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1 (This is a sample cover image for this issue. The actual cover is not yet available at this time.) This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier s archiving and manuscript policies are encouraged to visit:

Available online at www.sciencedirect.com Advances in Mathematics 30 (0) 97 978 www.elsevier.

2 Available online at Advances in Mathematics 30 (0) Propagation of holomorphic extendibility and non-hypoellipticity of the -Neumann problem in an exponentially degenerate boundary Luca Baracco, Tran Vu Khanh, Giuseppe Zampieri Dipartimento di Matematica, Università di Padova, via Trieste 63, 35 Padova, Italy Received 5 May 0; accepted 5 March 0 Communicated by Takahiro Kawai Abstract We prove that CR lines in an exponentially degenerate boundary are propagators of holomorphic extendibility. This explains, in the context of the CR geometry, why in this situation the induced Kohn Laplacian b is not hypoelliptic (Christ (000) []). c 0 Published by Elsevier Inc. MSC: 3F0; 3F0; 3N5; 3T5 Keywords: -Neumann problem; Propagation of holomorphic extendibility; Analytic discs with singular boundary. Introduction Kohn noticed in [8] that analytic discs in the boundary of a pseudoconvex domain Ω C n prevent from the C -hypoellipticity of the -Neumann problem: the canonical solution is not smooth exactly at the boundary points where the datum is. On the other hand, it has been explained by Hanges and Treves in [5] that discs sitting in Ω are propagators of holomorphic extendibility from Ω across Ω. Thus, propagation and hypoellipticity appear to be in contrast one to another. Christ proved in [3] that on the hypersurface in C defined by Corresponding author. addresses: baracco@math.unipd.it (L. Baracco), khanh@math.unipd.it (T.V. Khanh), zampieri@math.unipd.it (G. Zampieri) /$ - see front matter c 0 Published by Elsevier Inc. doi:0.06/j.aim

3 x = e y s, L. Baracco et al. / Advances in Mathematics 30 (0) one does not have hypoellipticity for the induced Kohn Laplacian b when s. Note that for s < this is hypoelliptic as well as the complex Laplacian : in fact, in this case, one has superlogarithmic estimates which are sufficient for hypoellipticity of b and according to [0, Theorems.6 and 8.3 respectively] (cf. also []). (In [0], hypoellipticity of is deduced from microlocal hypoellipticity of b but it could be proved in a direct way as in [7, Theorem.].) It is worth remarking that superlogarithmicity does not entirily rule hypoellipticity. The pseudoconvex domain whose boundary is defined by the same equation as (.) but with y replaced by z, that is, x = e z s, has the same range s < for superlogarithmic estimates and, nonetheless, there always is hypoellipticity, for any value of s both for b (Kohn [9, Main Theorem in Section 3]) and for [6, Theorem.]. Here the matter is of a genuinely geometric type: there are no curves running in complex tangential directions along which the manifold is flat and which are, therefore, possible propagators (see also [4]). Coming back to the tubolar domain with boundary (.), we show here that the x -axis ia a propagator of holomorphic extendibility when s. Our result, Theorem.6 below, applies in fact to more general domains, not necessarily rigid. More precisely, our argument consists in exhibiting a family of discs squeezed along the x -axis, singular at x = 0 and with boundary in Ω apart from x = where they enter in x < 0. We show that they point down at x = 0 if and only if s. These discs propagate the extendibility down; as it has already been said, the proof does not go through when s <. There is no surprise about it because, for s <, there cannot be propagation of smoothness at the boundary. In fact, let χ = χ(x ) be C and satisfy χ 0 at and χ at 0, and consider the -closed form f := χ(x ) z. If s < the -Neumann problem is hypoelliptic and hence the equation u = f has a solution u in Ω which is smooth at 0 and ; thus the difference u χ(x ) z is holomorphic in Ω, singular at x = 0 but smooth at x =.. Squeezing discs along lines Analytic discs with Lipschitz boundary have been studied in several papers such as, for instance, [, 4]; we introduce here discs with logarithmic singularity at the boundary. In the standard disc of the complex plane C with variable τ = re iθ for θ [0, π] or θ [ π, +π] according to the need, we consider the family of holomorphic mappings ( = discs) depending on a small real parameter : ϕ (τ) =. log 4 τ These discs are squeezed along the interval 0, log 4 as 0 with the points + and interchanged with the left and right bounds respectively and they are singular at τ =. Moreover, the most of their mass concentrates at τ =. We have ϕ (τ) log τ, τ, τ close to. With the notation τ = e iθ we also have τ sin θ arg = arctg cos θ (.)

4 974 L. Baracco et al. / Advances in Mathematics 30 (0) and finally, at τ = log Im ϕ = arctg cos θ sin θ sin θ = arctg cotg θ 4 τ log 4 τ = log 4 + log 4 = π θ + π θ π θ + O() log τ Thus, for fixed as in next proposition, we have Proposition.. We have (i) e ϕ s = O ( τ) for s >, (ii) e Im ϕ s = O ( τ) for s >. Proof. As for (i), we have to notice that e ϕ s e s log τ s τ + log + O(). Im ϕ log τ at τ =. = τ s log τ s = O ( τ ) for s >. As for (ii), this follows from e Im ϕ s e s log τ s = τ s log τ s = O ( τ ) for s >. This concludes the proof of the proposition. In particular, the two functions in the statement of the proposition are C, and thus also C,β, at τ = for s > and s > in the two respective cases. We have a basic result about composition of ϕ with flat functions more general than e z s or e y s. For this, let h η (z, y ), (z, y ) C R, be a function sufficiently smooth depending on a parameter η. Proposition.. Let η h η, R C 3 be C k and satisfy η h η 0 in a neighborhood of z = 0. Assume that all (mixed) derivatives up to order in τ and k in η are O e y s for s. Then, the function (η, v) h η(ϕ, v) has the properties: (i) it sends R C,β C,β, (ii) it is C k with respect to η, (iii) it is differentiable with respect to v at v = 0 and its differential has small norm.

5 L. Baracco et al. / Advances in Mathematics 30 (0) Proof. (i): For a function g of a real variable t, the assumptions g = O e t s, g = O e t s, imply g( Im ϕ ) = O ( τ ) when s > (Proposition.), τ Im ϕ log 3 ( τ) τ, τ (g( Im ϕ )) = g (Im ϕ ) log 3 ( τ) τ = O ( τ ) (again, Proposition.). This concludes the proof of (i). (ii): Since η h η 0 when ϕ is singular, then the C k dependence of h η (ϕ, v) on η is a standard fact: if η g η, R C (R) is C and σ C,β, then η g η (σ ), R C,β is C k. (iii): It is convenient to use a more general setting. Thus, let g η be C 3. Then, the mapping G η : C,β C,β, v g η (v) is C at v = 0 and its differential G η satisfies G η v=0 L(C,β,C,β ) g η C 3. Note that, in our application, g η = h η (ϕ, ); thus g η C 3 is small near v = 0. Now, we can set up a Bishop s equation in the unknown v C,β v T (h η (ϕ, v)) = 0, (.) where T is the Hilbert transform normalized by taking value 0 at τ =. We rewrite the Eq. (.) in the functional space C,β as G η (v) = 0. By (iii) of Proposition., we have v=0 id L(C,β,C,β ) h η(im ϕ, 0) C 3. (.) G η By the implicit function theorem, we readily get Corollary.3. For small η, the Eq. (.) has a unique solution v C,β and this depends in a C k -fashion on η. We write v = v,η for the solution of (.) and also write u = T v and u = u,η. We also denote by A = A η the disc A = (ϕ, u + iv). When only dependence on is relevant, we write v = v, u = u and A = A. Note that under our assumption h = O e y s we have u = O e y s. For τ = re iθ and for a function in C 0 ( ), such as u, the harmonic extension of u from to, that we still denote by u, has a radial derivative which is given by r u τ= = π π u dθ, (.3) cos θ where the integral is taken in the sense of the principal value. We first show that the values of θ for which ϕ is not contained in the -neighborhood of log 4 is very small.

6 976 L. Baracco et al. / Advances in Mathematics 30 (0) Lemma.4. We have the inclusion θ : ϕ (θ) log > 4 In other words, the whole circle e iθ, θ [ π, π], except from θ via ϕ into the -neighborhood of ϕ ( ). Proof. We have log 4 τ + log 4 = e, e. (.4) log τ log 4 + log 4 log τ Now, the denominator is. Hence, the set in the left of (.4) is contained in τ : τ log >, which is in turn contained in τ = e iθ : θ < e.. e, e, is mapped By renaming, we neglect the irrelevant constant. Taking into account of Lemma.4, we decompose the integration in (.3) as π e r u = = π e e π π e We approximate, near θ = 0, cos θ by θ and define. F := e e e Im ϕ s θ dθ. Proposition.5. (i) For s, we have lim 0 F = 0. (ii) For s <, we have lim 0 F = +. Proof. (i): Note that τ θ on the unit circle near τ =. We have F e e e e e e e Im ϕ θ dθ (since s ) e log 4 + log 4 log θ+ log θ e log 4 log θ θ θ dθ dθ e e This proves (i). dθ e.

7 L. Baracco et al. / Advances in Mathematics 30 (0) (ii): We assume now s < and also suppose, without loss of generality, s >. By using the substitution log θ = t, we get F = e 0 + e s log s θ log θ dθ e s t s +t dt. Now, we remark that s t s + t > 0 if and only if t < + s s = s dt s where the last conclusion follows from +, s s >. s s. Thus, Theorem.6. Let Ω C be a domain defined by x > h(z, y ) with h satisfying z j h = O e y s for j =,. In particular, the boundary contains the real lines L defined by y = 0, x = 0 and y = const. Assume s ; then each line L is a propagator of holomorphic extendibility. Namely, if f hol(ω) extends to a full neighborhood of a point z L, then it also extends to a neighborhood of any other point z o L. log Proof. We may assume that z o = (0, 0), z =, 0 and that f extends to B (z ), the -neighborhood of z. Recall that the points z o and z correspond to τ = and τ = respectively under the map ϕ. We also remark that ϕ [ π, +π] \ e, e B (z ). We deform h by allowing a -bump at z. Thus, we define 4 h = h outside B (z ), on B (z ), (.5) continued smoothly on B (z ) \ B (z ). Attach a disc A = (ϕ, ũ + iṽ ) over ϕ to the hypersurface defined by x = h according to Proposition.; we have r ũ = π π e e Since s, then e ϵ r ũ > 0. ũ cos θ dθ e e Im ϕ s π cos θ dθ + e e Im ϕ s cos θ cos θ dθ. dθ 0 according to Proposition.5(i); thus

8 978 L. Baracco et al. / Advances in Mathematics 30 (0) In other terms, ũ points down at τ = ; in particular, ũ ( r) < 0 for r < close to r =. (.6) We fix for which (.6) is fulfilled and do not keep track of it in the notations which follow. If we replace ϕ by ϵ + ϕ, for a fixed ϵ, and substitute in (.5) by η for any η [, ], we get a family of discs {A η } η = {( ϵ + ϕ, ũ η + iṽ η )} η such that A η Ω B (z ), Ω (.7) A η contains a neighborhood of 0. η At this point, we move up in x -direction our discs so that A η Ω B (z ) still keeping the second of (.7). By Cauchy s formula, f extends from the boundaries A η to the full discs A η whose union is a neighborhood of 0. Acknowledgments We are grateful to Alexander E. Tumanov for important advice. We wish also to dedicate the result of this paper to Joseph J. Kohn in his 80-th birthday. References [] L. Baracco, D. Zaitsev, G. Zampieri, Rays condition and extension of CR functions from manifolds of higher type, J. Anal. Math. 0 (007) 95. [] M. Christ, Hypoellipticity: geometrization and speculation, in: Progress in Math., vol. 88, Birkhäuser, Basel, 000, pp [3] M. Christ, Hypoellipticity of the Kohn Laplacian for three-dimensional tubular Cauchy Riemann structures, J. Inst. Math. Jussieu (00) [4] V.S. Fedii, On a criterion for hypoellipticity, Math. USSR Sb. 4 (97) [5] J. Hanges, F. Treves, Propagation of holomorphic extandibility of CR functions, Math. Ann. 63 () (983) [6] T.V. Khanh, G. Zampieri, Hypoellipticity of the -Neumann problem in an exponentially degenerate boundary, J. Funct. Anal. (00). [7] T.V. Khanh, G. Zampieri, Necessary geometric and analytic conditions for general estimates in the -Neumann problem, Invent. Math. (0). [8] J.J. Kohn, Methods of partial differential equations in complex analysis, Proc. Sympos. Pure Math. 30 (977) [9] J.J. Kohn, Hypoellipticity at points of infinite type, Contemp. Math. 5 (000) [0] J.J. Kohn, Superlogarithmic estimates on pseudoconvex domains and CR manifolds, Ann. of Math. 56 (00) [] Y. Morimoto, A criterion for hypoellipticity of second order differential operators, Osaka J. Math. 4 (987) [] A. Tumanov, Thin discs and a Morera theorem for CR functions, Math. Z. 6 () (997) [3] A. Tumanov, Analytic discs and the extendibility of CR functions, in: Integral Geometry, Radon Transforms and Complex Analysis (Venice, 996), in: Lecture Notes in Math., vol. 684, Springer, Berlin, 998, pp [4] D. Zaitsev, G. Zampieri, Extension of CR functions on wedges, Math. Ann. 36 (4) (003)

Loss of derivatives in the infinite type

University of Wollongong Research Online Faculty of Engineering and Information Sciences - Papers: Part A Faculty of Engineering and Information Sciences 015 Loss of derivatives in the infinite type Tran