cid:17) Z Ω | ζ | | x | d x − Z B ( ) | ζ | | x | k ∑ i = Y i ( | x | / D ) d x ≥ C ( n , p ) (cid:16) Z B ( ) | x | p ⋆ ( p − ) / p Y + p ⋆ / pk + ( | x | / D ) | ζ | p ⋆ / p d x (cid:17) p / p ⋆ . Proof.
We perform the change of variables ζ ( r ) = g k , D ( r ) w ( r ) , where g k , D ( r ) : = r − n / Y − / k ( r / D ) , r = | x | . (4.1)Then by Theorem 2.7 for p = Z B ( ) | x | − n Y − k ( | x | / D ) | ∇ w | d x ≥ c ( n , p ) (cid:16) Z B ( ) | x | − n Y k ( | x | / D ) X + p ⋆ / pk + ( | x | / D ) | w | p ⋆ / p d x (cid:17) p / p ⋆ . Since w is radially symmetric, the above inequality is equivalent to Z rY − k ( r / D ) (cid:0) w ′ ( r ) (cid:1) d r ≥ c ( n , p ) (cid:16) Z r − Y k ( r / D ) X + p ⋆ / pk + ( r / D ) | w ( r ) | p ⋆ / p d r (cid:17) p / p ⋆ . The proof of this readily follows from [FT, Lemma 7.1] for q = p ⋆ / p there, or by [Mz, Theorem 3 - pg47] for d ν = rY − k ( r / D ) χ ( , ) d r and d µ = r − Y k ( r / D ) X + p ⋆ / pk + ( r / D ) χ ( , ) d r there. Proof of Theorem A for p > . As in the case p ≤
2, we can assume Ω = B ( ) . Applying thetransformation u = f k , D v we get by using both (2.5) and (2.8) that I k [ u ] ≥ c p (cid:16) Z Ω f pk , D ( x ) | ∇ v | p d x + Z Ω g k , D ( x ) | v | p − | ∇ v | d x (cid:17) , (4.2)where g k , D is given by (4.1). Note that g k , D is f k , D with p = x = ( r , θ ) ( r = | x | and θ = x / | x | ) to decompose v ( x ) into spherical harmonics. For this purpose, let { h l } l ∈ N ∪{ } be the orthonormal basis of L ( S n − ) that iscomprised of eigenfunctions of the Laplace-Beltrami operator − ∆ S n − (the angular part of the Laplacianwhen expressed in spherical coordinates). This has corresponding eigenvalues λ l = l ( l + n − ) , l ∈ N ∪ { } (see [Schn, Appendix]). Thus − ∆ S n − h l = λ l h l on S n − , and 1 n ω n Z S n − h l ( θ ) h m ( θ ) d H n − ( θ ) = δ lm for all l , m ∈ N ∪ { } . With these definitions we have the decomposition of v ∈ C ∞ c ( B ( )) in its spherical harmonics v ( x ) = ∞ ∑ l = v l ( r ) h l ( θ ) . h ( θ ) = v on ∂ B r ( ) , that is v ( r ) = n ω n r n − Z ∂ B r ( ) v ( x ) d H n − ( x ) = n ω n Z S n − v ( r θ ) d H n − ( θ ) . We now estimate the first term on the right hand side of (4.2). We have Z Ω f pk , D ( x ) | ∇ v | p d x = Z f pk , D ( r ) r n − Z S n − (cid:16) ( ∂ r v ) + r | ∇ θ v | (cid:17) p / d H n − ( θ ) d r ≥ Z f pk , D ( r ) r n − Z S n − | ∂ r v | p d H n − ( θ ) d r + Z f pk , D ( r ) r n − Z S n − r p | ∇ θ v | p d H n − ( θ ) d r | {z } = : J , (4.3)by the fact that ( κ + λ ) q ≥ κ q + λ q , for all κ , λ ≥ q ≥
1. To estimate the first term on the rightof (4.3) we use (2.15) to get Z S n − | ∂ r v | p d H n − ( θ ) ≥ Z S n − | ∂ r v | p d H n − ( θ ) + c p Z S n − | ∂ r ( v − v ) | p d H n − ( θ )+ p Z S n − | ∂ r v | p − ( ∂ r v ) ∂ r ( v − v ) d H n − ( θ ) . (4.4)But since { v l } l ∈ N ∪{ } are radial Z S n − | ∂ r v | p − ( ∂ r v ) ∂ r ( v − v ) d H n − ( θ ) = | v ′ ( r ) | p − v ′ ( r ) Z S n − ∂ r ( v − v ) d H n − ( θ )= | v ′ ( r ) | p − v ′ ( r ) ∞ ∑ l = v ′ l ( r ) Z S n − f l ( θ ) d H n − ( θ ) = , and so Z S n − | ∂ r v | p d H n − ( θ ) ≥ c p Z S n − | ∂ r ( v − v ) | p d H n − ( θ ) , where we have cancel also the first term on the right hand side of (4.4). Plugging this to (4.3) we deduce Z Ω f pk , D ( x ) | ∇ v | p d x ≥ c p Z f pk , D ( r ) r n − Z S n − (cid:16) | ∂ r ( v − v ) | p + r p | ∇ θ v | p (cid:17) d H n − ( θ ) d r + ( − c p ) J ≥ − p / c p Z f pk , D ( r ) r n − Z S n − (cid:16) | ∂ r ( v − v ) | + r | ∇ θ v | (cid:17) p / d H n − ( θ ) d r + ( − c p ) J = − p / c p Z Ω f pk , D ( x ) | ∇ ( v − v ) | p d x + ( − c p ) J , (4.5)by the fact that κ q + λ q ≥ − q ( κ + λ ) q , for all κ , λ ≥ q ≥
1. To estimate J observe first that Z S n − ( v − v ) d H n − ( θ ) = ∞ ∑ l = v l ( r ) Z S n − f l ( θ ) d H n − ( θ ) = , v is radial, we may use the Poincar´e inequality on S n − (see forexample [H, Theorem 2.10]) Z S n − | ∇ θ v | p d H n − ( θ ) = Z S n − | ∇ θ ( v − v ) | p d H n − ( θ ) ≥ C P ( n , p ) Z S n − | v − v | p d H n − ( θ ) . Inserting this in the definition of J , we get from (4.5) the existence of a positive constant C = C ( n , p ) such that Z Ω f pk , D ( x ) | ∇ v | p d x ≥ C ( n , p ) (cid:16) Z Ω f pk , D ( x ) | ∇ ( v − v ) | p d x + Z Ω f pk , D ( x ) | v − v | p | x | p d x (cid:17) ≥ C ( n , p ) (cid:16) Z Ω | x | − n Y k ( | x | / D ) X + p ⋆ / pk + ( | x | / D ) | v − v | p ⋆ d x (cid:17) p / p ⋆ , (4.6)where in the last inequality we have used the Sobolev inequality and the fact that X i ≤ i ∈ N .Next we estimate the second term on the right hand side of (4.2). Setting w = | v | p / we have Z Ω g k , D ( x ) | v | p − | ∇ v | d x = Z Ω g k , D ( x ) | ∇ w | d x . (4.7)Now we assert that the function ζ = g k , D w belongs to H ( Ω ) . Indeed ζ = g k , D w = g k , D | v | p / = g k , D f − p / k , D | u | p / = | x | − p / | u | p / ∈ H ( Ω ) , since 2 ≤ p < n . Thus by Remark 2.12 we have that the following equality is valid Z Ω g k , D ( x ) | ∇ w | d x = Z Ω | ∇ζ | dx − (cid:16) n − (cid:17) Z Ω | ζ | | x | d x − Z Ω | ζ | | x | k ∑ i = Y i ( | x | / D ) d x . (4.8)Taking the decomposition of ζ in its spherical harmonics we know that (see [FT, eq. (7.6)]) Z Ω | ∇ζ | d x − (cid:16) n − (cid:17) Z Ω | ζ | | x | d x − Z Ω | ζ | | x | k ∑ i = Y i ( | x | / D ) d x ≥ Z Ω | ∇ζ | d x − (cid:16) n − (cid:17) Z Ω | ζ | | x | d x − Z Ω | ζ | | x | k ∑ i = Y i ( | x | / D ) d x ≥ C ( n , p ) (cid:16) Z Ω | x | p ⋆ ( p − ) / p Y + p ⋆ / pk + ( | x | / D ) | ζ | p ⋆ / p d x (cid:17) p / p ⋆ , (4.9)where in the last inequality we have used Lemma 4.1 since ζ is radial. In particular, we have that ζ ( r ) = n ω n r n − Z ∂ B r ( ) ζ ( x ) d H n − ( x ) , ζ is nonnegative since ζ is nonnegative) ζ ( r ) = g k , D ( r ) n ω n r n − Z ∂ B r ( ) w ( x ) d H n − ( x )= g k , D ( r ) n ω n r n − Z ∂ B r ( ) | v ( x ) | p / d H n − ( x ) ≥ g k , D ( r ) (cid:12)(cid:12)(cid:12) n ω n r n − Z ∂ B r ( ) v ( x ) d H n − ( x ) (cid:12)(cid:12)(cid:12) p / = g k , D | v ( r ) | p / . This applied to (4.9) gives together with (4.8) and (4.7) that Z Ω g k , D ( x ) | v | p − | ∇ v | d x ≥ c ( n , p ) (cid:16) Z Ω | x | − n Y k ( | x | / D ) X + p ⋆ / pk + ( | x | / D ) | v | p ⋆ d x (cid:17) p / p ∗ . (4.10)Inserting (4.10) and (4.6) in (4.2), inequality (1.8) follows. Moreover, it is proved in the previous sectionthat the exponent 1 + p ⋆ / p on X k + , k ∈ N , cannot be decreased. The local estimate of Theorem 5.2 below is the key estimate in order to establish the series improve-ment to the Hardy-Morrey inequality that appears in Theorem B. To establish it we need the followingweighted Hardy inequality with trace term.
Lemma 5.1.
Let γ ∈ R \{ } and U be a bounded domain in R n , n ≥ , having locally Lipschitz boundary.Denote by ν ( x ) the exterior unit normal vector defined at almost every x ∈ ∂ U . Then for all D ≥ R U : = sup x ∈ U | x | , q ≥ , k ∈ N , all s = n and any v ∈ C ∞ c ( R n \ { } ) , there holds (cid:12)(cid:12)(cid:12) qn − s (cid:12)(cid:12)(cid:12) q Z U | ∇ v ( x ) | q | x | s − q Y γ k ( | x | / D ) d x + qn − s Z ∂ U | v ( x ) | q | x | s Y γ k ( | x | / D ) x · ν ( x ) d H n − ( x ) ≥ Z U | v ( x ) | q | x | s Y γ k ( | x | / D ) h + γ qn − s Z k ( | x | / D ) i d x . (5.1) Proof.
Integration by parts together with Lemma 2.2 give − Z U | v | Y γ k ( | x | / D ) div n x | x | s o d x = Z U ∇ | v | · x | x | s Y γ k ( | x | / D ) d x + γ Z U | v || x | s Y γ k ( | x | / D ) Z k ( | x | / D ) d x − Z ∂ U | v | Y γ k ( | x | / D ) x | x | s · ν d H n − ( x ) , and since div {| x | − s x } = ( n − s ) | x | − s , x =
0, we get Z U | ∇ v || x | s − Y γ k ( | x | / D ) d x − Z ∂ U | v || x | s Y γ k ( | x | / D ) x · ν d H n − ( x ) ≥ Z U | v || x | s Y γ k ( | x | / D ) (cid:2) s − n − γ Z k ( | x | / D ) (cid:3) d x , if s > n , or Z U | ∇ v || x | s − Y γ k ( | x | / D ) d x + Z ∂ U | v || x | s Y γ k ( | x | / D ) x · ν d H n − ( x ) ≥ Z U | v || x | s Y γ k ( | x | / D ) (cid:2) n − s + γ Z k ( | x | / D ) (cid:3) d x , s < n , where we have also used the fact that | ∇ | v ( x ) || ≤ | ∇ v ( x ) | for a.e. x ∈ U . We may write bothinequalities in one as follows1 | n − s | Z U | ∇ v || x | s − Y γ k ( | x | / D ) d x + n − s Z ∂ U | v || x | s Y γ k ( | x | / D ) x · ν d H n − ( x ) ≥ Z U | v || x | s Y γ k ( | x | / D ) h + γ n − s Z k ( | x | / D ) i d x . This is inequality (5.1) for q =
1. Substituting v by | v | q with q >
1, we arrive at q | n − s | Z U | ∇ v || v | q − | x | s − Y γ k ( | x | / D ) d x + n − s Z ∂ U | v | q | x | s Y γ k ( | x | / D ) x · ν d H n − ( x ) ≥ Z U | v | q | x | s Y γ k ( | x | / D ) h + γ n − s Z k ( | x | / D ) i d x . (5.2)The first term on the left of (5.2) can be written as follows q | n − s | Z U | ∇ v || v | q − | x | s − Y γ k ( | x | / D ) d x = Z U n q | n − s | | ∇ v || x | s / q − on | v | q − | x | s − s / q o Y γ k ( | x | / D ) d x ≤ q (cid:12)(cid:12)(cid:12) qn − s (cid:12)(cid:12)(cid:12) q Z U | ∇ v | q | x | s − q Y γ k ( | x | / D ) d x + q − q Z U | v | q | x | s Y γ k ( | x | / D ) d x , by Young’s inequality. Thus (5.2) becomes1 q (cid:12)(cid:12)(cid:12) qn − s (cid:12)(cid:12)(cid:12) q Z U | ∇ v | q | x | s − q Y γ k ( | x | / D ) d x + n − s Z ∂ U | v | q | x | s Y γ k ( | x | / D ) x · ν d H n − ( x ) ≥ Z U | v | q | x | s Y γ k ( | x | / D ) h q + γ n − s Z k ( | x | / D ) i d x . Multiplying by q we get (5.1). Theorem 5.2.
Let ≤ p = n and ≤ q < p. There exist constants B = B ( n , p , q ) ≥ and C = C ( n , p , q ) > such that for all u ∈ C ∞ c ( Ω \ { } ) , any ball B r of radius r ∈ (cid:0) , diam ( Ω ) (cid:1) that contains the origin, anyD ≥ B diam ( Ω ) and all k ∈ N Z B r | u | q | x | q h − q n ( p − q ) Z k ( | x | / D ) i d x ≤ Cr n ( − q / p ) Y − q / pk + ( r / D ) (cid:0) I k [ u ; D ] (cid:1) q / p . (5.3) Proof.
Given u ∈ C ∞ c ( Ω \ { } ) we define as usual v ∈ C ∞ c ( Ω \ { } ) through the transform u ( x ) = f k , D ( x ) v ( x ) , where D ≥ diam ( Ω ) and 1 < p = n . Then with q ∈ [ , p ) and r ∈ ( , diam ( Ω )) , we have for any ballcontaining the origin that Z B r | u | q | x | q h − q n ( p − q ) Z k ( | x | / D ) i d x = Z B r | v | q | x | nq / p Y − q / pk ( | x | / D ) h − q n ( p − q ) Z k ( | x | / D ) i d x ≤ (cid:16) pqn ( p − q ) (cid:17) q Z B r | x | q ( p − n ) / p | ∇ v | q Y − q / pk ( | x | / D ) d x | {z } = : M r + pqn ( p − q ) Z ∂ B r | v | q | x | nq / p Y − q / pk ( | x | / D ) x · ν d H n − ( x ) | {z } = : P r , (5.4)0where we have used Lemma 5.1 for U = B r , s = nq / p and γ = − q / p . By H ¨older’s inequality M r ≤ (cid:0) ω n r n (cid:1) − q / p (cid:16) Z B r | x | p − n | ∇ v | p Y − k ( | x | / D ) d x (cid:17) q / p (by (2.5)) ≤ C ( n , p , q ) r n ( − q / p ) (cid:0) I k [ u ; D ] (cid:1) q / p ≤ C ( n , p , q ) r n ( − q / p ) Y − q / pk + ( r / D ) (cid:0) I k [ u ; D ] (cid:1) q / p , (5.5)for any D ≥ b ′ diam ( Ω ) , b ′ = b ′ ( n , p ) ≥
1. The last inequality is true since 0 < Y k + ( t ) ≤ t ∈ ( , ] . For P r , noting that x · ν ≥ x ∈ ∂ B r ( B r is star-shaped with respect to any of it’s points;thus 0 particular), we may also apply H ¨older’s inequality as follows P r = Z ∂ B r (cid:8) Y − q / pk + ( | x | / D ) (cid:9)n | v | q | x | nq / p X q / pk + ( | x | / D ) o x · ν d H n − ( x ) ≤ (cid:16) Z ∂ B r Y − q / ( p − q ) k + ( | x | / D ) x · ν d H n − ( x ) | {z } = : S r (cid:17) − q / p (cid:16) Z ∂ B r | v | p | x | n X k + ( | x | / D ) x · ν d H n − ( x ) | {z } = : T r (cid:17) q / p . (5.6)By the divergence theorem we have S r = Z B r div n Y − q / ( p − q ) k + ( | x | / D ) x o d x = n Z B r Y − q / ( p − q ) k + ( | x | / D ) d x − qp − q Z B r Y − q / ( p − q ) k + ( | x | / D ) Z k ( | x | / D ) d x ≤ n Z B r ( ) Y − q / ( p − q ) k + ( | x | / D ) d x , since this integral increases if we change the domain of integration from B r to B r ( ) . Thus S r ≤ n ω n Z r t n − Y − q / ( p − q ) k + ( t / D ) d t ≤ C ( n ) r n Y − q / ( p − q ) k + ( r / D ) , (5.7)for any D ≥ η diam ( Ω ) , η ≥ n , p , q , by a direct application of Lemma 2.4 for α = n and β = q / ( p − q ) . To estimate T r we also employ the divergence theorem to get T r = Z B r div (cid:8) | x | − n X k + ( | x | / D ) x (cid:9) | v | p d x + Z B r | x | − n X k + ( | x | / D ) x · ∇ ( | v | p ) d x . A direct computation using Lemma 2.2 shows thatdiv (cid:8) | x | − n X k + ( | x | / D ) x (cid:9) = | x | − n Y k ( | x | / D ) X k + ( | x | / D ) , x ∈ Ω \ { } . Returning then to the original function u in the first integral and taking the absolute value in the second,we arrive at T r ≤ Z Ω | u | p | x | p Y k + ( | x | / D ) d x + p Z Ω | v | p − | x | n − | ∇ v | X k + ( | x | / D ) d x | {z } = : J . (5.8)1Now we apply the Cauchy-Schwarz inequality in the second integral above as follows J = Z Ω n | v | p / − | x | n / − | ∇ v | Y − / k ( | x | / D ) on | v | p / | x | n / Y / k ( | x | / D ) X k + ( | x | / D ) o d x ≤ (cid:16) Z Ω | v | p − | x | n − | ∇ v | Y − k ( | x | / D ) d x (cid:17) / (cid:16) Z Ω | v | p | x | n Y k ( | x | / D ) X k + ( | x | / D ) d x (cid:17) / ≤ c ( p ) (cid:0) I k [ u ; D ] (cid:1) / (cid:16) Z Ω | u | p | x | p Y k + ( | x | / D ) d x (cid:17) / , for all D ≥ b ′′′ D , where b ′′′ ≥ n , p . Here we have used (2.8) to estimate the first factorand returned to the original function u in the second factor. Estimate (5.8) is now T r ≤ Z Ω | u | p | x | p Y k + ( | x | / D ) d x + c ( p ) (cid:0) I k [ u ; D ] (cid:1) / (cid:16) Z Ω | u | p | x | p Y k + ( | x | / D ) d x (cid:17) / . According to Theorem 2.6, there exist constants b = b ( n , p ) ≥ c ( n , p ) >
0, both depending only on n , p , such that for any D ≥ bD , the common integral appearing on the right hand side is bounded aboveby c ( n , p ) I k [ u ; D ] . It follows that T r ≤ C ( n , p ) I k [ u ; D ] , (5.9)for any D ≥ max { b , b ′′′ } D . Setting b ′′ = max { b , b ′′′ , η } and since 0 ∈ Ω implies D ≤ diam ( Ω ) , wemay insert (5.9) into (5.6) and taking into account (5.7) we end up with P r ≤ C ( n , p , q ) r n ( − q / p ) Y − q / pk + ( r / D ) (cid:0) I k [ u ; D ] (cid:1) / p , for any D ≥ b ′′ diam ( Ω ) . The last inequality together with (5.5), when applied to estimate (5.4) gives(5.3) for any D ≥ B diam ( Ω ) with B = max { b ′ , b ′′ } . We start with (1.9) when one point in the H ¨older semi-norm taken to be the origin.
Proposition 6.1.
Let p > n. There exist constants ˜ B = ˜ B ( n , p ) ≥ and C = C ( n , p ) > such that for anyk ∈ N ∪ { } , all D ≥ ˜ B diam ( Ω ) and all u ∈ C ∞ c ( Ω \ { } ) sup x ∈ Ω n | u ( x ) || x | − n / p Y / pk + ( | x | / D ) o ≤ C (cid:0) I k [ u ; D ] (cid:1) / p . (6.1) Proof.
Let B r be a ball of radius r ∈ ( , diam ( Ω )) and set u B r : = | B r | Z B r u ( z ) d z . By the local version of Sobolev’s integral representation formula (see [GTr]-Lemma 7.16), we have | u ( x ) − u B r | ≤ n n ω n Z B r | ∇ u ( z ) || x − z | n − d z , x ∈ B r . u ( z ) = f k , D ( z ) v ( z ) , we get n ω n n | u ( x ) − u B r | ≤ Z B r | z | − n / p Y − / pk ( | z | / D ) | ∇ v ( z ) || x − z | n − d z + Z B r Y − / pk ( | z | / D ) | A , k ( | z | / D ) || v ( z ) || z | n / p | x − z | n − d z = : K r ( x ) + L r ( x ) , (6.2)with A , k given by (2.11) with a =
0. By H ¨older’s inequality K r ( x ) ≤ (cid:16) Z B r | x − z | ( n − ) p / ( p − ) d z (cid:17) − / p (cid:16) Z B r | z | p − n Y − k ( | z | / D ) | ∇ v | p d z (cid:17) / p ≤ (cid:16) Z B r ( x ) | x − z | ( n − ) p / ( p − ) d z (cid:17) − / p (cid:16) Z Ω | z | p − n Y − k ( | z | / D ) | ∇ v | p d z (cid:17) / p . Using now (2.5) we obtain the following estimate on K r K r ( x ) ≤ C ( n , p ) r − n / p (cid:0) I k [ u ; D ] (cid:1) / p ≤ C ( n , p ) r − n / p Y − / pk + ( r / D ) (cid:0) I k [ u ; D ] (cid:1) / p , x ∈ B r , (6.3)for any D ≥ diam ( Ω ) , where the last inequality is a consequence of 0 < Y k + ( t ) ≤ t ∈ ( , ] .Next we fix 0 < ε < ( p − n ) / n and estimate L r ( x ) . By H ¨older’s inequality L r ( x ) ≤ (cid:16) Z B r | A , k ( | z | / D ) || x − z | ( n − ) p / ( p − − ε ) d z | {z } = : M r , D ( x ) (cid:17) − ( + ε ) / p (cid:16) Z B r | v | p / ( + ε ) | z | n / ( + ε ) | A , k ( | z | / D ) | d z (cid:17) ( + ε ) / p , x ∈ B r . (6.4)Assumption ε < ( p − n ) / n guarantees M r , D ( x ) < ∞ for all x ∈ B r . More precisely, recalling first Remark2.8, we may restrict D so that D ≥ b ′′′ D with some b ′′′ = b ′′′ ( n , p ) ≥ A , k ( | z | / D ) ≥ x ∈ B r . Then we have M r , D ( x ) ≤ p − np Z B r ( x ) | x − z | ( n − ) p / ( p − − ε ) d z = C ( n , p ) r ( p − n − n ε ) / ( p − − ε ) , x ∈ B r . Returning to the original function u on the right of (6.4), we obtain for all D ≥ b ′′′ D that L r ( x ) ≤ C ( n , p ) r − n / p − n ε / p (cid:16) Z B r | u | p / ( + ε ) | z | p / ( + ε ) A , k ( | z | / D ) d z (cid:17) ( + ε ) / p , x ∈ B r . (6.5)At this point we use Theorem 5.2 with q = p / ( + ε ) ; that is Z B r | u | p / ( + ε ) | x | p / ( + ε ) h − pn ε ( + ε ) Z k ( | z | / D ) i d x ≤ C ( n , p ) r n ε / ( + ε ) Y − / ( + ε ) k + ( r / D ) (cid:0) I k [ u ; D ] (cid:1) / ( + ε ) . (6.6)To couple this with (6.5) we need a positive constant λ = λ ( n , p ) such that A , k ( | z | / D ) ≤ λ h − pn ε ( + ε ) Z k ( | z | / D ) i , ∀ z ∈ Ω . λ such that λ > ( p − n ) / p , keeping in mind that ε < ( p − n ) / n and recalling the definitionof A , k , this is the same as Z k ( | z | / D ) ≤ λ − p − np λ pn ε ( + ε ) − p , ∀ z ∈ Ω . (6.7)which is satisfied after a possible further restriction on D . More precisely, note once more that becauseof Remark 2.3 we can achieve (6.7) for sufficiently large ¯ b = ¯ b ( n , p ) and all D ≥ ¯ bD . Plugging (6.6) to(6.5) we obtain L r ( x ) ≤ C ( n , p ) r − n / p − n ε / p (cid:16) Z B r | u | p / ( + ε ) | z | p / ( + ε ) h − pn ε ( + ε ) Z k ( | z | / D ) i d z (cid:17) ( + ε ) / p , x ∈ B r , for all D ≥ max { b ′′′ , ¯ b } D . Using Theorem 5.2 with q = p / ( + ε ) , L r ( x ) ≤ C ( n , p ) r − n / p − n ε / p (cid:16) r n ε / ( + ε ) Y − / ( + ε ) k + ( r / D ) (cid:0) I k [ u ; D ] (cid:1) / ( + ε ) (cid:17) ( + ε ) / p = C ( n , p ) r − n / p Y − / pk + ( r / D ) (cid:0) I k [ u ; D ] (cid:1) // p , (6.8)for any D ≥ ˜ B diam ( Ω ) , where ˜ B depends only on n , p .Applying estimates (6.8) and (6.3) to estimate (6.2), we conclude | u ( x ) − u B r | ≤ C ( n , p ) r − n / p Y − / pk + ( r / D ) (cid:0) I k [ u ; D ] (cid:1) / p , (6.9)for all x ∈ B r and any D ≥ ˜ B diam ( Ω ) . Since 0 ∈ B r , it follows from (6.9) that | u B r | ≤ C ( n , p ) r − n / p Y − / pk + ( r / D ) (cid:0) I k [ u ; D ] (cid:1) / p . Hence | u ( x ) | ≤ | u ( x ) − u B r | + | u B r |≤ C ( n , p ) r − n / p Y − / pk + ( r / D ) (cid:0) I k [ u ; D ] (cid:1) / p , for all x ∈ B r and any D ≥ ˜ B diam ( Ω ) . Now if x ∈ Ω we may consider a ball B r of radius r = | x | / x and the previous inequality yields (6.1). Proof of Theorem B.
Let x , y ∈ Ω , x = y , and consider a ball B r of radius r : = | x − y | that contains x , y .Then r ∈ ( , diam ( Ω )) and we have | u ( x ) − u ( y ) | ≤ | u ( x ) − u B r | + | u ( y ) − u B r |≤ n n ω n n Z B r | ∇ u ( z ) || x − z | n − d z | {z } = : J ( x ) + Z B r | ∇ u ( z ) || y − z | n − d z | {z } = : J ( y ) o , (6.10)where we have used Sobolev’s integral representation formula (see [GTr]-Lemma 7.16) twice. In whatfollows we estimate J ( x ) independently of x . Applying the transform u ( z ) = f k , D ( z ) v ( z ) , we get J ( x ) ≤ Z B r | z | − n / p Y − / pk ( | z | / D ) | ∇ v ( z ) || x − z | n − d z | {z } = :K r ( x ) + Z B r Y − / pk ( | z | / D ) | A , k ( | z | / D ) || v ( z ) || z | n / p | x − z | n − d z | {z } = : Λ r ( x ) . (6.11)4Working as we did to get (6.3) we obtainK r ( x ) ≤ C ( n , p ) r − n / p Y − / pk + ( r / D ) (cid:0) I k [ u ; D ] (cid:1) / p , (6.12)for any D ≥ diam ( Ω ) . Next we rewrite Λ r ( x ) with the original function u to get Λ r ( x ) = Z B r | A , k ( | z | / D ) || u ( z ) || z || x − z | n − d z . We insert (6.1) in Λ r ( x ) to deduce Λ r ( x ) ≤ C ( n , p ) (cid:0) I k [ u ; D ] (cid:1) / p Z B r Y − / pk + ( | z | / D ) | A , k ( | z | / D ) || z | n / p | x − z | n − d z , for any D ≥ ˜ B diam ( Ω ) . Recalling once more Remark 2.8, we can further restrict D so that D ≥ max { b ′′′ , ˜ B } diam ( Ω ) and then Λ r ( x ) ≤ C ( n , p ) (cid:0) I k [ u ; D ] (cid:1) / p Z B r Y − / pk + ( | z | / D ) | z | n / p | x − z | n − d z , Letting n < Q < p we may use H ¨older’s inequality to obtain Λ r ( x ) ≤ C ( n , p ) (cid:0) I k [ u ; D ] (cid:1) / p (cid:16) Z B r Y − Q / pk + ( | z | / D ) | z | nQ / p d z (cid:17) / Q (cid:16) Z B r | x − z | ( n − ) Q ′ d z (cid:17) / Q ′ ≤ C ( n , p ) (cid:0) I k [ u ; D ] (cid:1) / p (cid:16) Z B r ( ) Y − Q / pk + ( | z | / D ) | z | nQ / p d z (cid:17) / Q (cid:16) Z B r ( x ) | x − z | ( n − ) Q ′ d z (cid:17) / Q ′ . Both integrals above are finite since n < Q < p implies nQ / p < n and ( n − ) Q ′ < n . By a simplecomputation Λ r ( x ) ≤ C ( n , p ) (cid:0) I k [ u ; D ] (cid:1) / p (cid:16) Z r t n − − nQ / p Y − Q / pk + ( t / D ) d t (cid:17) / Q r n / Q ′ − n + , (6.13)for any D ≥ max { b ′′′ , ˜ B } diam ( Ω ) . Lemma 2.4 for α = n − − nQ / p and β = Q / p ensures the existenceof constants η ≥ c > n , p , Q (and thus only on n , p ) , such that Z r t n − − nQ / p Y − Q / pk + ( t / D ) d t ≤ cr n − nQ / p Y − Q / pk + ( r / D ) , for any D ≥ e η diam ( Ω ) . Thus (6.13) becomes Λ r ( x ) ≤ C ( n , p ) r − n / p Y − / pk + ( r / D ) (cid:0) I k [ u ; D ] (cid:1) / p , (6.14)for any D ≥ B diam ( Ω ) , where B : = max (cid:8) max { b ′′′ , ˜ B } , η (cid:9) . Altogether, (6.12) and (6.14) when insertedin (6.11) give J ( x ) ≤ C ( n , p ) r − n / p Y − / pk + ( r / D ) (cid:0) I k [ u ; D ] (cid:1) / p , for any D ≥ B diam ( Ω ) . The proof of (1.9) follows since the same estimate holds true for J ( y ) .5To show the exponent 1 / p on X k + cannot be decreased, assume in contrary there exists ε ∈ ( , ] such that for all u ∈ C ∞ c ( Ω \ { } ) we have (cid:0) I k [ u ; D ] (cid:1) / p ≥ c sup x , y ∈ Ω x = y n | u ( x ) − u ( y ) || x − y | − n / p Y ( − ε ) / pk + (cid:16) | x − y | D (cid:17)o , for some constants c > D ≥ diam ( Ω ) . Choosing y = (cid:0) I k [ u ; D ] (cid:1) / p ≥ c | u ( x ) || x | − n / p Y ( − ε ) / pk + ( | x | / D ) , ∀ x ∈ Ω \ { } . This readily implies that Z Ω | u ( x ) | p | x | p Y − ε / k + ( | x | / D ) d x ≤ c − p I k [ u ; D ] Z Ω | x | − n X + ε / ( | x | / D ) d x . (6.15)Clearly, since ε > n , p , ε and Ω . Thuswe have violated the optimality of the exponent 2 of the remainder term (1.4). Acknowledgements
The first author was partially supported by Fondecyt grant 3140567 and by Mil-lenium Nucleus CAPDE NC130017.
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ONSTANTINOS
T. G
KIKAS
Centro de Modelamiento Matem´atico (UMI 2807 CNRS)Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile; [email protected] G EORGIOS P SARADAKIS