Rend. Sem. Mat. Univ. Poi. Torino Fascicolo Speciale 1989 Nonlinear PDE's M. Pierre ON THE EQUIVALENCE ON SOME CAPACITARE STRONG TYPE ESTIMATES The results described here have been obtained in collaboration with D.R. Adams (see [3]). They are presented in a particular and significalit situation and a more direct proof is given. They deal with some extension of capacitary estimates obtained previously in [1], [2], [6], [7]. These extensions are directly motivated by the study of some semilinear elliptic problems with data measures which we describe next. Let us consider the problem { AM = u7 + //, on Q u > 0 on fi, a = 0 on <9fi where 7 > 1, fi is a bounded regular open set in ÌR N and /i isa given finite nonnegative measure on fi. It is establishecl in [4] that (1) has a solution if and only if (2) /^<( 7 -l)/7 7 ' /la^lv- 7 ' JSÌ JSÌ forali <P.WQ,CO (Q) sudi that <p>0 and A<p is compactly supported. Condition (2) is at the sanie Urne a size condition on fi and a regularity condition as well. Olir goal here is to better understand this regularity condition and make it more explicit. Indeed, it turns out that it is equivalent to some other "simpler" conditions appearing for different reasons in the literature, for instance (assuming // is compactly supported in fì) (3) There exists k > 0 sudi that in JSÌ
130 for the same <p as above. This property characterizes those measures // for which an imbedding from W 2 ' y '(Q) into L 7 '(fì,<fy«) holds. It was proved in [10], [1] that (3) is also equivalent to the apparently "weaker" version (4) ( There exists k > 0 such that \J dfi<k C 2, V (K) for ali compact K e fi where Ci t y is the capacity associated with the Sobolev space W 2^'(ÌK N ). The fact that (4) implies (3) is not obvious although it can be deduced through elementary arguments involving measure theory and smooth truncation techniques. The proof that (3) implies (2) is quite more delicate and relies on deep results from the theory of singular integrals with /4 p -weights. Here we want to explain how (3) follows more or less easily from (2) and, on the contrary, where the main diflìculty lies in deducing (2) from (3). We will also describe how it can be overcome. We will not state here the most general result. It can be found in [3] with an extension to general elliptic operators. We will concentrate on a particular but significànt case which allows a more direct proof. For simplicity, we will work on the whole space 1R N and assume (5) 1 < p < N/2, in order to be able to deal with Riesz-potentials, that is functions defined by (6) / 2 /(*)= / \x-y?- N f(y)dy /in" for / G //(IR^). We will also use the Riesz-capacity defined by R 2>p (A) = inf { / JP (RW) : / lf(ir N ) t f > 0,/ 2 / > 1 on A] for any set A in IR N. THEOREM: Assume 1 < p < N/2 and let //bea nonnegative Radon measure on IR N. Then the following condìtions are equivalent There exists ibi > 0 such that for ali (p CS (W, N ) with y? > 0 [ <pdft<ki! la^iv 1 "' Jm. N Jn N
131 (8) Thereexists k 2 > 0 sudi that for ali <pecs (ÌR N ) f M p d»<k 2 f Av» " (9) { There exists k 3 > 0 sudi that for ali K compact in IR N /i(k) < k 3 R 2%p (K). Proof of the theorem About (8) => (9). The equivalence between (8) and (9) is proved in [10], [1], Note that the hard part is (9) => (8). Indeed, we could use (io / M p dfj.= n P t p - i >i{[\<p\>ti}dt. Assuming (9) we obtain that (11)»{[M>tì}<k 3 R.2 tp {[\<p\>t]}. Let tf = J 2 ( Ap ). Then # > <p and from (11) (12) /i{[? > t]} < k 3 R 2)P [* >t)< k 3 t-<>\\am p < k 3 r*>\\a<p\\l P. Therefore, the function to be integrated in (10) satisfies p'"'f{lm>t]}<pk 3 r l \\A V >\\l r, from which we cannot conclude the estimate (8). The difrculty is the same as trying to pass from a weak type estimate to a strong type. A more careful analysis described in [10], [1] allows to conclude. PROOF of (7) => (8). It is sufficient to prove (8) for nonnegative <p. Indeed if <p CS (ÌR N ) and * = I 2 (\A<p\) we bave (13) <p < # and / A#» = / \A<p\ p. If (8) is true for nonnegative C functions, it extends by density to the Itiesz-potential *, whence the expected estimate for tf, then for <p.
132 Let now <p e C$ (\ìl N ),<p > 0. We apply (7) with <p replaced by y? to obtain (14) < k x C p f A^ P + \V<p\ 2p <p~ p. As in Maz'ya [10] and Adams [1], we treat the last term above by integration by parts: (15) { ^ = -pp-i)fv l -*\v, t \, '- t v. l. l +pjv-'\v. i \ ì '- This computation cali be easily justifìed by starting with \*Pxi\ 2r (<p + e)~ r letting e tend to 0. From (15), we get and ( P - i)y"v _p i^.i 2p =(2^- < (2 P -1) [/1^, r] 'y v -"\ v *f ll/p* Therefore (16) /p-' P.<l*<[(2j>-l)/0»-l)], '/K. i r'. We now use the classical //-estimates for singular integrals (see e.g. [9]) (17) j\<p*i*tf<cip,n)j\ò*p\>. Then (14), (16), (17) yield which is (8). j<frdp<c(k ììpt N)J\ *py PROOFof(8) => (7). Let (p^c^(m N ) ì (p>0. by (p 1^ to obtain We apply (8) with <p replaced (18) [<fidp< ki /, p"v- 1+,/p A > + p-, (l -p-'v'+'^ivy»! 2!"
133 or (19) J'tpdfi < k 2 C(p) jwlv~ p + V^ V" 2p The difficulty comes from the last terni of this inequality. An integration by part like in the previous proof gives a bound of the kind (20) jfa, V- 2p < C(p)J\ 9xiIi IV"" but one is not able to directly estimate this last integrai in terms of / la^pv 1- ''... except if <p is a Riesz-potential... and it is suflicient to prove the estimate for Riesz-potentialsL. This is what we will show now. Let us check the last point. For this, we introduce tf = / 2 ( Ay? ). Then, as already noticed, if (21) [$dn<k /la*!"* 1 -'' then, silice <p < # and A* = \A<p\ (22) J<pdp <J^!dfi< kj\a*\r* l -r < k J Let us prove (21). We apply (8) with <p replaced by VJ/Ì/P ((8) can be extended by density). We repeat the computations (18), (19), (20) but with ty instead of <p. At this stage, according to our analysis, to complete the proof, it is suflicient to prove the existence of e = C(p, N) sudi tliat (23) [\y XiX J* l -r<c flavl»* 1 -''. This corresponds to L p -estimate (17) but with the weight * 1_p. We will use the following result about ^,-weights of Muckenhoupt [8]. LEMMA (Coifman and Fefferman [5]). Let it> e,^(111^),^ > 0 sudi that (24) sup[/ J [ / w- l^p~ l) IP-I < A' < co, 1 < p < co, where the supremum is taken over ali cubes Q and / denotes the average JQ over Q. Then, there exists C = C{K,p, N) sudi that for ali / e 2/(lR N ), / > 0 and # = hi (25) / \9, ixi \ p (x)w(x)dx<c! f p (x)w(x)dx. JìR N./ili"
134 We will apply this lemma with w = ^f lmmp which turns out to satisfy (24) with a Constant K independent of $ due to the fact that # is a Riesz-potential. Indeed, since / > 0 and \P = / 2 /> by Harnack's inequality, there exists C = C(N) such that for ali cubes Q (26) inf«>c/«. Q JQ Therefore, since ^0 V* G Q * l ~ p (x) < (inf *) 1-p <(cf*j which after integration on i Q implies or / &-p<c l - p [ / ^ 'Q UQ y-p (/o^m/*)"^ This says exactly that w = y l -P satisfies (24) with K = C l ~ p. The estimate (23) is now a consequence of (25) in the lemma. REFERENCES [1] Adams D.R., On the existence of capacitary strong type estimates in IR^. Arkiv for Matematik, voi. 14 (1976), n 1, p. 125-140. [2] Adams D.R., Lectures on p -potential theory. Umea Univ. report 2 (1981). [3] Adams D.R., Pierre M., Capacitary strong type estimates in semilinear problems, to appear. [4] Baras P., Pierre M., Critère d'existence de solutions positives pour des équations semi-linéaires non monotones. Ann. I.H.P., 2 n 3 (1985), p. 185-212. [5] Coifman R.R., FelTerman C, VVèighted norm inequalities for maximal functions and singular integrala. Studia Mathematica, 51 (1974), p. 241-250. [6] Hansson K., Imbedding theorems of Sobolev type in potential theory. Math. Scand.,45 (1979),77-102.
[7] Mazya V.G., On some integrai inequalities for functions of several variables. Problerns in Math. Analysis, n 3 (1973) Leningrad (Russian). [8] Muckenhoupt B., Weighted norm inequalities for the Hardy maximal function. Trans. A.M.S., 165 (1972), p. 207-226. [9] Stein E.M., Singular integrals and difterentiability properties of functions, Princeton Univ. Press, Princeton N.J. (1970). 135 MICHEL PIERRE Département de Mathématiques, B.P. 239 Université de Nancy I 54506 Vandoeuvre les Nancy Cedex - France