Capacity and blow-up for the dimensional wave operator

Transcription

1 Capaciy and blow-up for he dimensional wave operaor by David R. Adams Universiy of Kenucky This paper is dedicaed o he memory of L.I. Hedberg ( )

2 1. Inroducion. ThewaveoperaorinEuclideanIR space is = 2 2 where is he Laplacian on IR 3 ; IR = IR3 (, ). In his noe we begin a poenial heory developmen for ha is analogous o he familiar poenial heories for ; see [AH]. Here a poenial heory developmen means ha we wan o sudy he poinwise behavior of weak soluions o w = F (x, ), (x, ) IR 3+1 +, w(x, ) = w(x, ) =, x IR3 (1) where F belongs o a cerain funcion space. And by he poinwise behavior, we mean a sudy of he ses where w is disconinuous and/or w becomes infinie. Developing a poenial heory based on (1) is a reasonable possibiliy here, because classically here is a very simple formula for he soluion o (1), namely he so called rearded poenial : w(x, ) =RF (x, ) = y < F (x y, y ) y 1 dy. In paricular, he fundamenal soluion for is non-negaive, consequenly F implies w. This is no longer he case for higher space dimensions, which brings in added difficulies for a corresponding developmen here; see [S]. We rea only he case of hree space dimensions in his noe. The poinwise propery we concenrae on in his noe is he naure of he blow-up se; he se {(x, ) :RF (x, ) =+ } (2) 1

3 for F and belonging he mixed norm funcion space L x Lp 2 hose F for which f L = [ ( IR 3 ) ] p2 F (x, ) p //p2 1 1 dx d (IR ), i.e. is finie. Here 1,p 2 < wih he usual modificaion when eiher or p 2 is infinie. The sandard mehod of measuring a se like (2) is wih some sor of capaciy based on he differenial operaor and he given funcion space. In his paper, we propose such a capaciy, a wave capaciy, and sudy is properies and is relaionships o ses like (2), he blow-up se for he soluion w = RF. The reader may already be familiar wih he various L p L q esimaes for soluions o problems (1), esimaes of he form ( ) w L q x, c F L p x,, and similar esimaes for he soluion o he companion problem u =, IR u(x, ) = f(x), u (x, ) = g(x), x IR3+1 + inerms of norms of he iniial daa f and g. Such esimaes are generally used o prove exisence, uniqueness, and funcion space regulariy resuls for various nonlinear versions of he wave equaion, e.g. { u = u k, IR 3 (,T) plus iniial daa. Here k>1andt ; see [S]. However, esimaes like ( ) play only a minor role in his noe hey always give a lower bound on our wave capaciy (defined below), bu hey 2 (3) (4)

4 are no in general sharp enough nor numerous enough o do much good in esimaing he wave capaciy. Also, hey never give upper bounds on he wave capaciy. Bu because wave capaciy is ranslaion invarien in he space variable, we can obain a sharp a priori esimae of ( ) ype for radial (radial in he space variable) rearded poenials ha helps give precise resuls for he wave capaciy of a ball in 4-space. Also, cerain race ype esimaes are needed o give a priori esimaes for he wave capaciy on lower dimensional ses. We organize our discussion as A. defining a wave capaciy associaed wih operaor (1) and he mixed norm Lebesgue spaces, B. esablishing basic properies of our wave capaciy C. compuing upper and lower maching growh esimaes for he wave capaciy of a 4-ball, and hen a 3-ball, D. comparing wave capaciy o Hausdorff measures, especially in he case of he blow-up se (2). A. Wave capaciy. 1. Definiion. Le K be a compac subse of IR and se D p1 p 2 (K) =inf{ f p 2 L : f andrf 1 where p 2 =min{,p 2 } and 1,p 2. D p1 p 2 ( ) can be exended o all subses of IR in he sandard way; see [AH]. I is clear now ha 3

5 various a priori esimaes of he ype Rf q L 1 x L q 2 c f p L 1 will give lower bounds for D p1 p 2 (K). In paricular, he classical Sricharz esimae yields Rf L 4 x, c Rf L 4/3 x, L 4 (K) 1/3 c D 4/3,4/3 (K). (5) (Here and hroughou, he leer c will denoe a consan independen of he relevan quaniies in (5), c is independen of K, in Sricharz s esimae i is independen of F,ec.).HereL 4 denoes Lebesgue measure on IR If we apply (5) o K = Q r (x, )={(y, s) : y x <r, s <r}, >, a 4-ball in IR 3+1 +,henwehave cr 4/3 D 4/3,4/3 (Q r (x, )), (6) for all r>. To es he sharpness of (6), consider y < χ Q2r (x y, y ) y 1 dy y <r y 1 dy = c r 2 since if x x <rand <rand y <r< /2, hen x y x < 2r, y < 2r and r<, i.e. Rχ Q2r cr 2 on Q r (x, ). (7) Here Q 2r denoes Q 2r (x, ). Hence by he definiion of wave capaciy χ D 4/3,4/3 (Q r ) Q2r 4/3 cr 2 = cr 4/3, (8) 4 L 4/3 x L 4/3

6 for all r< /2, maching (6). This calculaion ses he one for his noe. However, here are difficulies. There are no enough embeddings of he Sricharz ype or oherwise o ge good lower bounds, as in (6), and he simple esimaes of ype (7) and (8) are no sharp for some,p 2 (even among hose for which = p 2 ). Hence i will be necessary o resor o oher devices. One such is o formulae he wave capaciy in is dual mode; cf. [AH]. 2. The adjoin rearded poenial. In order o formulae he wave capaciy in is dual mode, we need he adjoin rearded poenial of a Borel measure, denoed as R µ.wegehis by firs noing ha he adjoin rearded poenial applied o a non-negaive funcion is jus a soluion of he backwards wave equaion, i.e. wih Cauchy daa a ime = T>. So here we denoe S T = {(x, ) :x IR 3, <<T}. Now noe ha if F, G C (IR3+1 + ), hen where S T R G(x, ) = RF Gdxd= y <T S T F R Gdxd G(x y, + y ) y 1 dy, he adjoin rearded poenial of G. To exend R o he class of Borel measures µ wih compac suppor in S T, we proceed as follows: consider RF dµ ct 2 F L x(s T ) µ 1 S T for all F C (S T ), µ 1 denoing he oal variaion of µ; in his case jus µ(k). Thus by he Riesz represenaion heorem, here is a measure ν on 5

7 S T such ha S S RF dµ = Fdν, F. Consequenly, we say R µ = ν on S T. 3. The dual wave capaciy. If K is a compac se in S T,le d p1,p 2 (K) =sup{ µ 1 : µ M + (K) and R µ 1}, L p 1 x L p 2 (S T ) i.e. we are considering all hose Borel measures µ suppored by K and for which R µ = ν<<l 4 and he Radon-Nikodym derivaive dν/dl 4 belongs o L p 1 x L p 2 (S T ) wih mixed norm 1. Here p ha for all compac ses K S T, because µ(k) RF dµ = S T = p/(p 1). I is easy o see d p1,p 2 (K) p 2 D p1 p 2 (K) (9) FR µdxd F p L 1 S T R µ. L p 1 x L p 2 And if we resor o he min-max principle as in [AH], i follows ha equaliy acually holds in (9), and hen, ha he equaliy can be exended o all Borel ses in S T ; again see [AH]. B. Some basic properies of wave capaciy. 4. Wave capaciy is counably subaddiive. Here we see he reason for using he power p 2 i ensures ha he wave capaciy is counably subaddiive and i produces he correc 6

8 Hausdorff dimension for ses K. So, le {K j } be a sequence of subses of IR + 3+1, hen we claim D p1,p 2 ( j K j ) j D p1 p 2 (K j ). Case 1. p 2. Choose F j so ha RF j 1onK j, hen sup j F j F has he propery ha RF 1on j K j and F p 2 L j F j p 2 / dx d ( j F j dx) p2 / d. Case 2. <p 2.Now F L j F j dx 2 /p /p 2 1 p d j [ ( F j dx) p2 / d ] p1/p2, by Minkowski s inequaliy. 5. Wave capaciy is ranslaion invarien in IR 3. Here we are claiming D p1 p 2 (K +(x, )) = D p1 p 2 (K) (1) for all x IR 3. Indeed, (1) follows from he fac ha F x = F, where he norms are he mixed L Lebesgue norms and F x (x, ) =F (x+x,). Below, we noe ha his no longer works for ranslaions in he ime variable. 7

9 6. Wave capaciy of an iniial ball. Here we calculae he wave capaciy of he iniial ball { ˆQ r (, ) = {(x, ) : x < r, <<r} using dialaion. We prove D p1 p 2 ( ˆQ r (, )) = c(r 3/+1/p 2 2 ) p 2, (11) for all r>. Indeed, if RF (x, ) 1for(x, ) ˆQ r (, ), hen he change of variables ξ = x/r, s = /r produces a funcion G = r 2 F r for which RG 1 on ˆQ 1 (, )); F r (ξ,s) =F (rξ, rs). Thus D p1 p 2 ( ˆQ 1 (, )) r 2 F r p 2 L = r (2 3/ 1/p 2 ) p 2 F L. Now inerchange he roles of ˆQ r and ˆQ A relaion beween wave capaciies. ha Here we noe ha here is a consan c independen of he se K such D p1 p 2 (K) 1/ p 2 cd p1 p 2 (K) p 2 (12) for all compac ses K ˆQ R (, ) for some fixed R sufficienly large and wih 1/ 1/, 1/p 2 1/ p 2. In fac, o see (12) we firs noice ha for (x, ) K and R sufficienly large R(FχˆQ R (,))(x, ) =RF (x, ) for all (x, ) K. The resul now follows by applying Holder s inequaliy o he mixed norm of F χ ˆQr and recalling ha hese norms end o he norm of F as R. 8

10 8. An explici form for R µ. I would aid he calculaions of wave capaciy if we had an explici represenaion for he expression R µ when µ is a Borel measure on IR We don have such a represenaion in he general case, bu if dµ(x, ) = dν(x)g() d, wecanwrie R µ(x, ) =c x y 1 g( x y + ) dν(y) (13) IR 3 where ν is a Borel measure on IR 3 wih compac suppor, g is a locally inegrable funcion on IR + 1,andc is some consan. To see (13), we compue he averages of R µ over balls in IR o he limi in a sandard way. Thus firs consider R µ(x, ) dx d = Rχ Qr dµ Q r(x, ) S T = x z 1 dz dµ(x, ) S T M and pass where M = {z IR 3 : x z <, z x <r, x z ( ) <r}. And so he above inegral is = x z 1 IR 3 z x <r x z + +r x z + r g() d dz dν(x) provided r<. Now divide by L 4 (Q r (x, )) and pass o he limi as r. This gives he desired resul. C. Wave capaciy of balls and recangles 9. D p1 p 2 (Q R (x, )), The moivaion for his secion is: if we compue he wave capaciy of he ball Q r (x, )asr for all,p 2 such ha ( 1, 1 p 2 ) (, 1) (, 1), hen 9

11 we can ge a Hausdorff measure esimae on D p1 p 2 (K) and consequenly, a Hausdorff measure esimae for he nonhomogeneous wave equaion blow up se, which clearly saisfies D p1 p 2 ([RF =+ ]) = for F L, F. So, for our resul below, we divide he square (, 1) (, 1) ino he regions j,j=1, 2, 3, 4, where 1 : 1/p 2 1/, 1/p 2 < 2 2/ ; 2 :1/p 2 1/, 1/p 2 < 2 2/ ; 3 :1/p 2 1/, 1/p 2 > 2 2/ ; 4 :1/p 2 1/, 1/p 2 > 2 2/. We now prove 1

12 Theorem 1. If σ(,p 2 )= hen for any > p 2 /, for ( 1, 1 p 2 ) 1 1, for ( 1, 1 p 2 ) 2 ( p 2 2)p 2, for ( 1, 1 p 2 ) 3 ( p 2 2), for ( 1, 1 p 2 ) 4 D p1 p 2 (Q R (x, )) R σ(/p 2 ), as R. Also, when 2 2/ =1/p 2 R p 2/ (log 1/R) 1 p 2, for 1/p 2 1/ D p1 p 2 (Q R (x, )) R(log (1/R) (1 p 2 )/p 2, for 1/p 2 1/. Here means ha he raio is bounded above and below by some posiive consans for all R<r,forsomefixedr > ; r is independen of R, bu depends on, and p 2. Remark 1. According o Theorem 1, when 2 2/ =1/p 2 holds, embeddings of he form RF q L 1 x L q 2 c F p L 1 are impossible for 3 q q 2 = 1. In paricular, here is no embedding of he form RF L 6 x, c F 3/2 L. x, The ineresed reader should compare his wih Theorem 2.5 of reference [RS]. ProofofTheorem1. We consider he upper and lower bounds for D p1 p 2 (Q R (x, )) as R separaely. Clearly, we need only consider x =. Thelower 11

13 bounds can hen be achieved by looking a only radial F, i.e. F (x, ) = F ( x,). For he upper bounds, i seems o be necessary o look a wha we shall call mixed nonlinear rearded poenials. Lower bounds. We firs recall ha he radial case can be represened (see [S]) as: x = r, 2rRF(x, ) = r+( s) r ( s) F (ρ, s)ρdρds F (,s) L x ( [r +( s)] α r ( s) α α ) 1/p 1 ds where α =(1 2/ )p 1 +1,p 1 = /( 1). Nex, Holder s inequaliy and a change of variables, gives he upper bound ( T/r c r α/p 1 +1/p 2 ) 1/p (u +1) α u 1 α 2 p 2 /p 1 du. The inegrand in he above inegral is clearly locally inegrable when 3 3/ 1/p 2 >. Thus we need only consider r (α 1)1/p 1 1/p 2, 2 2/p1 > 1/p 2 T/r (u +1) α u 1 α p 2 /p 1 du c 1, 2 2/ < 1/p 2 2 (log 1/r) 1/p 2, 2 2/p1 =1/p 2. Consequenly, we have r α/p 1 +1/p 2, 2 2/p1 < 1/p 2 rrf(x, ) c F L r 1/p 1, 2 2/p1 > 1/p 2 r 1/p 1 (log 1/r) 1/p 2, 2 2/p1 =1/p 2. which gives he desired lower bound for D p1 p 2 (Q R (, )) provided 3 3/ 1/p 2 >. 12

14 When 3 3/ 1/p 2, we recognize he bound rrf(x, ) c HencewehaveheRieszpoenialesimae F (, u) p r L 1 x u 3/p 1 1 du. rrf(x, ) c I 3/p g(r) (14) 1 where g(u) = F (, u) L x and we are seing F (p, s) =fors<; I β g(r) = r u β 1 g(u) du. We now use he Hardy-Lilewood convoluion inequaliy on (14) when 3 3/ < 1/p 2 i.e. 3/p 1 p 2 < 1, o ge ( [rrf(x, )] q 1 dr) 1/q1 c g L p 2 = c F L for 1 q 1 = 1 p 2 3. And his gives he desired lower bound in his case. p 1 And finally, he case 3 3/ =1/p 2 is resolved using he well known exponenial Sobolev esimae (see Theorem 3.14 of [AH]) 1 R R [ rrf(x, ) exp b F ] p 2 dr c which gives D p1 p 2 (Q R ) c R p 2 resul. in his case, which again is our desired Upper bounds. Before we bie ino he general siuaion, we should noe ha here are some special cases where he desired upper bound can be achieved wih very lile effor. However, hese consideraions do no seem o work in general, especially in he roublesome case ha occurs wih 2. 13

15 Firs, noice ha esimae (7) can be used (as in (8)) o ge he correc upper bounds for 3 and 4. Secondly, he diagonal case 1/ =1/p 2 is raher easily resolved by noicing ha R χ Q p = R[R χ L Q ] 1/(p 1) dx d. (15) x, Q One simply needs o show ha on Q = Q R (, ), R 3/(p 1), 1/p < 2/3 R[R χ Q ] 1/(p 1) (x, ) c R 6 (log 1/R), 1/p =2/3 R 2p, 2/3 < 1/p. for all R<R. We will no discuss his here since our consideraions below will subsume his. Thirdly, one can noice ha via he relaions inequaliy (12), one can achieve he correc upper bounds for 1 from he diagonal case, i.e. D p1 p 2 (Q R ) 1/p 2 cd p1 (Q R ) 1/ cr 1/. This does no work for 2 nor for poins (1/, 1/p 2 ) on he line 2 2/ = 1/p 2. To rea he general case upper bound on D p1 p 2 (Q R (, )), we find i necessary o inroduce he following mixed nonlinear rearded poenials: J Q (x, ) =R[(R χ Q ) 1/(p1 1) R χ Q λ ](x, ) L p 1 x where λ = 1 p Noiceha Q J Q (x, ) dx d = R χ Q p 2 L p 1 x L p 2, 14

16 he p 2 analogue of (15). Noice furher, ha if we can show for (x, ) Q = Q R (, ), hen J Q (x, ) R B, R < R, D p1 p 2 (Q R ) 1/ p 2 R 4/p 2 B/p 2. (16) And hen we only need prove ha B = ( ) 4 p 2 1 p 2 on 1 and 2 and B = ( 3 p ) p 2 on 3 and 4 (and wih an appropriae modificaion wih log 1/R on he line 2 2/ =1/p 2 ) o achieve our goals. Sowebeginwihhecaseλ> and we look firs a R χ Q (, y ) p 1 = L p 1 x z ξ <R = R χ Q (z, y )(R χ Q (z, y )) 1/(p1 1) dz (17) IR 3 ξ 1 dξ ξ <T + y y + ξ <R (R χ Q (z, y )) 1/ 1 dz. Now noice ha if <R/2and y ξ <R/2henwege y + ξ <R,and <Rwih y + ξ <Rimplies y ξ < 2R. Hence he above inegral is equivalen o z ξ <R y ξ <αr ξ 1 dξ (R χ Q (z, y )) 1/( 1) dz for some choices of α>. Here we have omied he condiion ξ <T + y by choosing T sufficienly large (relaive o and R). Thus (17) is equivalen o y ξ <αr ξ 1 z ξ <R η 1 dη 15 y η <α z η < 1 1/( 1) dz dξ (18)

17 Now we change variables: ξ = Rξ, η = Rη, z = Rz and y = Ry,which make (18) ino 1/( 1) R 5+2/ 1 y ξ <α ξ 1 z ξ <1 η 1 dη y η <α z η < 1 dz dξ = R 5+2/( 1) W ( y ). Also, i is easy o see ha W ( y ) y 1 1/( 1) as y,andw is bounded for y 2. So now pu his all ino J Q and hen upon performing he same ricks as before here, we finally ge ha J Q (x, ) is bounded above on Q and below on 1 Q by a consan muliple of 2 R 2/(p 1)+2 (R 5+1/(p1 1) ) λ/p 1 y 1/(p1 1) ( y 1 1/(p1 1) ) λ/p 1 y 1 dy 2< y <β /R (19) for some choices of β>. The inegral in (19) has p 1 +(1 1/( 1)) λ p 1 +3=( 1) p 2 [2 2/ 1/p 2 ] hence when 2 2/ 1/p 2 <, (19) behaves like R p 2 (2+3/p 2 3/ ), as R, exacly as needed! When 2 2/ 1/p 2 >, we ge R (4/p 2 1/ )p 2, as R, 16

18 And when 2 2/ 1/p 2 =, R p 2 (2+3/p 2 3/ ) (log 1/R), as R. The desired esimaes for wave capaciy now follow. However, when λ<, we need o be more careful. To ge an upper bound on J Q (x, ), (x, ) Q, we clearly need a lower bound on (17), bu now we are no allowed o reduce from Q o 1 Q since ulimaely we need an upper 2 bound on J Q dx d. Q To ge around his problem, we firs ge esimaes for J Q (x, ) wih(x, ) Q + = upper half of Q: x <Rand << + R. This allows for he se where y + ξ <Rin (17) o be replaced by he smaller se where y R < ξ < y and his accomplishes he same goal. The lower half of Q is reaed similarly (for we are only ineresed in an esimae as a funcion of R, asr ). This concludes our proof of Theorem 1. Remark 2. Reexamining he proof of Theorem 1 reveals ha a similar argumen also implies ha he capaciy D p1 p 2 (B r (x ) { }) has he same growh in R as does D p1 p 2 (Q R (x, )), for each >. D. Esimaes for wave capaciy in erms of Hausdorff measures. 1. Hausdorff measure, Hausdorff capaciy, and Hausdorff dimension If φ(r) is a coninuous non-decreasing funcion of r, wih φ() =, 17

19 hen we se H φ ɛ (K) =inf j φ(r j ) (2) where K is a compac subse of IR and he infimum in (2) is aken over all covers of K by balls {Q rj (x j, j )} wih r j ɛ. For each fixed ɛ, we will refer o H φ ɛ as a Hausdorff capaciy associaed wih he measure funcion φ. The Hausdorff measure associaed wih φ is lim ɛ Hφ ɛ (K) H φ (K). When φ(r) =r d,forsomed>, we shall wrie more simply H d ɛ and Hd. The Hausdorff dimension of he se K, wrien as H-dim (K), is inf {d : H d (K) =}, K IR Now using he counably subaddiiviy of wave capaciy and Theorem 1, we easily have Corollary (o Theorem 1). Seing φ(r) =r σ(,p 2 ) for ( 1, 1 p 2 ) i, i = 1, 2, 3, 4, and r p 2/ (log 1/r) 1 p 2, 1/p 2 1/ φ(r) = r(log 1/r) (1 p 2 )/p 2, 1/p 2 1/ when ( 1, 1 p 2 ) lies on he line 2 2/ =1/, hen here is a consan c independen of he Borel se K IR such ha D p1 p 2 (K) ch φ (K). (21) Also noe ha from Theorem 1 we could bound D p1 p 2 (K) by he Hausdorff capaciy H φ ɛ (K), which in a sense is beer due o he fac ha Hausdorff 18

20 capaciy is finie on bounded subses of IR and sill Hφ and Hɛ φ have he same null ses; see [AH]. The Corollary gives a Hausdorff measure upper bound for he wave capaciy of he blow-up se [Rf =+ ], f L x Lp 2, f, and we can hen immediaely make he following conclusions: H dim ([Rf =+ ]) < 2, ( 1, 1 p 2 ) (, 1) (, 1), H dim ([Rf =+ ]) < 1, ( 1, 1 p 2 ) 1, p 2, H dim ([Rf =+ ]) 1, ( 1, 1 p 2 ) 2. From he firs of hese conclusions, i is clear ha he blow-up se canno conain a se of he ype E {, },wheree IR 2, L 2 (E) >. This can also be seen simply from he esimae Rf(,,, ) L x 1,x 2 c f L (22) where he lef hand inegral is aken over IR 2. To see (22), jus use Minkowski s inequaliy geing f(,, y 3, y ) p L 1 y 1 dy x y < which does no exceed 2π s ( s) f(,,u,s) du ds c f L upon changing o spherical coodinaes. Thus L 2 (E) 1/ cd p1 p 2 (E {, }) 1/ p 2. (23) 19

21 11. Wave capaciy of a space secion of he blow-up se We nex prove Theorem 2. For( 1, 1 p 2 ) 2, 1/ < 2/3, here is a consan c such ha 1 c L 1(E) D p1 p 2 ({x } E) cl 1 (E), (24) i.e. he wave capaciy behaves like Lebesgue 1-dimensional measure on ime lines. In paricular, L 1 ([Rf =+ ] x )= where K x is he x -secion; here f L x Lp 2 as given above. Proof. Wihou loss of generaliy, we can assume x =andhaf is radial. Then from he radial formula (see lower bounds esimaes of Theorem 1) Rf(,)= f( s, s)( s) ds. Hence T Rf(,)ψ() d f L ( T T S ) p ψ() p 1 ( s) (1 2/ )p 2 /p 1 1 d 1/p 2 ds. Bu since p 2 p 1 on 2, he above righ hand inegral does no exceed c ψ p when 1/ < 2/3. Dualiy hen gives L 1 T ( Rf(,) d) 1/ c f L. This gives he lef inequaliy in (24); he righ side follows from Theorem 1 in ha H 1 L 1 on a ime line. 2

22 Now consider he following example, which is mean o es he sharpness of Theorem 2, i.e. how big can he x -secion of he blow-up se be? For his we se f(x, ) = x + s α 1 dµ(s) wih x < 1and<<1and<<T.Noeha T 1 ( 2 ρ + s dµ(s)) α 1 ρ 2 dρ d = = c T 1 1 ( T )( ρ + s α 1 dµ(s) ) ρ + s α 1 dµ(s ) ρ + s α 1 ρ + s α 1 ddµ(s) dµ(s )ρ 2 dρ s s 2α 1 dµ(s) dµ(s )=c I αµ 2 L 2, ρ 2 dρ d by he Riesz convoluion heorem. Here I α is he Riesz convoluion operaor; see [AH]. Also, we need <α<1/2. Thus f is square inegrable over IR locally when I α µ is. Nex, noe Rf(x, )= x y +( y ) s α 1 dµ(s) y 1 dy y < x + s α 1 dµ(s) y 1 dy. y < So Rf(,) c 2 I α µ(). We now choose E IR 1 such ha he Riesz capaciy C α,2 (E) =, < α<1/2, bu C α,2 (E) >, α < α < 1/2. Then here exiss a measure µ such ha I α µ L 2 < and I 2α µ =+ on E; see [AH]. Bu hen I α µ =+ on E. 21

23 Finally, aking α and α arbirarily close o zero, we can deduce ha he x secion of he blow-up se, as in Theorem 2, can have 1 ɛ<h-dim 1 for any ɛ>. 12. Wave capaciy and produc ses Here we look for furher esimaes on ses like E F, E IR + 3+1,F IR1, and confine our aenion o he diagonal case = p 2 = p for simpliciy. Our firs observaion is ha [F], Theorem implies ha H 4 2p (E F ) ch 3 2p (E) L 1 (F ), whenever p<3/2. Consequenly, if H 3 2p (E) < and L 1 (F )=,wehaved pp (E F ) =. In paricular, if F = { },wege Theorem 3. Ifp<3/2 andh 3 2p (E) <, hen D pp (E { })=. Finally, we use our explici form for R µ given in (13) o deduce Theorem 4. Le E and F be bounded Borel ses of IR and IR 1 respecively, hen here is a consan c such ha 1 c C 2,p(E) L 1 (F ) D pp (E F ) ch 4 2p ɛ (E F ) (25) for some ɛ > ; c independen of E,F and ɛ. Here C 2,p is he sandard Riesz capaciy, now on subses of IR 3.Noeha C 2,p (E) ch 3 2p+δ (E) for any δ>, p 3/2; see [AH], Theorem The upper bound in (25) is clear from Theorem 1. For he lower bound we esimae (13) as follows: R µ L p x, y 1 χ F L p dν(y) L p 22

24 wherewehavesedµ(y, ) =dν(y)χ F ds. Thus ν 1 L 1 (F )= µ 1 fr µdxd f L p R µ x, L p x, Thus if I 2 ν L p 1, hen f L p x, L 1(F ) 1/p I 2 ν L p ν 1 L 1 (F ) 1/p D pp (E F ) 1/p. And since ν 1 C 2,p (E) 1/p we ge he desired resul. Remark 3. If he Riesz kernel I 2 (x) = x 2 3 is properly resriced o x < 1, say, hen esimae (25) can be considered a special case of (24) since C 2,p ({x }), p 3/2. Bu (25) is our aemp o ge a lower bound on he wave capaciy ha closely maches he upper bound: 1 (E) L 1 (F ) D pp (E F ) chɛ 4 2p (E F ), c H3 2p+δ 1/p > 2/3 andδ>. References [AH] Adams, D.R. and Hedberg, L.I., Funcion Spaces and Poenial Theory, Springer, [F] Federer, H., Geomeric Measure Theory, Springer, [S] Sogge, C., Lecures on Nonlinear Wave Equaion, Monographs in Analysis, Vol. II., Inernaional Press [RS] Ricci, F. and Sein, E., Harmonic analysis on nilpoen groups and singular inegrals. III. Fracional inegraion along manifolds, J. Funcional Anal. 86a (1989),