ON SOME GENERALISATIONS OF MEYERS-SERRIN THEOREM SU ALCUNE GENERALIZZAZIONI DEL TEOREMA DI MEYERS-SERRIN

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1 ON SOME GENERALISATIONS OF MEYERS-SERRIN THEOREM SU ALCUNE GENERALIZZAZIONI DEL TEOREMA DI MEYERS-SERRIN DAVIDE GUIDETTI, BATU GÜNEYSU, DIEGO PALLARA Abstract. We present a generalisation of Meyers-Serrin theorem, in which we replace the standard weak derivatives in open subsets of R m with finite families of linear differential operators defined on smooth sections of vector bundles on a (not necessarily compact) manifold X. Sunto. Presentiamo una generalizzazione del teorema di Meyers-Serrin, in cui sostituiamo le derivate deboli in sottoinsiemi aperti di R m con famiglie finite di operatori differenziali lineari, definiti su sezioni regolari di fibrati vettoriali su una varietà (non necessariamente compatta) X. 21 MSC. Primary 58J99; Secondary 46F99. Keywords. Meyers-Serrin theorem, Differential operators on manifolds, Vector bundles. Let us denote by W m,p (Ω) the class of elements f in L p (Ω) whose derivatives α f of order less than or equal to m (in the sense of distributions) belong to L p (Ω). W m,p (Ω) becomes a Banach space if it is equipped with its natural norm (1) f W m,p (Ω) := ( α f p L p (Ω) )1/p. α m Let us also denote by H m,p (Ω) the closure of C (Ω) W m,p (Ω) in W m,p (Ω). Then, in a quite famous paper ([7]), N.G. Meyers and J. Serrin proved that, in case 1 p <, H m,p (Ω) = W m,p (Ω). Bruno Pini Mathematical Analysis Seminar, Vol. 1 (215) pp Dipartimento di Matematica, Università di Bologna ISSN

2 On some generalisations of Meyers-Serrin theorem 117 This identity holds for every open subset Ω of R n, regardless of the regularity of its boundary. Observe that, as the inclusion H m,p (Ω) W m,p (Ω) is obvious, Meyers-Serrin theorem can be also stated by saying that C (Ω) W m,p (Ω) is dense in W m,p (Ω). The main aim of this seminar is to illustrate a generalisation of this result that we have recently obtained (see [5]), which is applicable to many different situations. A first possible generalisation of the class of spaces W m,p (Ω) could be obtained in the following way: take a finite family of linear partial differential operators P = {P 1,..., P s } with smooth coefficients in Ω. Then we could consider the class W P,p (Ω) of elements f in L p (Ω) such that P j f (in the sense of distributions) belongs to L p (Ω) for each j {1,..., s}. It is straightforward to check that W P,p (Ω) is a Banach space if it is equipped with the norm f W P,p (Ω) := ( f p L p (Ω) + s P j f p L p (Ω) )1/p. Analogous spaces and related Meyers-Serrin theorems, were considered in connection to the so called Lavrentiev phenomenon (see, for example, [3], [4]). j=1 Sobolev spaces constructed as domains of (even fractional) powers of sublaplacians for stratified Lie groups were also studied in detain in [2]. More generally, we could consider also differential operators in smooth manifolds, not only defined in spaces of scalar functions, but also defined on sections of vector bundles. So we begin by recalling some well known definitions (see, for example, [8]): let X be a smooth abstract m dimensional manifold without boundary. We always assume that its topology has a countable basis of open sets, in order to have at disposal the tool of partitions of unity. Suppose that (E, π E ) and (F, π F ) are vector bundles on X. This means that: (i) E is a smooth manifold, π E C (E; X); (ii) E x := π 1 ({x}) has the structure of an l E dimensional vector space on C (l E N ); (iii) x X there exists a al chart (Φ, U) in X around x (U open in X, Φ : U Φ(U) R m diffeomorphism) and a smooth diffeomorphism Ψ E : π 1 E (U) Φ(U) Cl E, such that, for x U, if v E x, Ψ E (v) = (Φ(x), L E (x)v)

3 118 Guidetti, Güneysu, Pallara with L E (x) linear isomorphism between E x and C l E. We denote by Γ C (X, E) the class of smooth sections with values in E, that is, the class of elements f C (X, E) such that f(x) E x x X, and with Γ C (X, E) the class of elements in Γ C (X, E) with compact support. Given φ : Φ(U) C l E, we can consider the following section f φ : U E: (2) f φ (x) = L E (x) 1 (φ(φ(x))). If φ C (Φ(U); C l E ), then f φ, extended with zero outside U, belongs to Γ C (X, E). Let P be a linear mapping from Γ C (X, E) to Γ C (X, F ). We define the following operator P Φ in C (Φ(U), C l E ), with values in C (Φ(U), C l F ): (3) P Φ (φ)(y) := L F (Φ 1 (y))p (f φ )(Φ 1 (y)). We say that P is a differential operator of order less than or equal to k if, for every system (Φ, U, L E, L F ), P Φ is a differential operator in Φ(U), that is, P Φ (φ)(y) = A α (y)dy α φ(y), α k with A α (y) l F l E matrix, depending smoothly on y. We say that P is elliptic of order k if P Φ is elliptic for every (Φ, U), that is, if l E = l F and the matrix α =k ξα A α (y) is invertible y Φ(U), ξ R m \ {}. Suppose now that the manifold X is equipped with a smooth density. We recall that this means that a positive Borel measure is fixed in X, so that, (Φ, U) al chart in X there exist Φ C (Φ(U)), positive, such that, if A is a Borel subset of U (A) = Φ (y)dy. Φ(A) Let (E, π E ) be a vector bundle on X. We fix a Hermitian structure (smoothly varying inner product (, ) Ex in each E x ). Let p [1, ). We denote by Γ L p (X, E) the class of measurable section of E, i.e., measurable functions f : X E, such that f(x) E x for almost every x and f p Γ p L (X,E) := f(x) p E x d(x) <. As usual, we do not distinguish sections coinciding almost everywhere. X

4 On some generalisations of Meyers-Serrin theorem 119 Let P be a differential operator between the vector bundles (E, π E ) and (F, π F ), both equipped with Hermitian structures (of course, this means that P maps Γ C (X, E) into Γ C (X, F )). Then one can show (see [9], Proposition IV.2.8) that there exists a unique differential operator P (the adjoint of P ) between (F, π F ) and (E, π E ) such that, f Γ C (X, E), g Γ C (X, F ) (P f(x), g(x)) Fx (dx) = X X (f(x), P g(x)) Ex (dx). By simple computation, one can draw the following al expression for P : if f = f φ as in (2) and P Φ is the operator defined in (3), one has, for x U, (4) P g(x) = with P Φ adjoint of P Φ, 1 Φ (Φ(x)) L E(x) P Φ(Mψ)(Φ(x)), (5) ψ(y) = L F (Φ 1 (y))g(φ 1 (y)), (y Φ(U)), (6) M(y) = Φ (y)l F (Φ 1 (y)) 1 L F (Φ 1 (y)) 1. This suggests the possibility of defining P f in cases when f is not smooth. So we denote by Γ L 1 (X, E) the class of ally summable sections. It is not difficult to see that this class does not depend on, as far as we limit ourselves to considering smooth densities. We can give the following definition: Definition 1. Let P be a differential operator between the vector bundles (E, π E ) and (F, π F ), both equipped with Hermitian structures. Let f Γ L 1 (X, E) and h Γ L 1 (X, F ). We write P f = h if, g Γ C (X, F ), (h(x), g(x)) Fx (dx) = One can show that, although P definition of P f is independent of them. X X (f(x), P g(x)) Ex (dx). depends on the Hermitian structures and, this Lemma 1. Using the notation (2) and (3), if f = f φ Γ L p (U, E) and P f Γ L p (U, E) in the sense of Definition 1, P Φ φ L p (Φ(U), Cl F ), where, of course, P Φ φ has a meaning in the sense of distributions.

5 12 Guidetti, Güneysu, Pallara Proof. Let g Γ C (U, F ). We set ψ(y) := L F (Φ 1 (y))g(φ 1 (y)). Of course, ψ C (Φ(U), C l F ). Employing (4), we have U (f(x), P g(x)) Ex (dx) = Φ(U) (φ(y), P Φ (Mψ)(y)) C l E dy = (M(y) P Φ φ(y), ψ(y)), with (, ) standing for the duality (D (Φ(U), C l F ), C (Φ(U), C l F )). On the other hand, (P f(x), g(x)) Fx (dx) = ( Φ (y)l F (Φ 1 (y)) 1 P f(φ 1 (y)), ψ(y)) C l F dy, so that U Φ(U) P Φ φ(y) = Φ (y)m(y) 1 L F (Φ 1 (y)) 1 P f(φ 1 (y)) L p (Φ(U); Cl F ). Finally, let f Γ L 1 (X, E). We write f Γ W n,p (X, E) if, for all systems (Φ, U, L E), y L E (Φ 1 (y))f(φ 1 (y)) is in W n,p (φ(u); Cl E ). This definition also does not depend on and the Hermitian structures. After these preliminaries, we introduce our assumptions: (A1) X is a smooth m dimensional manifold without boundary, with a countable basis of open sets. is a fixed smooth density in X. (A2) (E, π E ), (F 1, π F1 ),..., (F s, π Fs ) are vector bundles on X. Each of them is equipped with a Hermitian structure. For simplicity we write E, F 1,..., F s instead of (E, π E ), (F 1, π F1 ),..., (F s, π Fs ). (A3) For each j {1,..., s} P j is a differential operator between E and F j. Suppose that (A1)-(A3) are satisfied. We define, p [1, ), (7) Γ W P,p(X, E) := {f Γ L p (X, E) : P j f Γ L p (X, F j ) j {1,..., s}}.

6 On some generalisations of Meyers-Serrin theorem 121 It is straightforward to check that Γ W P,p(X, E) becomes a Banach space if it is equipped with the norm (8) f ΓW P,p(X,E) := ( f p Γ p L (X,E) + s P j f p Γ p L (X,F j ) )1/p. Now we are able to state the following generalisations of Meyers-Serrin theorem: j=1 Theorem 1. Suppose that (A1)-(A3) are fulfilled and that the operators P 1,..., P s are of order less than or equal to k, with k N. Γ C (X, E) W P,p (X, E) is dense in W P,p (X, E). Then, if W P,p (X, E) Γ W k 1,p(X, E), We show some examples and applications. The first result immediately follows from Theorem 1. Corollary 1. Suppose that (A1)-(A3) hold and the operators P 1,..., P s are of order not exceeding one. Then Γ C (X, E) W P,p (X, E) is dense in W P,p (X, E). A less obvious result is the following: Proposition 1. Suppose that (A1)-(A3) hold. Moreover, P s is elliptic of order k and P 1,..., P s 1 are of order less than or equal to k + 1 in case p (1, ), of order less than or equal to k in case p = 1. Then Γ C (X, E) W P,p (X, E) is dense in W P,p (X, E). Proof. If p (1, ), it is a consequence of standard elliptic theory in R m and Lemma 1 that from u Γ L p (X, E) and P s u Γ L p (X, E) it follows u Γ W k,p(x, E) (see [8], Theorem 1.3.6). In case p = 1, it is proved in [5] that, if P is an l l elliptic system of order k in R m, u L 1 (Rm ; C l ) and P u L 1 (Rm ; C l ), then φ C (R m ) φu belongs to the Besov space B k 1, (R m ; C l ), which contains W k,1 (R m ; C l ) and is contained in W k 1,1 (R m ; C l 2 ). As, if P s is elliptic, P sφ is elliptic for every system (Φ, U, L E, L Fs ), we deduce, again from Lemma 1, that, from u Γ L 1 (X, E) and P s u Γ L 1 (X, E), it follows u Γ W k 1,1(X, E), so that Theorem 1 is applicable. Remark 1. It is quite clear that it cannot be expected that Theorem 1 holds for p =. Nevertheless, one can replace Γ L (X, E) with Γ C L (X, E), the Banach space of bounded

7 122 Guidetti, Güneysu, Pallara continuous sections. The crucial condition becomes {u Γ C L (X, E) : P j u Γ C L (X, F j ) j {1,..., s}} Γ C k 1(X, E). Proposition 1, in a weak form, analogous to the one valid in case p = 1, can be extended to this case, because one can show that, if P s is elliptic of order k, from u Γ C L (X, E) and P s u Γ C L (X, E), it follows u Γ C k 1(X, E) Before considering our last example (Sobolev spaces in not necessarily compact Riemannian manifolds), we want to discuss the optimality of Theorem 1. First of all, it is clear that the condition W P,p (X, E) Γ W k 1,p(X, E) is not necessary to get the conclusion: for example, let X = R m, the standard Lebesgue measure, E = R m C the trivial bundle and assume that the operators P 1,..., P s have constant coefficients and (for simplicity) are defined in C (R m ). Then Γ C (X, E) W P,p (X, E) is dense in W P,p (X, E). To show this, it suffices to consider the usual regularisation procedure: fix ω C (R m ), with R m ω(x)dx = 1 and, set, for ɛ >, ω ɛ (x) := ɛ m ω( x ɛ ). Suppose, for simplicity, that u W P,p (X, C). Then, as 1 p <, the convolutions (ω ɛ u) ɛ> converge to u in L p (R m ; C) (as ɛ ) and, for each j {1,..., s}, (P j (ω ɛ u)) ɛ> = (ω ɛ P j u) ɛ> converge to P j u in L p (R m ; C), regardless of the condition W P,p (X, C) Γ W k 1,p(X, C). Neverthless, this condition is, in some sense, optimal, because it may happen that W P,p (X, C) Γ W k 2,p (X, C) and Γ C (X, E) W P,p (X, E) is not dense in W P,p (X, E). In fact let us consider the following example. Example 1. We consider the following operator P in R, equipped with the Lebesgue measure: P u(x) = xu (3) (x) + (x 1)u (2) (x). We set P := {P }. We take p (1, ), write W P,p instead of W P,p (R, C), and start by observing that W P,p = {u L p (R) : xu W 1,p (R)}. In fact, P = (1 ) x 2. Let u W P,p and set v = xu. Then v is a tempered distribution and v v = P u L p (R). Employing the Fourier transform F, we see that

8 On some generalisations of Meyers-Serrin theorem 123 v = F 1 ((1 iξ) 1 FP u), which belongs to W 1,p (R) because P u L p (R), on account of the formula We set f := P u. We have that v(x) = x e x y P u(y)dy. x u (x) = x 1 v(x) = x 1 v() + x 1 v (y)dy = x 1 v() + h(x), x R \ {}, with h(x) = x 1 x v (y)dy. By Hardy s inequality, h L p (R). It follows that (9) u (x) = v()p.v.( 1 ) + h(x) + k(x), x with k distribution with support in {}. From xu = v it follows xk(x) =, which implies k(x) = cδ(x). So from (9) we deduce u (x) = v() ln( x ) + x h(y)dy + ch(x) + const, where we have denoted by H(x) the Heaviside function. We infer that u L p (R), so that W P,p W 1,p (R). Nevertheless, W P,p is not contained in W 2,p (R). In fact, set u(x) := φ(x)x ln( x ), with φ C (R), φ(x) = 1 in some neighbourhood of. It is easily seen that u W P,p, as (1) xu (x) = 1, in some neighbourhood of. u does not belong to W 2,p (R), because u (x) = p.v.( 1 x ) in some neighbourhood of. Now we check that C (R) W P,p is not dense in W P,p. We argue by contradiction: we suppose that there exists a sequence (u k ) k N in C (R) W P,p, such that u k u L p (R) + P u k P u L p (R) (k ). We set v := xu, v k := xu k. Then, as v k = F 1 ((1 iξ) 1 FP u k ), (v k ) k N converges to v in W 1,p (R), implying that (v k ()) k N converges to v(). However, evidently, v k () =, while, by (1), v() = 1. We have reached a contradiction.

9 124 Guidetti, Güneysu, Pallara We conclude by extending the theorem of Meyers-Serrin to (not necessarily compact) Riemannian manifolds. We begin by defining the Sobolev space W m,p (X), with X Riemannian manifold. We take the measure induced by the metric in X and denote by T (X) the tangent bundle. We recall that the Riemannian (or Levi-Civita) connection is the unique connection in X, such that, X, Y, Z Γ C (X, T (X)), D X (Y ) D Y (X) = [X, Y ], with [X, Y ] commutator of X and Y, and X(Y Z) = D X (Y ) Z + Y D X (Z), with (X Y )(x) := X(x) x Y (x), x X and x is, of course, the inner product in T x (X). In al coordinates, if X = m j=1 X j x j D X (Y ) = i,j=1 and Y = m j=1 Y j x j, we have Y j X i + x i x j i,j,k=1 Γ k ijx i Y j. x k Here D x i ( x j ) = k=1 Γ k ij x k where the functions Γ k ij are the so called Christoffel symbols. Given X Γ C (X, T (X)), we can define D X (α) α Γ C (X, T j (X)) in a unique way so that the following conditions are satisfied: (I) D X (α) Γ C (X, T j (X)) α Γ C (X, T j (X)); (II) if f C (X), D X (f) = X(f); (III) if α Γ C (X, T (X)) and Y Γ C (X, T (X)), (D X (α), Y ) = X((α, Y )) (α, D X (Y )), with (α, Y )(x) := (α(x), Y (x)) C (X); (IV) if α Γ C (X, T j (X)) and β Γ C (X, T k (X)), D X (α β) = D X (α) β + α D X (β).

10 On some generalisations of Meyers-Serrin theorem 125 So, we can easily deduce that, if X Γ C (X, T (X)), α Γ C (X, T (X)) and, in al coordinates, X = m i=1 X i x i, α = m j=1 α jdx j, we have, again in al coordinates, (11) D X (α) = i,j=1 X i α j x i dx j i,j,k=1 Γ k ijx i α k dx j. Now we introduce the operators j (j N), each of which is a differential operator from C (X) to Γ C (X, T j(x) ). If f C (X), we define j f recursively: given j f Γ C (X, T j(x) ), we define j+1 f Γ C (X, T (j+1)(x) ) setting, X 1, X 2,... X j+1 Γ C (X, T (X)), (12) j+1 f(x 1, X 2,..., X j+1 ) := D X 1( j f)(x 2,..., X j+1 ). So, for example, we have 1 f = df = m f j=1 x j dx j (in al coordinates) and, from (11), 2 f = i,j=1 2 f x i x j dx i dx j i,j,k=1 Γ k ij f dx i dx j. x k In general one can easily see that, in al coordinates, (13) j j f f = ( + P i1,...,i x i1... dx j (f))dx i1 dx ij, ij i 1,...,i j =1 with P i1,...,i j (f) operator of order strict less than j. An outline of this theory can be found in [6]. Now we equip each bundle T j (X) with a Hermitian structure in a natural way: if α, β T x (X), we set (α, β) x := (V (α), V (β)) x, with V (α) T x (X) such that (α, w) = (V (α), w) x, w T x (X). Next, we define (α, β) x, with α, β Tx j (X), employing the fact that, given linear spaces X and Y, equipped with inner products (, ) X and (, ) Y, there is a unique inner product (, ) in X Y such that, α 1, α 2 X, β 1, β 2 Y See for this [1], Chapter 1.3. ((α 1 β 1 ), (α 2 β 2 )) = (α 1, α 2 ) X (β 1, β 2 ) Y. Now we are able to define the Sobolev spaces W s,p (X):

11 126 Guidetti, Güneysu, Pallara Definition 2. Let X be a Riemannian manifold, equipped with the measure induced by the Riemannian structure. For each j N, we equip each space T (X) j with the Hermitian structures previously described and consider the differential operators j from C (X) to Γ C (X, T (X) j ). Let p [1, ) and s N. We set W s,p (X) := {f L p (X) : j {1,..., s} j f Γ L p (X, T (X) j )}. W s,p is a Banach space with the norm s (14) f W s,p (X) := ( f p + j f p L p (X) Γ p L (X,T (X) j ) )1/p. j=1 From Theorem 1, we easily deduce the following generalisation of Meyers-Serrin theorem: Theorem 2. Let W s,p (X) be the spaces described in Definition 2. Then C (X) W s,p (X) is dense in W s,p (X). Proof. Let (U, Φ) be a al chart. If x U, j N and F = T (X) j, we can take L F (x)( α i1,...,i j (x)dx i1... dx ij ) = (α i1,...,i j (x)) 1 i1,...,i j m. i 1,...,i j =1 It follows that, if y Φ(U), employing the usual notation (2)-(3), we have j Φ (φ)(y) = ( j f φ x i1...dx ij (Φ 1 (y)) + P i1,...,i j (f φ )(Φ 1 (y))) 1 i1,...,i j m j φ = ( r i1...dr ij (y) + Q i1,...,i j (φ)(y))) 1 i1,...,i j m, with each operator Q i1,...,i j of degree less than j. We deduce from Lemma 1 that, if f φ W j 1,p (X), φ W j,p (Φ(U); Cmj ). We conclude that each space W s,p (X) is contained in W s,p (X). So we can apply Theorem 1. Remark 2. The conclusions of Theorem 2 continue to hold if we replace, in the previous construction of Sobolev spaces W s,p (X), the Levi-Civita connection with any other smooth connection: of course, we obtain a different Sobolev space, for which we have a corresponding Meyers-Serrin type theorem.

12 On some generalisations of Meyers-Serrin theorem 127 References [1] A.V. Balakrishnan, Applied Functional Analysis, Applications of Mathematics 3, Springer-Verlag (1976). [2] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), [3] B. Franchi, R. Serapioni, F. Serra Cassano, Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields, Houston J. Math. 22 (1996), [4] B. Franchi, R. Serapioni, F. Serra Cassano, Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields, Boll. Un. Mat. It. B 7 (1997), [5] D. Guidetti, B. Güneysu, D. Pallara, L 1 elliptic regularity and H = W on the whole L p scale on arbitrary manifolds, Preprint. [6] E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Lecture Notes 5, American Mathematical Society Courant Institute of Mathematical Sciences (2). [7] N.G. Meyers, J. Serrin, H = W, Proc. Nat. Acad. Sci. U.S.A 51 (1964), [8] L.I. Nicolaescu, Lectures on the geometry of manifolds, World Scientific Publishing Co., (1996), 2 nd Edition (27). [9] R. O. Wells, Differential analysis on complex manifolds, Springer Graduete Texts in Mathematics 65 (1979). ACKNOWLEDGEMENT: The first author is grateful to Prof. Stefano Francaviglia of the University of Bologna, for a very helpful discussion on some of the contents of this paper. Davide Guidetti, Dipartimento di matematica, Piazza di Porta S. Donato 5, 4126 Bologna, Italy. address: davide.guidetti@unibo.it Batu Güneysu, Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany. address: gueneysu@math.hu-berlin.de Diego Pallara, Dipartimento di matematica e fisica Ennio De Giorgi, Universitá del Salento, Lecce, Italy. address: diego.pallara@unisalento.it