Series Expansion for L p Hardy Inequalities

Seres Expanson for L p Hardy Inequaltes G. BARBATIS, S.FILIPPAS, & A. TERTIKAS ABSTRACT. We consder a general class of sharp L p Hardy nequaltes n R N nvolvng dstance fro a surface of general codenson 1 N. We show that we can successvely prove the by addng to the rght hand sde a lower order ter wth optal weght and best constant. Ths leads to an nfnte seres proveent of L p Hardy nequaltes. 1. INTROUCTION Let be a bounded doan n R N contanng the orgn. Hardy s nequalty asserts that for any p>1 1.1 u p N p p u p dx p x p dx, u C c \{0}, wth N p/p p beng the best constant, see for exaple [9], [13], [6]. An analogous result asserts that for a convex doan R N wth sooth boundary, and dx = dstx,, there holds 1.2 p p 1 u p dx p u p d p dx, u C c, wth p 1/p p beng the best constant, cf [11], [10]. See [13] for a coprehensve account of Hardy nequaltes and [5] for a revew of recent results. In the last few years proved versons of the above nequaltes have been obtaned, n the sense that nonnegatve ters are added n the rght hand sde of 1.1 or 1.2. Iproved Hardy nequaltes are useful n the study of crtcal phenoena n ellptc and parabolc PE s; see, e.g., [3, 4, 10, 14]. In ths wor we obtan an nfnte seres proveent for general Hardy nequaltes, that nclude 1.1 or 1.2 as specal cases. 171 Indana Unversty Matheatcs Journal c, Vol. 52, No. 1 2003

172 G. BARBATIS, S.FILIPPAS & A. TERTIKAS Before statng our an theores let us frst ntroduce soe notaton. Let be a doan n R N, N 2, and K a pecewse sooth surface of codenson, = 1,..., N. In case =N, we adopt the conventon that K s a pont, say, the orgn. We also set dx = dstx, K, and we assue that the followng nequalty holds n the wea sense: C p, p d p /p 1 0, n \ K. Here p denotes the usual p-laplace operator, p w = dv w p 2 w. When = N, C s satsfed as equalty snce d p /p 1 = x p N/p 1 s the fundaental soluton of the p-laplacan. Also, f = 1, s convex and K =, condton C s satsfed. For a detaled analyss of ths condton, as well as for exaples n the nteredate cases 1 <<N, we refer to [2]. We next defne the functon: 1.3 and recursvely X 1 t = 1 log t 1, t 0,1, X t = X 1 X 1 t, = 2, 3,...; these are the terated logarth functons sutably noralzed. We also set 1.4 I [u] := u p dx H p u p d p dx p 1 2p H p 2 where H = p/p. Our an result reads as follows. u p X2 1 X2 2 X2 dx, Theore A. Let be a doan n R N and K a pecewse sooth surface of codenson, = 1,...,N. Suppose that sup x dx < and condton C s satsfed. Then: 1 There exsts a postve constant 0 = 0, p sup x dx such that for any 0 and all u W 1,p 0 \ K there holds d p 1.5 u p dx H p u p d p dx p 1 2p H p 2 u p d p X2 1 X 2 d dx. If n addton 2 p<, then we can tae 0 = sup x dx.

Seres Expanson for L p Hardy Inequaltes 173 2 Moreover, for each = 1, 2,... the constant p 1/2p H p 2 s the best constant for the correspondng -Iproved Hardy nequalty, that s, p 1 2p H p 2 = nf u W 1,p 0 \K I 1 [u] u p, X2 1 X2 2 X2 dx n ether of the followng cases: a = N and K ={0}, b = 1 and K =, c 2 N 1 and K. We also note that the exponent two of the logarthc correctons n 1.5 s optal; see Proposton 3.1 for the precse stateent. For p = 2, convex, and K =, the frst ter n the nfnte seres of 1.5 was obtaned n [3]. In the ore general fraewor of Theore A, the frst ter n the above seres was obtaned n [2]. On the other hand, when p = 2and K={0}the full seres was obtaned n [7] by a dfferent ethod. For other types of proved Hardy nequaltes we refer to [4, 8, 12, 14]; n all these wors one correcton ter s added n the rght hand sde of the plan Hardy nequalty. We next consder the degenerate case p = for whch we do not have the usual Hardy nequalty. In [2] a substtute for Hardy nequalty was gven n that case. The analogue of condton C s now: C p =, p ln d 0, n \ K. If C s satsfed, then for any sup dx there holds cf [2], Theores 4.2 and 5.4: 1.6 u dx 1 u d X 1 d p dx, u W 1,p 0 \ K, wth 1/ beng the best constant. In our next result we obtan a seres proveent for nequalty 1.6. We set 1 Ĩ [u] := u dx 1 1 1 2 u d X 1 dx u d X 1 X2 2 X 2 dx. We then have the followng result. Theore B. Let be a doan n R N and K a pecewse sooth surface of codenson, = 2,...,N. Suppose that sup x dx < and condton C s satsfed. Then,

174 G. BARBATIS, S.FILIPPAS & A. TERTIKAS 1 for any sup dx and all u W0 1, \ K there holds 1 1.7 u u dx d X 1 dx 1 1 1 u 2 d X 1 X2 2 X 2 dx. 2 Moreover, for each = 2, 3,... the constant 1 2 1/ 1 s the best constant for the correspondng -proved nequalty. That s, 1 Ĩ 1 [u] 1 2 1 = nf u W 1,p 0 \K u. X 1 X2 2 X2 dx n ether of the followng cases: a = N and K ={0}, b 2 N 1 and K. To prove parts 1 of the above theores, we ae use of sutable vector felds and eleentary nequaltes; ths s carred out n Secton 2. To prove the second parts, we use a local arguent and approprate test functons; ths s done n Secton 3. 2. THE SERIES EXPANSION In ths secton we wll derve the seres proveent that appear n part 1 of Theores A and B. We shall repeatedly use the dfferentaton rule 2.1 d dt Xβ t = β t X 1X 2 X 1 X 1+β, β 1,,2,..., whch s proved by nducton. For = 1 t follows edately fro the defnton of X 1 t cf. 1.3: d d dt Xβ 1 t = β t 1 log t β 1 = β t X1+β 1 t. Moreover, assung 2.1 for a fxed 1, we have d dt Xβ +1 t = d dt [Xβ 1 X t] = β X t X1+β 1 X t dx t dt = β X t X1+β +1 t1 t X 1t X 1 tx 2 t hence, 2.1 s proved. = β t X 1t X tx 1+β +1 t;

Seres Expanson for L p Hardy Inequaltes 175 Proof of Theore A 1. We wll ae use of a sutable vector feld as n [2]. If T s a C 1 vector feld n, then, for any u Cc \ K we frst ntegrate by parts and then use Hölder s nequalty to obtan dv T u p dx = p p T u u p 2 udx 1/p u p dx u p dx + p 1 p 1/p T p/p 1 u p dx T p/p 1 u p dx. We therefore arrve at 2.2 u p dx dv T p 1 T p/p 1 u p dx. For 1 we ntroduce the notaton ηt = X 1 t X t, Bt = X1 2 t X2 t. In vew of 2.2, n order to prove 1.5 t s enough to establsh the followng pontwse estate: 2.3 dv T p 1 T p/p 1 H p d p To proceed we now ae a specfc choce for T. Wetae p 2 dx Tx = H H d p 1 1 + p 1 dx x ph η 1 + p 1 2pH 2 B. + aη 2 dx, where a s a free paraeter to be chosen later. In any case a wll be such that the quantty 1 + p 1/pHηd/ + aη 2 d/ s postve on. Note that Tx s sngular at x K, butsnceu C c \K all prevous calculatons are legtate. When coputng dv T we need to dfferentate ηd/. Recallng 2.1, a straghtforward calculaton gves η t = 1 t X2 1 + X2 1 X 2 + X 2 1 X2 2 + +X2 1 X 2 X + +X 2 1 X2, fro whch the followng relaton follows: 2.4 tη t = 1 2 Bt + 1 2 η2 t.

176 G. BARBATIS, S.FILIPPAS & A. TERTIKAS On the other hand, observng that, snce d =1, p d p /p 1 = p p p 2 d p 1 p 1 d d + 1 d 2, condton C ples 2.5 p d d + 1 0. Usng 2.4 and 2.5, a straghtforward calculaton shows that 2.6 dv T H p d p p + p 1 H η + paη2 + p 1 2pH 2 B + η2 + a H B + η2 η. It then follows that 2.3 wll be establshed once we prove the followng nequalty p 1 + p 1 H η + pa + p 1 2pH 2 η 2 + a H Bη + a H η3 p 1 1 + p 1 p/p 1 ph η + aη2 0. We set for convenence fb,η =p 1 + p 1 H η + gη = 1 + p 1 ph η + aη2 pa + p 1 2pH 2 p/p 1, and the prevous nequalty s wrtten as 2.7 fb, η p 1gη 0. η 2 + a H Bη + a H η3, When η>0 s sall, the Taylor expanson of gη about η = 0, gves 2.8 gη = 1 + 1 H η + 1 2ap 2 p 1 + 1 ph 2 η 2 + 1 6a 6 p 1H + 2 p p 2 H 3 η 3 + Oη 4. Let us also note, that n the specal case a = 0, there holds 2.9 gη = 1 + 1 H η + 1 2pH 2 η2 + 2 p 6p 2 H 3 1 + p 1 3 2p/p 1 ph ξ η η 3, a =0,

Seres Expanson for L p Hardy Inequaltes 177 for soe ξ η 0,η, wthout any sallness assupton on η. In vew of 2.8, f η s sall, nequalty 2.7 wll be proved once we show: 2.10 a 2 pp 1 η 2 H 6p 2 H 3 + Oη B. Fro the defnton of η and B t follows easly that 2.11 η2 B 1. We wll show that for any choce of H and p>1, there exsts an a R,such that 2.7 holds true. We dstngush varous cases. a H>0, 1 <p<2. We assue that η s sall, whch aounts to tang bg. It s enough to show that we can choose a such that 2.10 holds. In vew of 2.11 we see that for 2.10 to be vald, t s enough to tae a to be bg and postve. b H >0, p 2. In ths case we choose a = 0. Notce that, under our current assuptons on H, p the last ter n 2.9 s negatve and therefore 2.12 gη 1 + 1 H η + 1 2pH 2 η2, a =0. On the other hand fb,η=p 1 + p 1 H η + p 1 2pH 2 η2, a =0, and therefore 2.7 s satsfed, wthout any sallness assupton on η. In partcular, we can tae 0 = sup x dx n ths case. c H<0, 1 <p<2. We assue that η s sall. In ths case, the rght hand sde of 2.10 s negatve. Hence, we can choose a = 0 and 2.10 holds true. d H<0, p 2. Argung as n case a we tae a to be bg and negatve, and 2.10 holds true. We next consder the degenerate case p =. Proof of Theore B 1. We assue that p = 2 and that condton C s satsfed. The proof s qute slar to the prevous one. An easy calculaton shows that condton C ples that 2.13 d d + 1 0.

178 G. BARBATIS, S.FILIPPAS & A. TERTIKAS We now choose the vector feld 1 1 d 2.14 Tx = d 1 X 1 1 + X 1 1 X 2 X. Tang nto account 2.13, a straghtforward calculaton yelds that 2.15 dv T p 1 T p/p 1 1 X1 1 d 1 + X 2 X + 1 1 j=2 1 X 2 2 X2 j X j+1 X / 1 1+ X 2 X. To estate the last ter n the rght hand sde of 2.15 we use Taylor s expanson to obtan the nequalty / 1 2. 1+ X 2 X 1+ X 1 2 X + 2 1 2 X 2 X It then follows that j=2 2.16 dv T p 1 T p/p 1 1 1 X1 1 d 1 + X2 2 X2 j X j+1 X 1 2 X 2 2 X. Expandng the square n the last ter n 2.16 we conclude that 1 1 X dv T p 1 T p/p 1 1 1 d + 1 X2 2 2 X2 and the result follows. 3. BEST CONSTANTS In ths secton we are gong to prove the optalty of the Iproved Hardy Inequalty of Secton 2. More precsely, for any 1 let us recall that I [u] = u p dx H p u p d p dx p 1 u p 2p H p 2 d p X1 2 + X2 1 X2 2 + +X2 1 X2 dx.,

Seres Expanson for L p Hardy Inequaltes 179 We have the followng result. Proposton 3.1. Let be a doan n R N. If 2 N 1,thenwetaeKto be a pecewse sooth surface of codenson and assue K. If = N,thenwetaeK={0}. If = 1, then we assue K =. Let sup dx be fxed and suppose that, for soe constants B>0and γ R, the followng nequalty holds true for all u W 1,p 0 \ K u 3.1 p I 1 [u] B d p X2 1 X 1 2 X γ dx. Then γ 2, If γ = 2, thenb p 1/2p H p 2. Proof. All our analyss wll be local, say, n a fxed ball of radus δ denoted by B δ centered at the orgn, for soe fxed sall δ. The proof we present wors for any = 1, 2,...,N. We note, however, that for = N dstance fro a pont the subsequent calculatons are substantally splfed, whereas for = 1dstance fro the boundary one should replace B δ by B δ. Ths last change entals soe nor odfcatons, the arguents otherwse beng the sae. Wthout any loss of generalty we ay assue that 0 K 1, or 0 f = 1. We dvde the proof nto several steps. Step 1. Let ϕ Cc B δ be such that 0 ϕ 1nB δ and ϕ = 1nB δ/2. We fx sall paraeters α 0, α 1,...,α >0 and defne the functons and wx = d H+α 0/p /p 1 ux = ϕxwx. X 1+α /p It s an edate consequence of 3.17 below that u W 1,p. Moreover,f <p, then H<0and therefore u K = 0. On the other hand, f >p, then a standard approxaton arguent usng cut-off functons shows that W 1,p 0 \ K = W 1,p \ K. Henceu W 1,p 0 \ K. To prove the proposton we shall estate the correspondng Raylegh quotent of u n the lt α 0 0, α 1 0,...,α 0 n ths order. It s easly seen that 3.2 w = d +α0/p /p 1 X 1+α /p H + ηx d, p

180 G. BARBATIS, S.FILIPPAS & A. TERTIKAS where 3.3 ηx = α 0 +1 α 1 X 1 + +1 α X 1 X 2 X, wth X = X d/. Snceδs sall the X s are also sall. Hence ηx can be thought as a sall paraeter n the rest of the proof. Now u = ϕ w + ϕw and hence, usng the eleentary nequalty 3.4 a + b p a p +c p a p 1 b + b p, a, b R N,p>1, we obtan 3.5 u p dx ϕ p w p dx + c p ϕ ϕ B p 1 w p 1 w dx + c p ϕ p w p dx δ B δ We cla that 3.6 I 2,I 3 =O1 unforly as α 0,α 1,...,α tend to zero. =: I 1 + I 2 + I 3. Let us gve the proof for I 2. Usng the defnton of wx and the regularty of ϕ we obtan I 2 c d 1 +α 0 1 X 1+α B δ H + ηx p 1 dx. p The appearance of d +1 together wth the fact that η s sall copared to H ples that I 2 s unforly bounded see Step 2. The ntegral I 3 s treated slarly. Step 2. We shall repeatedly deal wth ntegrals of the for 3.7 Q = ϕ p d +β 0 X 1+β 1 1 X 1+β dx, β R; we therefore provde precse condtons under whch Q<. Fro our assuptons on ϕ we have d +β 0 X 1+β 1 1 X 1+β dx Q B δ/2 Usng the coarea forula and the fact that c 1 r 1 {d=r } B δ d +β 0 X 1+β 1 1 X 1+β dx. B δ ds < c 2 r 1

Seres Expanson for L p Hardy Inequaltes 181 we conclude that δ/2 c 1 0 r 1+β 0 X 1+β 1 1 X 1+β dr Q c 2 δ 0 where X = X r /. Hence, recallng 2.1 we conclude that r 1+β 0 X 1+β 1 1 X 1+β dr, 3.8 Q< β 0 >0,or β 0 = 0andβ 1 >0, or β 0 = β 1 = 0andβ 2 >0, or. β 0 = β 1 = =β 1 =0andβ >0. Step 3. We ntroduce soe auxlary quanttes and prove soe sple relatons about the. For 0 j we defne A 0 = A = Γ 0j = Γ j = ϕ p d +α 0 1 X 1+α dx, ϕ p d +α 0 X 1+α 1 1 X 1+α X 1+α +1 +1 X 1+α dx, ϕ p d +α 0 X α 1 1 Xα X 1+α +1 +1 X 1+α dx, ϕ p d +α 0 X 1+α 1 1 X 1+α X α +1 +1 Xα j j X 1+α j+1 j+1 X 1+α dx, wth Γ = A. We have the followng two denttes. Let 0 1begven and assue that α 0 = α 1 = =α 1 =0. Then 3.9 3.10 α A = 1 α j Γ j + O1, j=+1 j α Γ j = α Γ j + 1 α Γ j + O1, =+1 =j+1 where the O1 s unfor as the α s tend to zero. Let us gve the proof for 3.9. We assue that >0, the case = 0 beng a straghtforward adaptaton. A drect coputaton gves 3.11 α d X 1 X 1 X 1+α =dvd +1 X α d d d d + 1 X α,

182 G. BARBATIS, S.FILIPPAS & A. TERTIKAS hence α A = ϕ p dvd +1 X α dx 1+α +1 +1 X 1+α dx ϕ p d d d + 1 X α X 1+α +1 +1 X 1+α dx =: E 1 E 2. It s a drect consequence of [1, Theore 3.2] that 3.12 d d + 1 = Od, as d 0, hence E 2 s estated by a constant tes ϕ p d +1 X α X 1+α +1 +1 X 1+α dx, and therefore s bounded unforly n α 0, α 1,...,α. To handle E 1 we ntegrate by parts obtanng E 1 = ϕ p dd +1 X α X 1+α +1 +1 X 1+α dx ϕ p d +1 X α d 1 X 1+α dx. The frst ntegral s of order O1 slarly to I 2, I 3 above, whle the second s equal to j=+1 1 α jγ j. Hence 3.9 has been proved. To prove 3.10 we use 3.11 once ore and proceed slarly; we ot the detals. Step 4. We proceed to estate I 1. It follows fro 3.2 that I 1 = ϕ p d +α 0 1 X 1+α H + η p p dx. Snce η s sall copared to H we ay use Taylor s expanson to obtan 3.13 H + η p p Usng ths nequalty we can bound I 1 by H p + H p 2 Hη + p 1 2p H p 2 η 2 + c η 3. 3.14 I 1 I 10 + I 11 + I 12 + I 13,

where 3.15 Seres Expanson for L p Hardy Inequaltes 183 I 10 = H p ϕ p d +α 0 1 X 1+α dx = H p u p B δ d p I 11 = H p 2 H ϕ p d +α 0 1 X 1+α ηx dx, B δ I 12 = p 1 2p H p 2 ϕ p d +α 0 1 X 1+α η 2 x dx, B δ I 13 = c ϕ p d +α 0 1 X 1+α ηx 3 dx. B δ Step 5. We shall prove that 3.16 I 11,I 13 = O1 unforly n α 0,α 1,...,α. Indeed, substtutng for η n I 11 we see by a drect applcaton of 3.9 for = 0 that I 11 = O1. To estate I 13 we observe that X 1 X cx 1 for soe c>0and thus obtan I 13 c 1 α0 3 ϕ p d +α 0 1 X 1+α dx + c 2 dx, ϕ p d +α 0 X 2+α 1 1 X 1+α 2 2 X 1+α dx. The second ntegral s bounded unforly n the α s due to the factor X1. 2 Moreover, usng the fact 0 ϕ 1and {d=r} B δ ds < cr 1 we obtan α 3 0 ϕ p d +α 0 1 X 1+α dx δ r cα0 3 0 δ cα0 3 = c α 0 α 2 0 = O1 r 1+α 0 1 0 r 1+α 0 X1 2 δ/ α0 0 r X 1+α r dr dr r = s 1/α 0 1 1 α 0 log s 2 ds unforly as α 0 0. Hence 3.16 has been proved. Cobnng 3.5, 3.6, 3.14, 3.15, and 3.16 we conclude that 3.17 u p dx H p u p d p dx I 12 + O1, unforly n the α s.

184 G. BARBATIS, S.FILIPPAS & A. TERTIKAS Step 6. Recallng the defnton of I 1 [ ] we obtan fro 3.17 3.18 I 1 [u] p 1 2p H p 2 1 ϕ p d +α 0 1 X 1+α η 2 x X 2 1 X2 dx + O1 =: p 1 2p H p 2 J + O1. Expandng η 2 x cf 3.3 and collectng slar ters we obtan 3.19 J = ϕ p d +α 0 1 X 1+α {α 0 2 + 1 α 2 X1 2 X2 1 1 +2 X 2 1 X2 2α 0 j=+1 1 α j X 1 X j j=1 1 α 1 α j X 2 1 X2 X +1 X j }dx = α0 2 A 0 + A + α 2 2α A 2α 0 1 α j Γ 0j + 1 j=+1 21 α 1 α j Γ j. j=1 Step 7. We ntend to tae the lt α 0 0 n 3.19. All ters have fnte lts except those contanng A 0 and Γ 0j whch, when vewed separately, dverge. When cobned however, they gve α0 2 A 0 2α 0 1 α j Γ 0j j=1 = α 0 1 α j Γ 0j +O1 by 3.9 j=1 j=1 j = 1 α j α Γ j + = 1 α α 2 A + j=+1 =j+1 1 α Γ j + O1 by 3.10 2α 11 α j Γ j + O1.

Seres Expanson for L p Hardy Inequaltes 185 All the ters n the last expresson rean bounded as α 0 0; hence tang the lt n 3.19 we obtan 3.20 J = A 1 α A + j=+1 1 α j Γ j + O1 α 0 =0, where the O1 s unfor wth respect to α 1,...,α. Step 8. We next let α 1 0 n 3.20. All ters have fnte lts except those nvolvng A 1 and Γ 1j, whch dverge. Usng 3.9 once ore ths te for = 1 we see that, when cobned, these ters stay bounded n the lt α 1 0. Hence 1 3.21 J = A α A + 1 α j Γ j + O1 α 0 = α 1 =0. j=+1 We proceed n ths way, and after lettng α 1 0 we are left wth 3.22 J = 1 α A + O1, α 0 = α 1 = =α 1 =0, unforly n α. Cobnng 3.1, 3.18 and 3.22 we conclude that 3.23 B p 1 1 α A + O1 2p H p 2 ϕ p d X 1 X 1 X γ 1+α. dx Suppose now that γ <2. Then lettng α 2 γ >0 we observe that the denonator n 3.23 tends to nfnty, whle the nuerator stays bounded. Ths ples B = 0, provng part of the proposton. Now, f γ = 2 then the denonator n 3.23 s equal to A. Hence lettng α 0wehaveA by 3.8 and hence Ths concludes the proof. B p 1 2p H p 2. We next consder the degenerate case p =. We have the followng result. Proposton 3.2. Let be a doan n R N. If 2 N 1,thenwetaeKto be a pecewse sooth surface of codenson and assue K ; f = N, thenwetaek={0}.

186 G. BARBATIS, S.FILIPPAS & A. TERTIKAS Let sup x dx be fxed and suppose that, for soe constants B>0and γ R, the followng nequalty holds true for all u C c \ K u 3.24 Ĩ 1 [u] B d X 1 Then: γ 2; f γ = 2,thenB 1 2 1/ 1. X2 2 X γ dx. Proof. The proof s slar to that of Proposton 3.1. Wthout any loss of generalty we assue that 0 K. As n the prevous theore we let ϕ be a non-negatve, sooth cut-off functon supported n B δ ={ x <δ}, equal to one on B δ/2 and tang values n [0, 1]. Gven sall paraeters α 1,...,α >0wedefne wx = X +1+α 1/ 1 X 1+α 2/ 2 X 1+α /, and ux = ϕxwx. Subsequent calculatons wll establsh that u W 1, see 3.29. We wll prove that u W 1, 0 \ K by showng that 3.25 d α 0/ u u n W 1,, as α 0 0. We have 3.26 d α0/ u u dx cα0 + d +α 0 u dx d α0/ 1 u dx. The second ter n the rght hand sde of 3.26 tends to zero as α 0 0bythe donated convergence theore. Moreover, there exsts a constant c α1 such that X α 1/2 1 X2 1 X 1 c α1. Hence the frst ter n the rght hand sde of 3.26 s estated by c α1 α0 d +α 0 X +1+α 1/2 1 dx. A drect applcaton of [2, Lea 5.2] shows that ths tends to zero as α 0 0. Hence u W 1,p 0 \ K.

Seres Expanson for L p Hardy Inequaltes 187 To proceed we use 3.4 obtanng 3.27 u dx ϕ w dx + c ϕ ϕ 1 w 1 w dx + c ϕ w dx =: I 1 + I 2 + I 3. Argung as n the proof of the prevous proposton cf Step 1 we see that I 2 and I 3 are bounded unforly wth respect to the α s. Hence 3.28 u dx unforly as α 1,...,α 0. Now, a drect coputaton yelds where ϕ w dx + O1 w = d 1 X 1+α 1/ 1 X 1+α 2/ 2 X 1+α / 1 d d ηx = α 1 + 1 α X 2 X. + ηx d, Fro 3.28 and recallng 3.13 we have u dx ϕ d X 1+α 1 1 X 1+α 2 2 X 1+α 3.29 { 1 1 1 + η + 1 1 1 } η 2 2 + c η 3 dx. The ter contanng η 3 s bounded unforly wth respect to α 1,..., α cf Step 4 n the prevous proposton. Moreover, t s edately seen that 3.30 ϕ d X 1+α 1 1 X 1+α 2 2 X 1+α = u d X 1, Hence

188 G. BARBATIS, S.FILIPPAS & A. TERTIKAS 3.31 Ĩ 1 [u] ϕ d X 1+α 1 1 X 1+α 2 2 X 1+α 1 1 α 1 + 1 α X 2 X + 1 1 1 2 α 2 1 + 1 α X 2 X 1 1 1 1 X2 2 2 X2 dx + O1, where the O1 s unfor wth respect to all the α s. Expandng the square and collectng slar ters we conclude that 3.32 Ĩ 1 [u] 1 1 1 J + O1, 2 unforly n α1,...,α, where 3.33 1 J = A + α 2 2α A + 2 1 α 1 α j Γ j. j=+1 We ntend to tae the lt α 1 0 n 3.33. All ters have a fnte lt except A 1 and Γ 1j, whch do not contan the factor X 1+α 2 2. When cobned they gve α1 2 2α 1A 1 + 2 1 α 1 1 α j Γ 1j j=2 = α1 2 A 1 2α 1 1 α j Γ 1j + O1 = 1 α j α 1 Γ 1j + O1 by 3.9 = = j=2 j 1 α j α Γ j j=2 1 α α 2 A + =j+1 j=+1 j=2 1 α Γ j + O1 by 3.10 2α 11 α j Γ j + O1. In ths expresson we can let α 1 0. Hence 3.33 becoes 1 3.34 J = A α A + 1 α j Γ j + O1, α 1 = 0. j=+1

Seres Expanson for L p Hardy Inequaltes 189 Ths relaton s copletely analogous to 3.20. For the rest of the proof we argue as n the proof of Proposton 3.1; we ot the detals. REFERENCES [1] L. AMBROSIO & H.M. SONER, Level set approach to ean curvature flow n arbtrary codenson,j.ff. Geoetry 43 1996, 693-737. [2] G. BARBATIS,S.FILIPPAS & A. TERTIKAS, A unfed approach to proved L p Hardy nequaltes wth best constants, subtted. [3] H. BREZIS & M. MARCUS, Hardy s nequaltes revsted, Ann. Scuola Nor. Psa 25 1997, 217-237. [4] H. BREZIS & J.-L. VÁZQUEZ, Blow-up solutons of soe nonlnear ellptc probles, Rev.Mat. Unv. Cop. Madrd 10 1997, 443-469. [5] E.B. AVIES, A revew of Hardy nequaltes, Oper. Theory Adv. Appl. 110 1998 55-67. [6] E.B. AVIES & A.M. HINZ, Explct constants for Rellch nequaltes n L p, Math. Z. 227 1998 511-523. [7] S. FILIPPAS & A. TERTIKAS, Optzng proved Hardy nequaltes, J. Funct. Anal. 192 2002, 186 233. [8] F. GAZZOLA, H.-CH. GRUNAU & E. MITIIERI, Hardy nequaltes wth optal constants and reander ters. Preprnt downloadable at http://athsun1.unv-treste.t/~tder/publcazon.htl. [9] G. HARY, G.PÓLYA & J.E. LITTLEWOO, Inequaltes. 2nd edton, Cabrdge Unversty Press 1952. [10] M. MARCUS, V.J.MIZEL & Y. PINCHOVER, On the best constant for Hardy s nequalty n R n, Trans. Aer. Math. Soc. 350 1998 3237-3255. [11] T. MATSKEWICH & P.E. SOBOLEVSKII, The best possble constant n generalzed Hardy s nequalty for convex doan n R n, Nonlnear Anal., Theory, Methods & Appl. 28 1997, 1601-1610. [12] V.G. MAZ JA, Sobolev Spaces, Sprnger, 1985. [13] B. OPIC & A. KUFNER,Hardy-type Inequaltes, Ptan Research Notes n Math. Volue 219, Longan 1990. [14] J.L. VAZQUEZ & E. ZUAZUA, The Hardy nequalty and the asyptotc behavor of the heat equaton wth an nverse-square potental, J.Funct.Anal.173 2000, 103-153. G. BARBATIS: epartent of Matheatcs Unversty of Ioannna 45110 Ionnna, Greece EMAIL: gbarbat@cc.uo.gr S. FILIPPAS: epartent of Appled Matheatcs Unversty of Crete 71409 Heralon, Greece E-MAIL: flppas@te.uoc.gr More nforaton on next page...

190 G. BARBATIS, S.FILIPPAS & A. TERTIKAS A. TERTIKAS: epartent of Matheatcs Unversty of Crete 71409 Heralon & Insttute of Appled and Coputatonal Matheatcs FORTH, 71110 Heralon, Greece http://www.ath.uoc.gr/~tertas/ E-MAIL: tertas@te.uoc.gr KEY WORS AN PHRASES: Hardy nequaltes; best constants; dstance functon; weghted nors. 2000 MATHEMATICS SUBJECT CLASSIFICATION: 35J20; 2610; 46E35, 35P. Receved: July 11th, 2001; revsed: January 16th, 2002.