ADAMS INEQUALITY WITH THE EXACT GROWTH CONDITION IN R 4

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1 ADAMS INEQUALITY WITH THE EXACT GROWTH CONDITION IN R 4 NADER MASMOUDI AND FEDERICA SANI Contents. Introduction.. Trudinger-Moser inequality.. Adams inequality 3. Main Results 4 3. Preliminaries Rearrangements A useful inequality involving rearrangements Otimal descending growth condition 9 4. Proof of the Main Theorem (Theorem. 4.. Estimate of the integral on the ball B R with u (R > 3 5. Sharness 4 6. From Adams inequality with the exact growth condition to Adams inequality in W, (R Aendix A: An Alternative Proof of Adams Inequality 7 8. Aendix B: An Alternative Proof of the Trudinger-Moser Inequality with the Exact Growth Condition 9 Acknowledgements References Abstract: Adams inequality is an extension of the Trudinger-Moser inequality to the case when the Sobolev sace considered has more than one derivative. The goal of this aer is to give the otimal growth rate of the exonential tye function in Adams inequality when the roblem is considered in the whole sace R 4.. Introduction Trudinger-Moser inequality is a subsitute to the well known Sobolev embedding theorem when the critical exonent is infinity since W, n ( L ( when R n, n, is a bounded domain. Adams inequality is an extension of the Trudinger-Moser inequality to the case when the Sobolev sace considered has more than one derivative. The goal of this aer is to give the otimal growth rate of the exonential tye function when the roblem is considered in the whole sace R 4... Trudinger-Moser inequality. Let R n, n, be a bounded domain. The Sobolev embedding theorem asserts that, for < n, W, ( L q ( q n n. If we look at the limiting Sobolev case, namely = n, then but it is well known that W, n ( L q ( q, W, n ( L (.

2 NADER MASMOUDI AND FEDERICA SANI In other words, in the limiting case the Sobolev embedding theorem does not give a notion of critical growth. In 97, J. Moser in [8] refined a result roved indeendently by V. I. Yudovich [39], S. I. Pohozaev [3] and N. S. Trudinger [38] and obtained a notion of critical growth for the Sobolev saces W, ( in the limiting case = n. Theorem. ([8], Theorem. There exists a constant C n > such that su e α u n n dx C n α α n (. u W,n (, u n where u n n = u n dx denotes the Dirichlet norm of u, α n := nω /(n n and ω n is the surface measure of the unit shere S n R n. Furthermore (. is shar, i.e. if α > α n then the suremum in (. is infinite. In the literature (. is known under the name Trudinger-Moser inequality. This inequality has been extended in various directions. Related results and variants can be found in several aers, see e.g. Y. Li [, 3] for generalizations to functions on comact Riemannian manifolds and A. Cianchi [3] for a shar inequality concerning functions without boundary conditions. For Trudinger-Moser tye inequalities in other function saces, in articular Orlicz and Lorentz saces, see e.g. A. Cianchi [, ], A. Alvino, V. Ferone and G. Trombetti [5], B. Ruf and C. Tarsi [35], H. Bahouri, M. Majdoub and N. Masmoudi [6, 7], Adimurthi and K. Tintarev [4] and O. Druet [3]. Concentration and comactness roerties of sequences bounded in W,n were also studied in [, 6, 9, 6]. We also mention that in a recent aer [5], D. G. De Figueiredo, J. M. Do Ó and B. Ruf gave an interesting overview of results concerning Trudinger-Moser tye inequalities with alications to related equations and systems [6]. In addition to alications in ellitic systems, this inequality was also alied to Klein-Gordon equation with exonential nonlinearities [9, ]. In this aer, firstly we will focus our attention on last develoments in the study of Trudiger-Moser tye inequalities which are domain indeendent, thus valid in the whole sace R n and, then we will consider ossible generalizations at the case of Sobolev saces involving second order derivatives. An interesting extension of (. is to construct Trudinger-Moser tye inequalities for domains with infinite measure. In fact, we can notice that the bound in (. becomes infinite, even in the case α α n, for domains R n with = + and consequently the original form of the Trudinger-Moser inequality is not available in these cases. One of the first results in this direction is due to S. Adachi and K. Tanaka []: Theorem. ([], Theorem. and Theorem.. Let ψ(t := e t n j= tj j!. For α (, α n there exists a constant C(α > such that ψ(α u n n dx C(α u n n u W, n (R n with u n, (. R n and this inequality is false for α α n. The limit exonent α n is excluded in (., which is quite different from Moser s result (see Theorem.. However, in the case n = (i.e. for W, ( with R, B. Ruf [33] showed that if the Dirichlet norm is relaced by the standard Sobolev norm, namely u n W, n := u n n + u n n, then the exonent α = 4π becomes admissible. Strengthening the norm in this way, the result of Moser (Theorem. can be fully extended to unbounded domains and the suremum in (. is uniformly bounded indeendently of the domain R :

3 ADAMS INEQUALITY WITH THE EXACT GROWTH 3 Theorem.3 ([33], Theorem.. There exists a constant C > such that for any domain R su (e 4πu dx C (.3 and this inequality is shar. u W, (, u W, In [4], Y. Li and B. Ruf extended Theorem.3 to arbitrary dimensions n 3, i.e. to W, n ( with R n not necessarily bounded and n 3. In short, the failure of the original Trudinger-Moser inequality (. on R n can be recovered either by weakening the exonent α = nω /(n n or by strengthening the Dirichlet norm u n. Then a natural question arises: What if we kee both the conditions α = nω /(n n and u n? In the -dimensional case, an answer to this question is due to S. Ibrahim, N. Masmoudi and K. Nakanishi [] and the idea is to weaken the exonential nonlinearity as follows Theorem.4 ([], Proosition.4. There exists a constant C > such that R e 4πu ( + u dx C u u W, (R with u. (.4 Moreover, this fails if the ower in the denominator is relaced with any <. Obviously this last inequality imlies (. and it is interesting to notice that it also imlies inequality (.3, which is not so obvious. This imroved inequality has some alication to the Klein-Gordon equation with exonential nonlinearity []. Remark. When one deals with Trudinger-Moser tye inequalities, as (., (., (.3 and (.4, the target sace is a Sobolev sace involving first order derivatives, i.e. W, n. In these saces it is easy to reduce the roblem to the radial case considering the rearrangements of functions. In fact, given a function u W, n ( with R n, the sherically symmetric decreasing rearrangement u of u (see Section 3. for a recise definition is such that u W, n (, u n = u n and g( u dx = g(u dx, for any Borel measurable function g : R R such that g. Moreover, accordingly to the Pólya-Szëgo inequality, u n u n. (.5 The reduction of the roblem to the radial case by mean of the rearrangements of functions is a key tool in the roof of all the revious results... Adams inequality. In 988 D. R. Adams [] obtained another interesting extension of (. for Sobolev saces involving higher order derivatives. Let R n with n, if we consider saces of the form W, ( then the limiting case in the Sobolev embedding theorem corresonds to the dimension n = 4 and in this articular case Adams result can be stated as follows. Theorem.5 ([], Theorem. Let R 4 be a bounded domain. Then there exists a constant C > such that su e 3π u dx C (.6 and this inequality is shar. u W, (, u

4 4 NADER MASMOUDI AND FEDERICA SANI We oint out that some authors (see [5], [37] and [34] have remarked that the roof of Adams can, in fact, be adated in order to obtain a stronger inequality involving a larger class of functions, i.e. the Sobolev sace W, ( W, (. More recisely, Theorem.6 ([5], [37] and [34]. Let R 4 be a bounded domain. Then there exists a constant C > such that su e 3π u dx C u W, W, (, u and this inequality is shar. In articular, we bring the reader s attention to the aer [37] where C. Tarsi suggests a roof of Theorem.6 which differs esserntially from Adams one and relies on shar embeddings into Zygmund saces. We recall that L. Fontana [7] roved that the comlete analogue of Adams theorem, Theorem.5, is valid for every comact smooth Riemannian manifold M. See also L. Fontana and C. Morurgo [8] for a generalization of Adams result to functions defined on arbitrary measure saces. As in the case of first order derivatives, one notes that the bound in (.6 becomes infinite for domains with = +. In [34] the authors show that, relacing the norm u by the full Sobolev norm u W, := u + u + u, the suremum in (.6 is bounded by a constant indeendent of the domain R 4. Theorem.7 ([34], Theorem.4. There exists a constant C > such that for any domain R 4 su (e 3π u dx C (.7 and this inequality is shar. u W, (, u W, Remark. Dealing with higher order derivatives, the roblem cannot be reduced directly to radial case. In fact, given a function u W, (, we do not know whether or not u still belongs to W, ( and, even in the case u W, (, no inequality of the form (.5 is known to hold for higher order derivatives. Adams aroach to this difficulty is to exress u as Riesz otential of its Lalacian and then aly a result of O Neil on non-increasing rearrangements for convolution integrals (see also Section 8. To overcome the same difficulty, in [34] the authors aly a suitable comarison rincile, which leads to comare a function u W, ( W, ( not necessarily radial with a radial function v W, ( W, (, reserving a full Sobolev norm equivalent to the standard one and increasing the integral, i.e. (e 3π u dx 3πv (e. Main Results dx. As in the case of first order derivatives, we ask the question: How can we modify the exonential nonlinearity in order to obtain an analogue of Theorem.4 for the Sobolev sace W, (R 4? The main result of this aer gives an answer to this question:

5 ADAMS INEQUALITY WITH THE EXACT GROWTH 5 Theorem.. (Adams inequality with the exact growth. There exists a constant C > such that R4 e 3π u ( + u dx C u u W, (R 4 with u. (. Moreover, this fails if the ower in the denominator is relaced with any <. We recall that T. Ogawa and T. Ozawa [3] roved the existence of ositive constants α and C such that dx C u u W, (R 4 with u. (. R 4 (e αu The roof of this result follows the original idea of N. S. Trudinger; making use of the ower series exansion of the exonential function, the roblem reduces to majorizing each term of the exansion in terms of the Sobolev norms in order that the resulting ower series should converge. But with these arguments the authors of [3] did not find the best ossible exonent for this tye of inequalities. However, as a simle consequence of Theorem., we have the following Adachi-Tanaka tye inequality in the sace W, (R 4 : Theorem.. For any α (, 3π there exists a constant C(α > such that dx C(α u u W, (R 4 with u, (.3 R 4 (e αu and this inequality is false for α 3π. Moreover, the arguments introduced in [], to show that the Trudinger-Moser inequality with the exact growth condition (Theorem.4 imlies the Trudinger-Moser inequality in W, (R (Theorem.3, will enable us to deduce from Theorem. a version of the Adams inequality in W, (R 4 involving the norm + instead of the full Sobolev norm W,, for a recise statement see Theorem 6.. Obviously, this newer version of Adams inequality in R 4 imlies Theorem.7. This aer is organized as follows. In order to rove Theorem., we, first, recall some reliminaries on rearrangements of functions (Section 3.. Second, we introduce a useful inequality involving rearrangements, Proosition 3.5, that will enable us to avoid the difficulty of the reduction of the roblem to the radial case (see Remark. Then, following [], a key ingredient is the study in Section 3.3 of the growth, in the exterior of balls, of rearrangements of functions belonging to W, (R 4. In Section 4, we rove inequality (. and Section 5 is devoted to the roof of the sharness of (.. Finally, in Section 6 we show that the Adams inequality with the exact growth condition exressed by Theorem. imlies inequality (.7 and, as already mentioned, we obtain an imroved version of Theorem.7. We comlete this aer with two Aendices. In the first one, we suggest an alternative roof of Adams inequality in its original form (Theorem.5 which, at the same time, gives a simlified version of Adams roof and involves the Sobolev sace W, ( W, (, where R 4 is a bounded domain. In the second Aendix, we roose a variant of the roof of the Trudinger-Moser inequality with the exact growth condition, namely of Theorem.4, which simlifies the critical art of the estimate.

6 6 NADER MASMOUDI AND FEDERICA SANI 3. Preliminaries 3.. Rearrangements. Let R n, n, be a measurable set, we denote by the oen ball B R R n centered at R n of radius R > such that = B R where denotes the n-dimensional Lebesgue measure. Let u : R be a realvalued measurable function in. Then the distribution function of u is the function µ u : [, + [, + ] defined as µ u (t := {x u(x > t} for t, and the decreasing rearrangement of u is the right-continuous non-increasing function u : [, + [, + ] which is equimeasurable with u, namely u (s := su{t µ u (t > s} for s. We can notice that the suort of u satisfies su u [, ]. Since u is non-increasing, the maximal function u of the rearrangement u, defined by u (s := s s is also non-increasing and u u. Moreover u (t dt for s, Proosition 3.. If u L (R n with < < + and + ( + [u (s] ds ( [u (s] dt In articular, if su u with domain in R n, then ( [u (s] ds ( [u (s] dt = then This is an immediate consequence of the following well known result of G. H. Hardy (see e.g. [8], Chater 3, Lemma 3.9 Lemma 3. (Hardy s inequality. If < < +, + = and f is a non-negative function, then ( + [ s s s ] ds f(t dt s ( + [ s.. f(s ] ds s Finally, we will denote by u : [, + ] the sherically symmetric decreasing rearrangement of u: u (x := u (σ n x n for x, where σ n is the volume of the unit ball in R n. 3.. A useful inequality involving rearrangements. Let R n, n, be a bounded oen set. As a consequence of the coarea formula and the isoerimetric inequality, we have the following well-known result Lemma 3.3 ([36], Inequality (3. If u W, ( then µ ( u(t d u dx [nσ n µ n n dt { u >t} for a.e. t >. u ].

7 ADAMS INEQUALITY WITH THE EXACT GROWTH 7 Sketch of the roof. For fixed t, h >, alying Hölder s inequality, we get ( u dx u dx h h {t< u t+h} and, letting h +, we obtain d u dx dt {t< u } {t< u t+h} ( µ u (t µ u (t + h h ( d u ( dx µ dt u (t. (3. {t< u } On the other hand, from the Coarea Formula and the isoerimetric inequality d u dx nσ n µ n u (t dt {t< u } and this inequality together with (3. gives the desired estimate. Let f L (, we consider the following Dirichlet roblem: { u = f L ( in, u = on. (3. It is well-known that Lemma 3.4 ([36], Inequality (3. Let u W, ( be the unique weak solution to (3., then d µu(t u dx f (s ds dt for a.e. t >. { u >t} Sketch of the roof. For fixed t, h >, we define if u t, φ(x := ( u t sign (u if t u t + h, h sign (u if t + h < u. Then φ W, ( and u φ dx = fφ dx, since u is a weak solution to (3.. Now, easy comutations show that u dx = f ( u t sign (u dx+ fh sign (u dx. (3.3 {t< u t+h} {t< u t+h} Dividing through by h in (3.3 and letting h +, we obtain d u dx d f ( u t dx = dt dt {t< u } {t< u } {t+h< u } {t< u } f dx µu(t f ds. If u W, ( is the unique weak solution to (3. then, combining Lemma 3.3 and Lemma 3.4, we get µ u(t F (µ [nσ n µ n u (t n ] u

8 8 NADER MASMOUDI AND FEDERICA SANI for a.e. t >, where F (ξ := ξf (ξ = ξ f (s ds. Let < s < s, integrating both sides of this last inequality from s to s and making the change of variable ξ = µ u (t, we obtain that s s µu(s F (ξ µu(s f (ξ dξ = dξ. (3.4 [nσ n ] µ u(s ξ n n [nσ n ] µ u(s ξ n From inequality (3.4 we deduce the following result. Proosition 3.5. Let u W, ( be the unique weak solution to (3. then u (t u t f (ξ (t dξ for a.e. < t [nσ n ] t ξ t. n Equivalently, if = B R then u (R u (R [nσ n ] BR B R f (ξ dξ for a.e. < R R R. ξ n Proof. Let < t < t, without loss of generality we may assume that u (t > u (t. Then there exists η > such that Arbitrarily fixed η [, η, we set u (t > u (t + η η [, η. s := u (t η and s := u (t + η, so that < s < s. Now alying inequality (3.4, we get u (t u (t η [nσ n ] µu(s µ u(s f (ξ dξ ξ n [nσ n ] t t f (ξ dξ, (3.5 ξ n in fact, recalling the definition of decreasing rearrangement of a function, since s < u (t then µ u (s > t and, similarly, since s > u (t then µ u (s < t. From the arbitrary choice of η [, η, inequality (3.5 holds for any η [, η and assing to the limit as η +, we obtain the desired estimate. Remark 3. It is easy to rove that an analogue of Proosition 3.5 holds also for functions in W, (R n. In fact, if u W, (R n then there exists a sequence {u k } k C (Rn such that u k u in W, (R n. Thus, in articular, f k f in L (R n where f k := u k, f := u and u k u in L (R n. Since the decreasing rearrangement oerator is nonexansive in L (see e.g. G. Chiti [4], we have f k f and u k u in L (R n. (3.6 In articular, u to the assage to a subsequence, we have also that u k u a.e. in R n and, from Proosition 3.5, it follows that for fixed < t t u (t u (t = lim k + u k (t u k (t t fk lim (ξ dξ. k + [nσ n ] t ξ n On the other hand, lim k + [nσ n n ] t t fk (ξ dξ = ξ n [nσ n n ] as a direct consequence of Hölder s inequality and (3.6. t t f (ξ dξ, ξ n

9 ADAMS INEQUALITY WITH THE EXACT GROWTH Otimal descending growth condition. The following exonential version of the radial Sobolev inequality will be a key tool in the roof of Theorem.. Proosition 3.6. Let u W, (R 4 and let R >. If u (R > and f := u in R 4 satisfies for some K >, then + B R e 3π K [u (R] [f (s] ds 4K, u [u (R] R 4 L C (R 4 \B R K where C > is a universal constant indeendent of u, R and K. Fix h, R, K >. Let { µ(h, R, K := inf u L (R 4 \B R u W, (R 4, u (R = h and + B R [f (s] ds 4K where f := u in R 4} and µ(h := µ(h,,. Using scaling roerties, it is easy to see that µ(h, R, K = R µ(h,, K = R ( h Kµ,, = R ( h Kµ. K K We will rove the following result. Proosition 3.7. If h > then µ(h e6π h. h Obviously, Proosition 3.6 is a direct consequence of Proosition 3.7. In order to rove Proosition 3.7, it suffices to consider the discrete version. Given any sequence a := {a k } k, following [], we introduce the notations and, for h >, we let a := + a k, a (e := + k= k= a k ek µ d (h := inf{ a (e a = h, a }. Lemma 3.8. For any h >, we have µ(h µ d ( 3π h. Proof. Fix h > and let u W, (R 4 be such that u ( = h and + B [f (s] ds 4, where f := u in R 4. Then we can define a sequence a := {a k } k as follows: a k := h k h k+, where h k := 3π u (e k 4.

10 NADER MASMOUDI AND FEDERICA SANI By construction a = 3π u ( = 3π h. In order to estimate a, we can notice that alying Proosition 3.5 (see also Remark 3 and Hölder s inequality, we get h k h k+ = 3π [u (e k 4 u (e k+ 4 ] 3π Be (k+/4 [f (s] ds 6π Be k/4 s ( Be (k+/4 [f (s] ds. Therefore a = Finally, + k= Be k/4 (h k h k+ 4 + k= Be (k+/4 [f (s] ds = 4 Be k/4 + B [f (s] ds. u L (R 4 \B = π + k= + k= e k+ 4 e k 4 + [u (r] r 3 dr π [u (e k+ 4 ] ek+ e k 4 [u (e k+ 4 ] e k+ = 3π + j= k= h je j 3π + j= a je j from which we deduce that a (e = a + + k= and the roof is comlete if we show that a k ek h + u L (R 4 \B, h u L (R 4 \B. This is again a consequence of Proosition 3.5 (see also Remark 3 and Hölder s inequality, in fact for r (, e 6 h 3π u (r Br B that is h u (r. Thus [f (s] ds ( + [f (s] ds s B ( B e /6 B s ds h, u L (R 4 \B π e 6 [u (r] r 3 dr h. Now, it is sufficient to follow the roof of [], Lemma 3.4 to obtain Lemma 3.9. For any h >, we have µ d (h e h h. This last lemma together with Lemma 3.8 comletes the roof of Proosition 3.7.

11 ADAMS INEQUALITY WITH THE EXACT GROWTH 4. Proof of the Main Theorem (Theorem. In this Section we will show that Theorem. holds, to this aim let us fix u W, (R 4 with u. We define R = R (u > as R := inf{r > u (r } [, + and the idea is to slit the integral we are interested in into two arts R4 e 3π u R4 ( + u dx = e 3π (u R4 e 3π (u ( + u dx = \BR ( + u dx. B R + Since R = R (u >, we have to obtain an estimate indeendent of R. By construction u on R 4 \B R, hence we have that Consequently R4 \BR e 3π (u e 3π (u ( + u 3π (u e 3π (u 3π e 3π (u on R 4 \B R. ( + u dx 3π e 3π (u dx 3π e 3π u 3π e 3π u R 4 \B R and without loss of generality we may assume that R >. From now on, we will focus our attention in the estimate of the integral on the ball B R. In order to simlify the notations, we set and f := u in R 4 β := + [f (s] ds. As a consequence of Proosition 3. with = and in view of the assumtion u, we have β 4. (4. Let R = R (u > be such that BR [f (s] ds = β( ε and + B R [f (s] ds = β ε (4. with ε (, arbitrarily fixed. We oint out that ε (, is fixed indeendently of u and this forces R to deend on u, so our final estimate has to be indeendent of R but may instead deend on ε. Alying Proosition 3.5 (see also Remark 3 and Hölder s inequality, we have the following estimate u (r u (r 6π ( B r [f (s] ds B r Hence, from (4. and (4.3, it follows that ( u (r u ε ( (r 3π log r4 u (r u (r ( ε 3π ( log r4 r 4 ( log r4 r 4 r 4 for a.e. < r r. (4.3 for a.e. < r r R, (4.4 for a.e. R r r. (4.5 In order to estimate the integral on the ball B R, we will distinguish between two cases: First case: < R R,

12 NADER MASMOUDI AND FEDERICA SANI Second case: < R < R. The first case is subcritical and easily estimated. In fact in the case R R, alying (4.4, we get for < r R since u (R =. Thus, recalling that we obtain Hence BR e 3π (u ( u ε ( (r + 3π log R4 r 4, (a + b ( + εa + [u (r] ε 3π log R4 r 4 + ( + ε ( + u dx π e 3π ( + ( + b a, b R, ε >, (4.6 ε ε R 4( ε R for a.e. < r R. r 3 4( ε dr = ( ( = π e 3π + ε R 4 = e 3π + ε dx B R e 3π ( + ε B R [u ] dx e 3π ( + ε u = e 3π ( + ε u. Therefore, from now on, we will always assume that < R < R. We can write e BR 3π (u e ( + u dx = \BR BR 3π (u ( + u dx B R + and again the estimate of the integral on B R \ B R is easier. In fact, alying (4.5 and (4.6, we get for R r R [u (r] ε ( + ε 3π log R4 ( r ε >. ε Consequently, for arbitrarily fixed ε >, we have BR \BR e 3π (u ( ( + u dx π e 3π + ε R 4 ε R (+ε r 3 4 ε(+ε dr. R Since ε <, we can choose ε > so that + ε < / ε and with this choice of ε > we obtain e \BR BR 3π (u ( ( + u dx π e 3π + R 4 ε R 4 ε (+ε R 4 4 ε (+ε 4 4 ε ( + ε (+ π e 3π R 4 ε R4 4 4 ε ( + ε (+ e 3π ε dx e 3π (+ ε 4 4 ε ( + ε B R \B R 4 4 ε ( + ε u. Hence to conclude the roof it remains to estimate the integral on the ball B R, assuming that < R < R and in articular u (R >.

13 ADAMS INEQUALITY WITH THE EXACT GROWTH Estimate of the integral on the ball B R with u (R >. We recall that u (R > where R > satisfies (4.. Since (4. holds, from the otimal descending growth condition (Proosition 3.6, we deduce that u [u (R ] R 4 L C (R 4 \B R e 3π ε [u (R ] ε C ε u = C ε u. (4.7 Alying (4.6, we can estimate for r R ( [u (r] ( + ε[u (r u (R ] + + [u (R ] ε >, ε hence B R e 3π(u ( + u dx [u (R ] e3π ( + ε [u (R ] e 3π [u ε (R ] [u (R ] 3π (u e B R [u (R ] dx B R e 3π (+ε[u u (R ] B R e 3π (+ε[u u (R ] dx dx rovided that ε ε ε. Therefore in view of (4.7, to conclude the roof it suffices to show that dx CR 4 (4.8 B R e 3π (+ε[u u (R ] for some ε ε ε. To this aim, we can notice that from Proosition 3.5 it follows that BR u (r u f (ξ ( B R dξ for a.e. < r < B R, 6π ξ consequently B R e 3π (+ε[u u (R ] r dx BR Now, making the change of variable r = B R e t, we get ( + +ε dx B R e B R e 3π (+ε[u u (R ] ( +ε R B R f (ξ e r ξ dξ dr. R B R f (ξ B R e t ξ dξ t dt (4.9 and the roof of (4.8 reduces to the alication of the following one-dimensional calculus inequality due to J. Moser [8]. Lemma 4. ([8]. There exists a constant c > such that for any nonnegative measurable function φ : R + R+ satisfying the following inequality holds where + + φ (s ds e F (t dt c, ( t F (t := t φ(s ds.

14 4 NADER MASMOUDI AND FEDERICA SANI Let φ(s := + ε B R f ( B R e s e s s. By construction φ and, in view of (4. and (4., + φ (s ds = B R + ε 4 = + ε 4 BR + [f ( B R e s ] e s ds = [f (r] dr ( + ε( ε rovided that ε ε ε. In articular, choosing ε = ε ε, we have that + φ (s ds. Hence, alying Lemma 4., we get where ( t F (t = t ( + ε = t + e F (t dt c, ( BR + ε φ(s ds = t BR f (ξ dξ. B R e t ξ This together with (4.9 ends the roof of (4.8. t f ( B R e s e s ds = Remark 4. It is interesting to notice that with these arguments (Proosition 3.5 and Lemma 4. we can obtain an alternative roof of Adams inequality in its original form, namely of Theorem.5, see Section 7. Remark 5. As we will see in Section 8, it is ossible to simlify the roof of Theorem.4. Besides, we oint out that one can estimate the integral on the ball B R in the case when u (R > following the longer arguments given in [] and a roer use of Proosition Sharness In order to rove that the inequality exressed in Theorem. is shar, we have to show that the following inequality g(u dx C u R 4 u W, (R 4 with u (5. fails if we relace the growth g(u := e3π u ( + u with the higher order growth g(u := eβu ( + u, (5. where either β > 3π and = or β = 3π and <. In articular, it suffices to rove that (5. fails if we consider an exonential nonlinearity of the form (5. with β = 3π and <. To this aim, for fixed <, we argue by contradiction assuming that R4 e 3π u ( + u dx C u u W, (R 4 with u. (5.3

15 ADAMS INEQUALITY WITH THE EXACT GROWTH 5 We consider the sequence of test functions {u k } k introduced in [7]: log 3π R k q r + q if r 4 R 8π R k log 8π R log k k R k u k (r := q log π log r if 4 R k < r R k η k if r > where {R k } k R +, R k + and η k is a smooth function satisfying for some R > η k ν = B η k B = η k BR =,, π log R k η k ν = BR and η k, η k are all O ( / log /R k. Then, by construction, ( ( u k = O log, u k = + O R k Let ũ k := u k u k from which we deduce that for any k. From (5.3, it follows that R4 e 3π ũ k ( + ũ k dx u k u k lim k + log R k R4 e 3π ũ log R k, log R k. k dx < +. (5.4 ( + ũ k On the other hand, for any k, we can estimate R4 e 3π ũ k ( + ũ k B dx e 3π ũ k e dx 4 Rk ( + ũ k B 3π ũ k 4 Rk ũ k dx ( u e k log ( R k log, R k since u k 3π log R k lim k + log R k which contradicts (5.4. on B 4 R k. Therefore e 3πũk dx lim (log R 4 ( + ũ k k + R k = +, 6. From Adams inequality with the exact growth condition to Adams inequality in W, (R 4 In this section we will show that our Adams inequality with the exact growth condition (Theorem. imlies Adams inequality in W, (R 4 (Theorem.7. More recisely we will rove the following result Theorem 6.. There exists a constant C > such that for any domain R 4 su (e 3π u dx C (6. and this inequality is shar. u W, (, u + u

16 6 NADER MASMOUDI AND FEDERICA SANI Since u W, := u + u + u u + u u W, (, inequality (6. imlies inequality (.7. Also, we oint out that the sharness of inequality (6. can be roved using the same arguments as in Section 5 or, equivalently, it follows from the sharness of (.7. In order to deduce inequality (6. from Theorem., we will follow the arguments introduced in [] to show that the Trudinger-Moser inequality with the exact growth condition (Theorem.4 imlies the Trudinger-Moser inequality in W, (R (Theorem.3. Firstly, we recall that as a direct consequence of our Adams inequality with the exact growth condition we have inequality (.3 and in articular (e u dx C u u W, (R 4 with u. (6. R 4 Using the ower series exansion of the exonential function and the Stirling s formula, from (6., we can deduce the existence of a constant C > such that for any integer k u k Ck u k u W, (R 4 with u. This Sobolev estimate can be extended to nonintegers alying an interolation inequality (see e.g. [9], Chaitre IV., Remarque. Hence, there exists a constant C > such that for any u C u u W, (R 4 with u. (6.3 Proof of Theorem 6.. Let u W, (R 4 \ {} be such that u + u. Then for some θ (,, we have We distinguish two cases: the case θ, and the case θ <. If θ, we define ũ := u so that u = θ and u θ. ũ and ũ, then the desired inequality follows from (.3 alied to ũ with α = 6π. If θ <, then we introduce the set and, we slit the integral into two arts dx = A 3πu (e R 4 A A := {x R 4 u(x } R 4 \A (e 3π u dx + A (e 3π u dx. The integral on R 4 \A can be easily estimated using the definition of A and recalling that e x x e x x R. Hence, to comlete the roof, we have to estimate the integral on A. To this aim, we can notice that ( (e 3π u [e 3π u ] ( dx ( + u dx + u A( dx ( 4 A e 3π u ( + u dx ( A u dx =: 4I u u,

17 ADAMS INEQUALITY WITH THE EXACT GROWTH 7 where > and we simly alied Hölder s inequality. So, choosing := θ, we have (e 3π u dx 4Iu θ u θ θ. (6.4 θ A Using the Sobolev estimate (6.3, we obtain ( θ θ u ( u θ θ θ θθ θ C = C θ. (6.5 θ θ θ On the other hand, if we define ũ := then ũ = u θ = θ θ u θ, so that and ũ = u θ, I u R4 e 3π ũ θ ( + ũ dx C θ ( θ (6.6 as a consequence of Theorem.. Combining (6.5 and (6.6 with (6.4, we deduce that (e 3π u θ θ dx ( θ θ A and the right hand side of this last inequality is uniformly bounded for < θ <. 7. Aendix A: An Alternative Proof of Adams Inequality The arguments introduced in Section 4. in order to rove an Adams tye inequality with the exact growth condition (Theorem. lead to an alternative roof of Adams inequality in its original form (Theorem.5. We recall that Adams aroach to the roblem is to exress u as the Riesz otential of its Lalacian and then aly the following result Theorem 7. ([], Theorem. There exists a constant c > such that for all f L (R 4 with suort contained in R 4 e I f(x π f c where I f is the Riesz otential of order, namely f(y I f(x := x y dy. R 4 The reason why it is convenient to write u in terms of Riesz otential is that one cannot use directly the idea of decreasing rearrangement u to treat the higher order case, because no inequality of the tye (.5 is known to hold for higher order derivatives. To avoid this roblem, Adams alied a result of R. O Neil [3] on nonincreasing rearrangements for convolution integrals, if h := g f then h (t tg (tf (t + + t g (sf (s ds. (7. Hence, a change of variables reduces the estimate to the following one-dimensional calculus inequality

18 8 NADER MASMOUDI AND FEDERICA SANI Lemma 7. ([], Lemma. Let a : R [, + R be a nonnegative measurable function such that a.e. ( + a(s, t when < s < t, and su a (s, t ds =: b < +. t> + Then there exists a constant c = c (b such that for any φ : R R satisfying φ and the following inequality holds where + + φ (s ds, e F (t dt c ( + F (t := t a(s, tφ(s ds. We oint out that the one-dimensional calculus inequality of Moser, Lemma 4., corresonds to the articular case { when < s < t, a(s, t = otherwise. Alternatively, it is ossible to rove Theorem.5 avoiding to consider the Riesz otential s reresentation formula and using the shar estimate given by Proosition 3.5 instead of O Neil s lemma (7.. Moreover, arguing in this way, it suffices to merely aly the one-dimensional inequality due to J. Moser, namely Lemma 4., instead of the refined version due to D. R. Adams, namely Lemma 7.. In fact, let R 4 be a bounded domain and let u W, W, ( be such that u L (. Our aim is to show that e 3π u dx C (7. for some constant C > indeendent of u. As a consequence of Proosition 3., if we set f := u on then [f (r] dr 4 t [f (r] dr = 4 u L ( 4. (7.3 Moreover, from Proosition 3.5 it follows that u f (ξ (r dξ for a.e. < r. 6π ξ Hence, we can estimate e 3π u dx = r e 3π [u ] dr and, making the change of variable r = e t, we get + ( e 3π u dx e Now, let φ(s := ( R f (ξ e r ξ dξ dr R f (ξ e t ξ dξ f ( e s e s s, t dt. (7.4

19 so that φ and, in view of (7.3, Defining + ADAMS INEQUALITY WITH THE EXACT GROWTH 9 φ (s ds = 4 ( t F (t := t = t ( + [f ( e s ] e s ds = 4 ( φ(s ds = t f (ξ e t ξ dξ and recalling (7.4, the roof of (7. reduces to showing that which is nothing but Lemma e F (t dt c t [f (r] dr. f ( e s e s ds = Remark 6. These arguments simlify Adams roof and at the same time allow to obtain a stronger inequality, i.e. an Adams-tye inequality for the Sobolev sace W, W, ( W, (. 8. Aendix B: An Alternative Proof of the Trudinger-Moser Inequality with the Exact Growth Condition In [], the authors roved the following exonential version of the radial Sobolev inequality. Theorem 8. ([], Theorem 3.. There exists a constant C > such that for any non-negative radially decreasing function u W, (R satisfying we have u(r > and u K for some R, K >, L (R \B R e 4π K u (R u u (R R L C (R \B R K. It is interesting to notice that this otimal descending growth condition is strong enough to lead us to reduce the roof of the Trudinger-Moser inequality with the exact growth condition (Theorem.4 to a simle alication of the Trudinger-Moser inequality in its original form (Theorem.. In order to rove that there exists a constant C > such that R e 4πu ( + u dx C u u W, (R with u, (8. the authors in [] introduced an enlightening method of roof that allows to reduce (8. to an estimate of an integral on a ball. We oint out that in Section 4 we followed exactly this method in order to reduce the roof of Theorem. to a similar estimate, but for the convenience of the reader we recall the sketch of the roof of (8. in []. Exloiting the roerties of rearrangements of functions, it suffices to show that (8. holds for any non-negative and radially decreasing function u W, (R with u. So let u be such a function and let R = R (u > be such that R + u dx = π u rr dr ε and u dx = π u rr dr ε, B R R \B R R

20 NADER MASMOUDI AND FEDERICA SANI where ε (, is arbitrarily fixed indeendently of u. Alying Hölder s inequality, we have the following estimate r ( r ( u(r u(r u r, dr u r r dr log r for a.e. < r r r r and, in view of the choice of R, we get: ( ε ( u(r u(r log r π r u(r u(r r ( ε ( log r π r for a.e. < r r R, for a.e. R r r. Since we are in the same framework as in Section 4, it is easy to see that the difficult art of the roof is to show that BR e 4πu ( + u dx C u (8. in the case when u(r >. To obtain (8., the authors in [] argue as follows. Alying the monotone convergence theorem, one can see that BR e 4πu dx C lim ( + u N N + j= e 4π ε u (r j u (r j r j, where C > and {r j } j is a monotone decreasing sequence of ositive real numbers such that r j as j + and r := R, more recisely r j := R e πj for any j. Thus, in view of the otimal descending growth condition (Theorem 8., the roof of (8. is comlete if one shows the existence of a constant C > such that for any N N j= e 4π u ε (r j u (r j r j C e 4π ε u (R u R (R. (8.3 But to succeed in the roof of this last inequality, the authors in [] need to aly a really delicate rocedure. It is interesting to notice that, to obtain (8., in fact we can avoid the assage through (8.3 merely using the otimal descending growth condition (Theorem 8. together with the Trudinger-Moser inequality in its original form (Theorem.. To do this, we start following the argument introduced by B. Ruf in [33] and we define for r R v(r := u(r u(r, so that v W, (B R and v L (B R = u L (B R ε. In this way, for arbitrarily fixed ε >, we can estimate for r R ( u (r = [v(r + u(r ] ( + εv (r + + u (R ε and thus ( + u e4π ε (R dx ( + u u (R BR e 4πu where w := + εv on B R. B R e 4πw dx, (8.4

21 ADAMS INEQUALITY WITH THE EXACT GROWTH Firstly, we can notice that in view of the choice of R and as a consequence of the otimal descending growth condition (Theorem 8. we have e 4π (+ ε u (R u (R u R L C (R \B R ε rovided that ε ε +ε. On the other hand, by construction w W, (B R and w L (B R = ( + ε v L (B R ( + ε( ε (8.5 rovided that < ε ε +ε, hence with this choice of ε we have dx CR (8.6 B R e 4πw as a consequence of the Trudinger-Moser inequality (Theorem.. In conclusion, choosing ε = ε +ε we have that both the inequalities (8.5 and (8.6 hold, and this allows us to deduce the desired estimate (8. from (8.4. Acknowledgements. N. M. was artially suorted by an NSF-DMS grant This work was done while F. S was visiting the Courant Institute, she would like to thank all the faculty and staff of CIMS for their kind hositality. References [] D. R. Adams, A shar inequality of J. Moser for higher order derivatives, Ann. of Math. ( 8 (988, No., [] S. Adachi, K. Tanaka, Trudinger tye inequalities in R N and their best exonents, Proc. Amer. Math. Soc. 8 (, No. 7, 5 57 [3] Adimurthi and O. Druet. Blow-u analysis in dimension and a shar form of Trudinger-Moser inequality. Comm. Partial Differential Equations, 9(-:95 3, 4. [4] Adimurthi and K. Tintarev. Cocomactness roerties of Moser-Trudinger functional in connection to semilinear biharmonic equations in four dimensions. Mat. Contem., 36:, 9. [5] A. Alvino, V. Ferone, G. Trombetti, Moser-tye inequalities in Lorentz saces, Potential Anal. 5 (996, No. 3, [6] H. Bahouri, M. Majdoub, and N. Masmoudi. On the lack of comactness in the D critical Sobolev embedding. J. Funct. Anal., 6(:8 5,. [7] H. Bahouri, M. Majdoub, and N. Masmoudi. Lack of comactness in the d critical sobolev embedding, the general case. COMPTES RENDUS MATHEMATIQUE, 35(3-4:77 8,. [8] C. Bennett, R. Sharley, Interolation of oerators, Pure and Alied Mathematics, 9. Academic Press, Inc., Boston, MA, 988, xiv+469. [9] H. Brezis, Analyse fonctionnelle. Théorie et alications. Collection Mathématiques Aliquées our la Maîtrise, Masson, Paris, 983, xiv+34. [] L. Carleson and A. Chang, on the existence of an extremal function for an inequality of J. Moser, Bulletin des Sciences Mathématiques, (986, 3 7. [] A. Cianchi, A shar embedding theorem for Orlicz-Sobolev saces, Indiana Univ. Math. J. 45 (996, No., [] A. Cianchi, Otimal Orlicz-Sobolev embeddings, Rev. Mat. Iberoamericana (4, No., [3] A. Cianchi, Moser-Trudinger inequalities without boundary conditions and isoerimetric roblems, Indiana Univ. Math. J. 54 (5, No. 3, [4] G. Chiti, Rearrangements of functions and convergence in Orlicz saces, Alicable Anal. 9 (979, No., 3 7 [5] D. G. De Figueiredo, J. M. Do Ó, B. Ruf, Ellitic equations and systems with critical Trudinger-Moser nonlinearities, Discrete Contin. Dyn. Syst. 3 (, No., [6] D. G. de Figueiredo and B. Ruf. Existence and non-existence of radial solutions for ellitic equations with critical exonent in R. Comm. Pure Al. Math., 48(6: , 995. [7] L. Fontana, Shar borderline Sobolev inequalities on comact Riemannian manifolds, Comment. Math. Helv. 68 (993, No. 3,

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