HARDY AND RELLICH INEQUALITIES WITH REMAINDERS

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1 HARDY AND RELLICH INEQUALITIES WITH REMAINDERS W. D. EVANS AND ROGER T. LEWIS Astract. I this paper our primary cocer is with the estalishmet of weighted Hardy iequalities i L p () ad Rellich iequalities i L () depedig upo the distace to the oudary of domais R with a fiite diameter D(). Improved costats are preseted i most cases.. Itroductio Recetly, cosiderale attetio has ee give to extesios of the multi-dimesioal Hardy iequality of the form u(x) u(x) dx µ() dx + λ() u(x) dx, u H δ(x) (), (.) where is a ope coected suset of R ad δ(x) := dist(x, ). It is kow that for µ() = there are smooth domais for which 4 λ(), ad for λ() =, there are smooth domais for which µ() < - see M. Marcus, V.J. Mizel, ad Y. Pichover [8] ad 4 T. Matskewich ad P.E. Soolevskii [9]. I [], H. Brezis ad M. Marcus showed that for domais of class C iequality (.) holds for µ() = 4 ad some λ() (, ) ad whe is covex λ() 4D() (.) i which D() is the diameter of. M. Hoffma-Ostehof, T. Hoffma-Ostehof, ad A. Laptev [6] aswered a questio posed y H. Brezis ad M. Markus i [] y estalishig the improvemet to (.) that (.) holds for a covex domai Date: March 8, Mathematics Suject Classificatio. Primary 47A63; Secodary 46E35, 6D. Key words ad phrases. Rellich iequality, Hardy iequality, Remaider terms.

2 W. D. EVANS AND R. T. LEWIS, with µ() = 4, K() λ(), ad K() := 4 [ s ] / (.3) i which s := S ad is the volume of. For a covex domai ad µ() = /4, a lower oud for λ() i (.) i terms of was also otaied y S. Filippas, V. Maz ya, ad A. Tertikas i [5] as a special case of results o L p Hardy iequalities. They prove that λ() 3D it (), where D it () = sup x δ(x), the iteral diameter of. Sice 3D it () 3 4 K()/ /, their result is a improvemet of (.3) for =, 3, ut the estimates do t compare for > 3. I this paper we show that (.) holds for (.3) replaced y µ() = 4 ad λ() 3K() as well as provig weighted versios of the Hardy iequality i L p () for p >. I the case p =, the followig are special cases of our results. If is covex ad σ (, ], the u(x) σ ( σ) dx 4D() σ B(, σ) δ(x) σ +3 ( s ) σ u(x) dx (.4) for p+ Γ( B(, p) := ) Γ( ) π Γ( +p ). (.5) If σ [, ] ad is covex, the δ(x) σ u(x) ( σ) dx B(, σ) δ(x) σ u(x) dx 4 + C H(, σ) δ(x) σ u(x) dx. ( σ) for C H (, σ) give i (3.4). Similar results for weighted forms of the Hardy iequality i L p () are give i sectio 4. Fially, we show that our oe-dimesioal iequalities i lead to improved costats for the Rellich iequality otaied y G. Baratis i [] for 4.. Oe-dimesioal iequalities As is the case i [6], our proofs are ased o oe-dimesioal Hardytype iequalities coupled with the use of the mea-distace fuctio itroduced y Davies to exted to higher dimesios; see [4]. The asic oe-dimesioal iequality is as follows:

3 HARDY AND RELLICH INEQUALITIES 3 Lemma. Let u C (, ), (t) := mit, t ad let f C [, ] e mootoic o [, ]. The for p > f ((t)) u(t) p dt p p f((t)) f() p f ((t)) p u (t) p dt. (.) Proof. First let u := vχ (,], the restrictio to (, ] of some v C (, ). For ay costat c [f(t) c] u(t) p dt = [f(t) c] u(t) p By choosig c = f(), we have that + [f(t) c] p [ u(t) ] p [ u(t) ] dt. f (t) u(t) p dt = p [f(t) f()] u(t) p Re[u(t)u (t)]dt. (.) Similarly, for u = vχ [,), v C (, ), we have f ( s) u(s) p ds = p [f( s) f()] u(s) p Re[u(s)u (s)]ds. Therefore, sice f is mootoic, for ay u C(, ) f ((t)) u(t) p dt = p f((t)) f() u(t) p Re[u(t)u (t)]dt p f ((t)) p p u(t) p f((t)) f() u (t) dt [ p f ((t)) u(t) p dt ] p p f ((t)) p p [ f((t)) f() p f ((t)) p u (t) p dt o applyig Hölder s iequality. Iequality (.) ow follows. The ext lemma provides the oe-dimesioal result eeded to improve (.3), which was proved i [6]. Lemma. Let σ ad defie µ(t) := (t). For all u C(, ) ( σ ) ( (t) ) σ ] u(t) (t) σ u (t) dt (t) [+k(σ) σ dt, µ(t) (.3) for [ k(σ) := ] σ σ, σ <,, σ [, ]. Proof. O settig f(t) = t σ i (.) we get σ p (t) σ u(t) p dt p p ] p (t) p+σ [(t) ] σ p u (t) p dt. (.4)

4 4 W. D. EVANS AND R. T. LEWIS With u C(, ), let p = ad sustitute v(t) = [ ( (t) ) σ]u(t) i (.4). We claim that this gives ( σ ) ( (t) ) σ ] v(t) σ (t) v (t) dt (t) [ σ dt (.5) for ay real umer σ. The sustitutio gives (t) σ/ v (t) = ( σ) σ (t) σ/ (t)u(t)+(t) σ/[ ( (t) ) σ ] u (t). Cosequetly, (t) σ v (t) = ( σ) σ (t) σ u(t) + (t) σ[ ( ) (t) σ ] u (t) ( σ) σ (t) [ ( (t) ) σ ] [ u ] which implies that (t)σ v (t) dt = (t)σ[ ( ) (t) σ ] u (t) dt + ( σ) σ (t) σ u(t) dt +( σ) [ σ d (t) [ ( (t) ) σ ] ] u dt dt = (t)σ[ ( (t) ) σ ] u (t) dt (.6) sice (t) = i (, ) ad i (, ). Therefore, (.5) follows from (.4). Sice = µ(t) + (t) for [ ( (t) k σ (x) := ) σ ] = [ + (t) σ = σ (t) σ ] [ + σ( (t) ) ] σkσ (.7) ( (t) ) µ(t) µ(t), x [, ), σ. ( + x) σ (x) σ For σ <, k σ (x) > i (, ), k σ () = ad k σ (x) as x. By examiig the derivative of k σ (x) we see that k σ(x) = ( σ)(( + x) σ σ x σ ) [( + x) σ (x) σ ] lim k σ(x) = x + For σ <, k σ (x) is miimized at Calculatios show that ( σ), σ <,,, σ =, < σ <. x σ := /( σ ) <. k σ (x σ ) = [ σ ] σ =: k(σ).

5 HARDY AND RELLICH INEQUALITIES 5 For σ [, ), k σ(x) is ever zero i (, ) idicatig that k σ (x) is miimized at x = for σ [, ) ad x [, ). The iequality (.3) ow follows. I order to treat the case i which p, we make use of the methods of Tidlom [] ad prove a weighted versio of Theorem. i []. Lemma 3. Let u C (, ), p (, ), ad σ p. The [ ] p (t)σ u (t) p p σ dt p (t)σ p + (p ) σ p u(t) p dt. Proof. We may assume that σ p sice otherwise the coclusio is trivial. Accordig to (.) for a mootoic fuctio f ad a positive fuctio g, f (t) u(t) p dt p f(t) f() u(t) p u (t) dt [ ] [ /p ( ) ] p g(t) u (t) p dt f(t) f() p /(p ) /p g(t) u(t) p dt. Cosequetly, p p g(t) u (t) p dt ( ( ) p f (t) u(t) p dt ) /(p ) p. u(t) p dt) ( f(t) f() p g(t) Now, as i [], usig a corollary to Youg s iequality, amely A p /B p pa (p )B, ) /p u(t) p dt, it fol- with A = f (t) u(t) p dt, B = lows that p p g(t) u (t) p dt p f (t) (p ) ( f(t) f() p g(t) ( f(t) f() p g(t) ) /(p ) u(t) p dt. Choose f(t) = t σ p+ ad g(t) = (p σ ) (p ) t σ. The ( f(t) f() p g(t) ) /(p ) = (p σ ) [ t σ p+ σ p+ p t σ ] p = (p σ )t [( σ p ( ) ) t p σ p ] p.

6 6 W. D. EVANS AND R. T. LEWIS Cosequetly, for t (, ) ( ) f(t) f() p /(p ) p f (t) (p ) g(t) [( = (p σ ) pt σ p (p )t σ p ( [ = (p σ )t σ p + (p ) (p σ )t σ p + (p ) ( ) t p σ ( t ( t ) ) p σ p ] p ) ] p ) p σ p (p σ ) t σ p + (p ) ( ). p σ ad the iequality follows. I the iequality aove we have used the fact that [ ( t p σ ] p ( p t ) p σ. ) The proof is completed y followig the last part of the proof of Lemma. For a certai rage of values take y σ, σ [ c σ, ) with c σ >, the iequality i L () give y Lemma gives a etter oud tha Lemma 3 with p =. I fact for σ < ( (t) ) σ ] (t) [ σ + k(σ) = (t) σ + σ k(σ) µ(t) (t)µ(t) + σ k(σ) σ (t) σ µ(t) σ with σ k(σ) + σ k(σ) (.8) (t)µ(t) σ (t) σ µ(t) σ 5 [ σ, σ [, ), σ σ + σ k(σ)(t) σ ] k(σ) σ, σ <. Sice k(σ) decreases to for σ < as σ ad k( 3)., the the left-had side of (.8) is greater tha σ for σ [ 3, ). 3. A Hardy iequality i L () We eed the followig otatio (c.f.[6]). For each x ad ν S, τ ν (x) := mis > : x + sν ; D ν (x) := τ ν (x) + τ ν (x); ν (x) := miτ ν (x), τ ν (x); µ ν (x) := maxτ ν (x), τ ν (x) = D ν (x) ν (x); D() := sup D ν (x); x, ν S x := y : x + t(y x), t [, ]. Note that D() is the diameter of ad x is the part of which ca e see from the poit x. The volume of x is deoted y x.

7 HARDY AND RELLICH INEQUALITIES 7 Let dω(ν) deote the ormalized measure o S (so that = dω(ν)) ad defie S (x; s) := ν (x) s dω(ν). S (3.) Hece / (x; ) = (x) the mea-distace fuctio itroduced y Davies i [4]. For B(, p) := cos(e, ν) p dω(ν) = Γ( p+ ) Γ( ) S π Γ( +p ), e R, (3.) it is kow that B(, p) (x; p) := dω(ν) (3.3) S ν (x) p δ(x) p for covex domais see Exercise 5.7 i [4], [3], ad []. Note that B(, ) =. This fact ca e applied to most of the results elow whe is covex. For a Hardy iequality i L () with weights we will eed to defie ( s ) ( σ) C H (, σ) := k(σ)[ σ + σ k(σ)]( σ) (3.4) for σ [, ] ad where as give i Lemma [ k(σ) := ] σ σ, σ <,, σ [, ]. Note that C H (, ) = 3 K() for K() defied i (.3). Theorem. If σ, the for ay u C () δ(x) σ u(x) ( σ) dx (x; σ ) u(x) dx 4 δ(x) σ + C H (, σ) u(x) dx. (3.5) x ( σ) If < σ, the u(x) dx σ ( σ) 4D() σ (x; σ ) + 3 ( s x ) σ u(x) dx. (3.6) If is covex, the for ay u C() δ(x) σ u(x) ( σ) dx B(, σ) δ(x) σ u(x) dx 4 + C H(, σ) δ(x) σ u(x) dx. ( σ)

8 8 W. D. EVANS AND R. T. LEWIS whe σ [, ] ad u(x) dx σ ( σ) 4D() σ whe σ (, ]. B(, σ)δ(x) σ + 3 ( s ) σ u(x) dx. Proof. Let ν u, ν S, deote the derivative of u i the directio of ν, i.e., ν u = ν ( u). It follows from Lemma that for σ (, ] σ ν(x) ν u dx ( σ ) ν(x) σ ( + k(σ) [ ν(x) µ ν(x) ] ( σ) ) u(x) dx. Expadig the itegrad i (3.7), we have ν (x) σ ( + k(σ) [ ν(x) ] ( σ) ) µ ν(x) = ν (x) σ + σ k(σ) ν(x) σ (τ ν(x)τ ν + ( σ) k(σ) ν(x) σ (x)) σ µ ν(x). ( σ) If σ ν (x) σ [ + k(σ) ( ν (x)) ( σ) ] µ ν (x) k(σ)δ(x) σ ν (x) σ + σ (τ ν(x)τ ν + ( σ) k(σ) δ(x) σ (x)) σ (3.7) (3.8) τ ν(x) ( σ) +τ ν (x) ( σ) sice ν (x) σ δ(x) σ i this case. As i [6], we ote that sice (τ S ν (x)τ ν (x)) σ dω(ν) (τ S ν (x)) ( σ) dω(ν) [ (τ S ν (x)) dω(ν) ] ( σ) = [ s x ] ( σ) for σ, the S (τ ν (x)τ ν dω(ν) [ (τ (x)) σ S ν (x)τ ν (x)) σ dω(ν) ] [ s x ] ( σ). For the third term i iequality (3.9) S (τ ν (x) ( σ) + τ ν (x) ( σ) )dω(ν) = implyig that for σ S (τ ν (x) ( σ) + τ ν (x) ( σ) ) dω(ν) S τ ν (x) ( σ) dω(ν) [ s x ] ( σ). (3.9) Cosequetly, for σ we have that S ν (x) σ [ + k(σ) ( ν (x)) ( σ) ] dω(ν) µ ν(x) (x; σ ) + C H (, σ)δ(x) σ / [ ] (3.) x ( σ).

9 HARDY AND RELLICH INEQUALITIES 9 Upo comiig this fact with (3.7) we have ( ) σ (x; σ ) + C H (,σ)δ(x) σ x ( σ)/ u(x) dx S ν (x) σ ν u(x) dω(ν)dx = δ(x)σ cos(ν, u(x)) dω(ν) (3.) S u(x) dx for σ. Sice cos(ν, α) dω(ν) = (3.) S for ay fixed α S (see Tidlom [], p.7), iequality (3.5) follows. For < σ, we cosider first the third term o the right-had side of (3.8). We have S ν (x) σ µ ν (x) ( σ) dω(ν) σ (τ S ν (x) + τ ν (x)) σ (τ ν (x) + τ ν (x)) ( σ) dω(ν) = σ τ ν (x) + τ ν (x) σ L σ (S ) σ[ ] σ τ ν (x) L σ (S ) + τ ν (x) L σ (S ) = ( σ) τ S ν (x) σ dω(ν) ( σ)[ (τ S ν (x)) dω(ν) ] σ = ( σ)[ s x ] σ for y the Mikowski ad Hölder iequalities. Therefore, the term ν (x) S σ dω(ν) µ ν (σ )( s ) σ (x) ( σ) x. Similarly, i the secod term of (3.8) S ν (x)µ ν (x) σ dω(ν) (τ S ν (x) + τ ν (x))(τ ν (x) + τ ν (x)) σ dω(ν) σ[ s x ] σ as efore implyig that dω(ν) S ν (x)µ ν σ ( s ) σ (x) σ x. For < σ < we ow have that S ν (x) σ [ + k(σ) ( ν (x)) ( σ) ] dω(ν) (x; σ ) + 3 ( s x ) σ µ ν (x) sice k(σ) = i this case. Cosequetly, S ν (x) σ cos(ν, u(x)) dω(ν) u(x) dx ( ) [ ( σ (x; σ ) + 3 s ) σ ] x u(x) dx.

10 W. D. EVANS AND R. T. LEWIS Accordig to (3.) it follows that S ν (x) σ cos(ν, u(x)) dω(ν) D()σ σ. Therefore, (3.6) holds. The iequalities i the statemet of the theorem for the case of a covex domai follow from (3.3) ad the fact that x = for all x. 4. A L p () iequality With the guidace of Tidlom s aalysis for the Hardy iequality i [], L p versios of the weighted Hardy theorem i the last sectio ca e proved y similar techiques. Whe σ =, the ext theorem reduces to Theorem. of []. Theorem. Let u C() ad p (, ). If σ, the for B(, p) defied i (3.) δ(x)σ u(x) p dx [ p σ /p] p B(,p) ad if σ [, p ], the u(x) p dx σ [ p σ /p] p B(,p) D() σ (x; σ p) + (p )[ s x (x; σ p) + (p )[ s x ] p σ u(x) p dx ] p σ u(x) p dx. (4.) (4.) If is covex, (x, σ p) ca e replaced i (4.) ad (4.) y the term B(, p σ)/δ(x) p σ (i view of (3.3)) ad x y. Proof. From Lemma 3 we have that for σ p, ay ν S, ad u C() ν(x) σ ν u(x) p dx [ p σ p ] p ν (x) σ p + (p )p σ D ν (x) u(x) p dx. p σ (4.3) If σ we oud ν (x) σ for ay ν S y δ(x) σ i the first itegral aove. If σ >, we oud it y D() σ / σ. As i [] we may use the fact that ν u(x) p dω(ν) = B(, p) u(x) p. (4.4) S After oudig ν (x) σ as descried aove, itegrate i (4.3) over S with respect to dω(ν). I order to evaluate the itegral of (/D ν (x)) p σ, we proceed as i []. Sice σ p, the f(t) = t σ p is covex for t > ad we have that ( ) p σdω(ν) ( D ν (x) ) σ p ( x ) p σ dω(ν) S D ν (x) S s (4.5)

11 HARDY AND RELLICH INEQUALITIES y Jese s iequality ad Lemma. of []. The coclusio follows. 5. Rellich s iequality The methods descried aove with Propositio elow ca e used to prove a weighted Rellich iequality which, for 4 ad without weights, improves the costat give i a Rellich iequality proved recetly y Baratis ([], Theorem.). A compariso is made elow. The methods used y Baratis depeds upo the idetity (5.) first proved y M.P. Owe ([], see the proof of Theorem.3). I order to icorporate weights, our proof requires the poit-wise idetity (5.) which does ot follow from the proof of Owe. Propositio. Let e a domai i R. The, for all u C (R ) νu(x) [ dω(ν) = u(x) + u(x) ], (5.) S ( + ) x i x j ad for all u C() S νu(x) dω(ν)dx = Proof. For ν = (ν,..., ν ) we have νu = (ν ) u = l,m= ν lν m u lm = l= ν l u ll + i,j= 3 u(x) dx. (5.) ( + ) l<m ν l ν m u lm i which u pq (x) := u(x) x p x q. Cosequetly, S νu dω(ν) = l,m= u llu mm (ν S l ) (ν m ) dω(ν) +4 m= Re(u mm u pq ) (ν S m ) ν p ν q dω(ν) p<q +4 Re(u pq u jk ) ν S p ν q ν j ν k dω(ν). j<k p<q (5.3) Let θ j [, π] for j =,...,, ad θ [, π]. Usig the covetio that Π p j=q = for p < q ad θ =, we have ν j = Π j k= si θ k cos θ j, j =,...,, ( )!! dω(ν) := Π k= (si θ k) k dθ k dθ, (5.4) γ for!! := ( ) ( 4) ad (π) ( )/ for odd, γ = (π) / for eve. Calculatios show that S (ν m ) ν p ν q dω(ν) =, m =,...,, p < q

12 W. D. EVANS AND R. T. LEWIS implyig that the secod term o the right-had side of (5.3) vaishes. A similar cosideratio for the third term o the right-had side of (5.3) shows that ν p ν q ν j ν k dω(ν), p < q, j < k, S oly if j = p ad k = q. Therefore, (5.3) reduces to S νu(x) dω(ν) = l,m= u llu mm (ν S l ) (ν m ) dω(ν) +4 u pq (ν S p ) (ν q ) dω(ν). (5.5) p<q However, further calculatios show that p < q, νpν q (+) dω(ν) = S p = q =,..., implyig that S νu dω(ν) = 3 (+) + (+) = (+) 3 (+) m= u mm [4 u pq + Re(u pp u qq )] p<q [ u(x) + i,j= u(x) ] x i x j which is (5.). Equality (5.) ow follows sice u(x) dx = u(x) dx. x i x j Defie ad i,j= δ(x) σ, σ <, d(x; σ) := ( D() ) σ, σ [, ]; β(, σ) := ( σ) (3 σ) ( + ) ; 6 C R (, σ) := 4 σ k(σ ) [ s for σ ad k(σ) defied i Lemma. Theorem 3. For σ ad u C (), d(x; σ) [ u(x) + holds whe 4 σ ad ] 4 σ ( ) + σ k(σ ) x i x j ]dx i,j= u(x) β(, σ) (x; σ 4) u(x) dx + 4 σ k(σ ) [ s ] 4 σ u(x) 4 σ x dx (5.6) (5.7) (5.8) (5.9)

13 HARDY AND RELLICH INEQUALITIES 3 d(x; σ) [ u(x) + x i x j ]dx i,j= u(x) β(, σ) (x; σ 4) u(x) dx + 4 σ k(σ ) [ s ] 4 σ + (3 σ) k(σ ) [ s ] 4+t σ u(x) 4 σ x dx δ(x) t u(x) 4+t σ x dx (5.) holds whe 4 + t σ ad t σ. Proof. For σ, it follows that (t) σ u (t) dt (t) σ[ ( (t) ) σ ] u (t) dt µ(t) ( σ ) (t) σ u (t) dt y (.4). Therefore, for σ ad u C (, ), (t)σ u (t) dt ( ( σ)(3 σ) 4 ) (t)σ 4[ + k(σ ) ( (t) µ(t) y (.3). From (5.) we have for u C () ν(x) σ νu(x) dx ( ) ( σ)(3 σ) ν(x) σ 4 + k(σ ) 4 for σ. As i (3.8) we write ( ν(x) ) 3 σ ] u(t) dt (5.) µ ν (x) ) 3 σ u(x) dx (5.) ( ) 3 σ ν (x) σ 4 + k(σ ) ν (x) µ ν(x) = ν (x) σ σ k(σ ) ν (x) µ ν + (3 σ) k(σ ) σ+ ν (x) (x) 3 σ µ ν. (x) (3 σ) (5.3) Sice ν (x)µ ν (x) = τ ν (x)τ ν (x), i the secod term o the right-had side of (5.3) we may write ν (x) µ ν (x) = 3 σ =: I(ν; x). [τ ν (x)τ ν (x)]µ ν (x) σ

14 4 W. D. EVANS AND R. T. LEWIS Thus S I(ν; x)dω(ν) = τ ν(x) τ ν (x) τ ν(x) σ 3 (x)τ ν (x) dω(ν) + τ ν (x) τ ν (x) τ ν(x) σ 3 (x)τ ν (x) dω(ν) τ ν (x) τ ν (x) τ ν(x) σ 4 (x)dω(ν) ad τ ν (x) dω(ν) σ 4 τ ν (x) τ ν (x) + τ ν (x) τ ν (x) τ ν(x) σ 4 (x)dω(ν) = τ ν (x) τ ν (x) τν (x)dω(ν) S ( ) (4 σ)/ x s τ ν (x) σ+4 dω(ν) (4 σ)/ for 4 σ. Therefore for the secod term o the right-had side of (5.3), for σ ad 4 σ, it follows that S ν (x)dω(ν) ( u(x) s ) 4 σ u(x) dx dx. (5.4) µ ν (x) 3 σ x 4 σ For ay t (, ), we may write the third term i (5.3) as σ+ ν (x) µ ν (x) = ν(x) t (τ (3 σ) ν (x)τ ν (x)) σ t µ(x) 8+3σ+t =: ν (x) t J(ν, x). If t σ S J(ν; x)dω(ν) As efore τ ν (x) τ ν (x) τ ν (x) 4+σ t dω(ν) τ ν (x) τ ν (x) τ ν(x) 4+σ t dω(ν) + τ ν (x) τ ν (x) τ ν(x) 4+σ t dω(ν). = τ ν (x) τ ν (x) τν (x)dω(ν) S ( x s τ ν (x) 4 σ+t dω(ν) ) (4 σ+t)/ (4 σ+t)/ if 4 σ + t. Associated with the third term o the right-had side of (5.3), we have for σ, t σ >, ad 4 σ + t S σ+ ν (x)dω(ν) ( u(x) s ) 4+t σ δ(x) t u(x) dx dx. µ ν (x) (3 σ ) x 4+t σ (5.5)

15 HARDY AND RELLICH INEQUALITIES 5 From (5.) (5.5) we otai S ν (x) σ νu(x) dω(ν)dx ( σ) (3 σ) (x; σ 4) u(x) dx + 4 σ k(σ ) [ s ] 4 σ + (3 σ) k(σ ) [ s ] 4+t σ u(x) x 4 σ dx 6 δ(x) t u(x) x 4+t σ dx provided σ, t σ, ad 4 + t σ. Note, that we may simply choose zero as a lower oud for the third term o the right-had side of (5.3) ad coclude that S ν (x) σ νu(x) dω(ν)dx ( σ) (3 σ) + 4 σ k(σ ) [ s ] 4 σ 6 u(x) x 4 σ for σ ad 4 σ. Now, it follows from Propositio that S ν (x) σ νu(x) dω(ν)dx [ u(x) d(x; σ) + (+) Thus, (5.9) ad (5.) are proved. (x; σ 4) u(x) dx dx x i x j ]dx. i,j= u(x) It follows from Theorem. of Baratis [] that for a covex ouded domai ad all u C () u(x) dx 9 u(x) [ dx + 6 δ(x) 4 48 ( + ) s ] 4/ u(x) dx. (5.6) As i Theorem, for a covex domai R, we may replace (x, σ 4) i Theorem 3 y B(, 4 σ)/δ(x) 4 σ ad x y to coclude from (5.9) that for 4 [ s ] 4/ u(x) dx 9 u(x) 6 δ(x) dx + c 4( + ) u(x) dx 4 (5.7) for all u C () i which c 4 = 3k( ).5. Therefore (5.7) improves the oud give y (5.6) for all 4. Refereces [] G. Baratis, Improved Rellich iequalities for the polyharmoic operator, Idiaa Uiversity Mathematics Joural 55(4) (6), 4 4. [] H. Brezis ad M. Marcus, Hardy s iequalities revisited, Dedicated to Eio De Giorgi, A. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (997, 998), [3] E.B. Davies, A Review of Hardy Iequalities. The Maz ya Aiversary Collectio, Vol., Oper. Theory Adv. Appl., Vol., pp , Birkäuser, Basel, 999. [4] E.B. Davies, Spectral Theory ad Differetial Operators, Camridge Studies i Advaced Mathematics, Vol. 4, Camridge Uiv. Press, Camridge, 995.

16 6 W. D. EVANS AND R. T. LEWIS [5] S. Filippas, V. Maz ya, ad A. Tertikas, O a questio of Brezis ad Marcus, Calc. Var. Partial Differetial Equatios 5(4) (6), [6] M. Hoffma-Ostehof, T. Hoffma-Ostehof, ad A. Laptev, A geometrical versio of Hardy s iequality, J. Fuct. Aal., 89 (), [7] E.H. Lie ad M. Loss, Aalysis, Graduate Studies i Mathematics, vol. 4, d editio, America Mathematical Society, Providece, R.I.,. [8] M. Marcus, V.J. Mizel, ad Y. Pichover, O the est costat for Hardy s iequality i R, Tras. Amer. Math. Soc. 35 (998), [9] T. Matskewich ad P.E. Soolevskii, The est possile costat i a geeralized Hardy s iequality for covex domais i R, Noliear Aalysis TMA, 8 (997), 6 6. [] M.P. Owe, The Hardy-Rellich iequality for polyharmoic operators, Proc. Royal Society of Ediurgh, A 9 (999), [] J. Tidlom, A geometrical versio of Hardy s iequality for W,p (), Proc. A.M.S., 3(8) (4), School of Mathematics, Cardiff Uiversity, 3 Segheydd Road, Cardiff CF4 4AG, UK address: EvasWD@cardiff.ac.uk Departmet of Mathematics, Uiversity of Alaama at Birmigham, Birmigham, AL , USA address: lewis@math.ua.edu