RIESZ MEETS SOBOLEV. In memoriam Nick Dungey and Andrzej Hulanicki.

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1 RIESZ MEETS SOBOLEV THIERRY OULHON AND ADAM SIKORA Abstract. We show that the L p boundedness, p > 2, of the Riesz transform on a complete non-compact Riemannian manifold with upper and lower Gaussian heat kernel estimates is euivalent to a certain form of Sobolev ineuality. We also characterize in such terms the heat kernel gradient upper estimate on manifolds with polynomial growth. In memoriam Nick Dungey and Andrzej Hulanicki. ontents. Introduction 2. Gradient estimates imply heat kernel bounds 4 3. Doubling volume 6 4. Polynomial volume growth 5. Applications 7 References 2. Introduction The present paper may be considered as a companion paper to [2], which gave criteria for the L p boundedness, for p > 2, of the Riesz transform on non-compact Riemannian manifolds. Here we reformulate these criteria in terms of certain Sobolev ineualities. That is, we deduce some L p to L p estimates from suitable L to L p estimates, for < p. Let M be a complete, connected, non-compact Riemannian manifold. The methods of this paper remain valid for other types of spaces endowed with a gradient, a metric which is compatible with this gradient, a measure, and finally an operator associated with the Dirichlet form constructed from the gradient and the measure. An interesting example is a Lie group endowed with a family of left-invariant Hörmander vector fields. We leave the details of such extensions to the reader. Let d be the geodesic distance on M; denote by B(x, r) the open ball with respect to d with center x M and radius r >. Date: August 9, Mathematics Subject lassification. 58J35, 42B2, 46E35. Key words and phrases. Heat kernels, Riesz transform, Sobolev type ineualities.

2 2 THIERRY OULHON AND ADAM SIKORA Denote by µ the Riemannian measure, by L p (M, µ), p, the corresponding L p spaces, and let V (x, r) = µ(b(x, r)). Let be the (non-negative) Laplace-Beltrami operator. One could consider another measure µ with positive smooth non-zero density with respect to µ, and the associated operator µ, formally given by ( µ f, f) = f 2 d µ. Again, for simplicity, we stick to the standard case. Let be the Riemannian gradient. We can now define formally the Riesz transform operator /2. Let p (, ). The boundedness of the Riesz transform on L p (M, µ) reads (R p ) f p p /2 f p, f (M), and if the reverse ineuality (RR p ) also holds, one has (E p ) f p /2 f p, f (M). One says that M satisfies the volume doubling property if there exists such that (D) V (x, 2r) V (x, r), r >, x M, more precisely if there exist ν, ν > such that V (x, r) ( r ) ν (D ν ) V (x, s) ν, r s >, x M. s The heat semigroup is the family of operators (exp( t )) t> acting on L 2 (M, µ), it has a positive and smooth kernel p t (x, y) called the heat kernel. In the seuel, we shall consider the following standard heat kernel estimates for manifolds with doubling : the on-diagonal upper estimate, (DUE) p t (x, x) V (x, t) for some >, all x M and t >, the full Gaussian upper estimate, (UE) p t (x, y) V (x, t) exp c d2 (x, y) t for some, c >, all x, y M and t >, the upper and lower Gaussian estimates, c (LY ) V (x, t) exp d2 (x, y) p t (x, y) ct V (x, t) exp d2 (x, y) t for some, c >, all x, y M and t >, and finally the gradient upper estimate (G) x p t (x, y) tv (y, t) M

3 RIESZ MEETS SOBOLEV 3 for all x, y M, t >. self-improves into x p t (x, y) It is known that, under (D) and (DUE), (G) exp d2 (x, y) tv (y, t) t for all x, y M, t >, see [5] and also [2, Section 4.4]). It will follow from Proposition 2. below that assumption (DUE) is not needed here. Recall that (DUE) plus (D) implies (UE) ([8, Theorem.], see also [2, orollary 4.6]), and (UE) plus (G) and (D) implies (LY ) ([2]). We shall see in Proposition 2. below that (G) implies (DUE), therefore (G) plus (D) implies (LY ). onversely, (LY ) implies (D) (see for instance [24, p.6]). The following is one of the two main results of [2] (Theorem.4 of that paper). Theorem.. Let M be a complete non-compact Riemannian manifold satisfying (D), (DUE), and (G). Then the euivalence (E p ) holds for < p <. Taking into account Proposition 2. below, one can skip condition (DUE), and formulate this result in the following simpler way. Theorem.2. Let M be a complete non-compact Riemannian manifold satisfying (D) and (G). Then the euivalence (E p ) holds for < p <. Let us now introduce an L p version of (G), namely (G p ) e t p p p t, t >. The other main result of [2] is the following (Theorem.3 and Proposition. of that paper). Theorem.3. Let M be a complete non-compact Riemannian manifold satisfying (D) and (LY ). Let p (2, ]. The following assertions are euivalent: (a) (R p ) holds for all p (2, p ). (b) (G p ) holds for all p (2, p ). (c) For all p (2, p ), there exists p such that p t (., y) p p [ ], t >, y M. t V (y, t) p According to Proposition 3.6 below, we will be able to add another euivalent condition in the above list, namely (d) For all p (2, p ), there exists p such that f 2 p p f p f p, f (M).

4 4 THIERRY OULHON AND ADAM SIKORA The two above results are the cornerstones of the present paper. Our main results are Theorems 4., 4.2 and orollary 4.3 below. In Theorem 4., using Theorem.3, we give a necessary and sufficient condition for (R p ) to hold for p in an interval above 2 on manifolds with polynomial volume growth satisfying (D) and (LY ), in terms of an L p L Sobolev type ineuality with a gradient in the left-hand side. In Theorem 4.2, we give a necessary and sufficient condition for (G) on manifolds with polynomial volume growth satisfying a mild local condition, in terms of a multiplicative L Sobolev type ineuality, with a gradient in the left-hand side. In orollary 4.3 we deduce from Theorem 4. and Theorem. that this L Sobolev ineuality alone implies (E p ) on manifolds with polynomial growth and the above local condition. Here is the plan we will follow. In section 2, we prove that (G) implies (DUE), together with a similar statement for some related kernels. In section 3, we give a first version of our results for manifolds with doubling and a polynomial volume upper bound. In section 4, we assume full polynomial growth and obtain more complete results. Finally, in section 5, we give applications of our methods to second order elliptic operators in R n. 2. Gradient estimates imply heat kernel bounds Note that the following result does not reuire assumption (D). Proposition 2.. (G) implies (DUE). Proof. For x M, t >, define We claim that K = K(x, t) = V (x, t)p t (x, x). 2 p t (y, x) K V (x, t) for all y B(x, K t ). Indeed, according to (G) and the mean value theorem, for such y, p t (y, x) p t (x, x) Thus, given the definition of K, hence the claim. Now d(y, x) tv (x, t) p t (y, x) p t (x, x) M p t (y, x) dµ(y) K dµ(y) B(x, K t ) V (x, t) K t tv (x, t) = K V (x, t). K V (x, t) = B(x, K t K KV (x, = V (x, t) K V (x, t), ) p t (y, x) dµ(y) t ).

5 RIESZ MEETS SOBOLEV 5 If K, this means that K. Hence K max (, ), and p t (x, x) 2 max (, ) V (x, t), t >, x M. As we noticed in the introduction, the following is a conseuence of Proposition 2. together with known results. orollary 2.2. Assume that M satisfies (G) and (D). Then M satisfies (LY ). It may be of interest to notice that the assumption in Proposition 2. can be replaced by a gradient estimate of some other kernels. Namely, for a >, denote by rt a (x, y) the (positive) kernel of the operator R a t = (I + t ) a = Γ(a) + s a exp( s(i + t )) ds. Similarly, for < a <, denote by p a t (x, y) the kernel of the operator = exp( (t ) a ). In the following statement, we assume doubling only P a t for simplicity, otherwise one has to include an additional constant in the outcome. Proposition 2.3. Assume (D). Suppose that a t = r a t for some a > or a t = p a t for some < a <. Next assume that M satisfies the gradient upper estimate (G a ) x a t (x, y) tv (y, t) for all x, y M, t >. Then M satisfies (DUE). Proof. First note that in all cases t a (x, y) dµ(y), x M. M Fix x M, t >. Define K = K(x, t) = V (x, t)t a (x, x). 2 Exactly as in the proof of Proposition 2., one shows that K max (, ). To finish the proof of Proposition 2.3, note that, for a >, p t (x, x) = p t/2 (, x) 2 r a/2 t/2 (, x) 2 = r a t/2(x, x).

6 6 THIERRY OULHON AND ADAM SIKORA Indeed, write so that exp( t ) = exp( t )(I + t ) a (I + t ) a, p t (, x) = exp( t )(I + t ) a r a t (, x), and since by spectral theory the operator exp( t )(I + t ) a is uniformly bounded on L 2 (M, µ), the claim is proved. Similarly, for < a <, writing one sees that exp( t ) = exp( t ) exp(t ) a exp( (t ) a ), p t (x, x) = p t/2 (, x) 2 p a t/2(, x) 2 = p a 2 a a (x, x). t 3. Doubling volume Recall that (LY ) implies (D). Thus the doubling volume assumption will be implicit in the first two statements of this section. Theorem 3.. Let M satisfy (LY ). Let ν > be such that (3.) V (x, r) r ν, r >, x M. Let p (2, ]. Assume (3.2) f p p ν 2( p)+ 2 f, f (M), for all p (2, p ) and some < < p. Then (R p ) holds for all p (2, p ). Proof. Let p be such that 2 < p < p and (, p) such that (3.2) holds. Taking f = p 2t (., y) = exp( t )p t (., y), t >, y M, in (3.2), one obtains p 2t (., y) p p ν 2( p)+ 2 exp( t )pt (., y), hence, by analyticity of the heat semigroup on L (M, µ), p 2t (., y) p t ν 2( p) 2 pt (., y), t >, y M. On the other hand, (UE) yields Hence p 2t (., y) p p t (., y) t 2 [ V (y, t) ] p [ ], t >, y M. V (y, t) [t ν 2 V (y, t) ] p, t >, y M,

7 RIESZ MEETS SOBOLEV 7 and, according to (3.), the uantity t ν 2 V (y, t) is bounded from above, therefore (3.3) p 2t (., y) p [ ], t >, y M. t V (y, t) p One concludes by applying [2], namely Theorem.3 above. Remarks: -Remember that it follows from [9] that, under the assumptions of Theorems 3., (R p ) also holds for p (, 2]. As a conseuence, (RR p ) also holds, therefore assumption (3.2) implies /2 f p p ν 2( p)+ 2 f, f (M), hence, by making the change of functions /2 f f, the Sobolev ineuality It follows that f p p ν 2( p) f, f (M). V (x, r) cr ν, r >, x M (see [6]). Thus, in fact, under the assumptions of Theorem 3., the volume growth of M has to be polynomial of exponent ν (in particular, ν has to coincide with the topological dimension of M). However, the fact that we do not use explicitly polynomial growth in the proof will allow us below some true excursions in the doubling volume realm. -An euivalent formulation of (3.2) is f p p α f, f (M), for all p (2, p ) and some α > /2, with = p + 2 ν (α 2) α = and = pν is a valid choice. See Section 5 below. ν+p -When p <, if one assumes (3.4) f p ν 2 + p 2 f, f (M),. In particular, instead of (3.2), one still obtains the same conclusion by interpolation. -One can also replace (3.2) by the following weaker ineuality f p p α/2 f θ f θ 2, f (M), where < θ <,, 2, < θ p + θ 2 < and ( θ αθ = ν + θ ) +. 2 p Here also, one can take p = p. In the next statement, we shall relax the volume upper bound assumption for small radii. This can be useful in situations where the volume growth is

8 8 THIERRY OULHON AND ADAM SIKORA polynomial, but with different exponents for small and large radii, say for instance the Heisenberg group endowed with a group invariant Riemannian metric. We shall say that the local Riesz ineuality (R p ) loc holds on M if f p p ( /2 f p + f p ), f (M). This is the case for instance if M has Ricci curvature bounded from below (see [3]). Theorem 3.2. Let M satisfy (LY ) and (R p ) loc. Let ν > be such that (3.5) V (x, r) r ν, r, x M. Let p (2, ]. Assume (3.2) for all p (2, p ) and some < < p. Then (R p ) holds for all p (2, p ). Proof. Given (3.5), the same proof as in Theorem 3. yields p 2t (., y) p [ ], t, y M. t V (y, t) p On the other hand, (R p ) loc easily implies, by analyticity of the heat semigroup on L p (M, µ), t e t p p +, for all t >, hence, following [2, p.944], p 2t (., y) p [ ], t, y M. t V (y, t) p One concludes as before. Remark: One way to ensure (3.5) is to assume (D ν ) and (3.6) sup V (x, ) < +. x M Let us consider now the limit case p = in ineuality (3.2). Theorem 3.3. Let M satisfy (D), (DUE), and (3.) for some ν >. Assume (3.7) f f ν+ α α/2 ν+ α f, f (M), for some [, ) and some α > ν +. Then (E p) holds for all p (, ).

9 RIESZ MEETS SOBOLEV 9 Proof. Taking again f = p 2t (., y) = exp( t )p t (., y), t >, y M, in (3.7), and using the fact that (UE) yields (3.8) p t (., y) one obtains p 2t (., y) [ ], t >, y M, V (y, t) ν+ [ ] ( V (y, t) )( ν+ α ) α/2 α exp( t )p t (., y), hence, by analyticity of the heat semigroup on L (M, µ) (when =, it follows from (UE), see for instance [4, Lemma 9], [3, Theorem 3.4.8, p.3] or [23]), p 2t (., y) hence, using (3.), p 2t (., y) α [ ] ( V (y, t) )( ν+ α ) t 2 ( ν+ ν+ α ) α pt (., y) ν+ [ ] ( V (y, t) ) t 2 tv (y, t) [t ν/2 V (y, t)] /, t >, y M, tv (y, t), t >, y M, that is, (G). One concludes by applying [2], namely Theorem. above. Theorem 3.4. Assume that M has Ricci curvature bounded from below. Let M satisfy (D), (DUE), (3.5) and (3.7) for some ν >, some [, ) and some α > ν +. Then (E p) holds for all p (, ). Proof. Given (3.5), the same proof as in Theorem 3.3 yields p 2t (., y) tv (y, t), t, y M, that is, (G) for large time. Since M has Ricci curvature bounded from below, it follows from [2] that (G) also holds for small time. One concludes as before. Note that ineuality (3.7) is known in R n, with ν = n. Remark : Let the space Lip(M) be the completion of (M) with respect to the norm f Lip = sup x y f(x) f(y). d(x, y)

10 THIERRY OULHON AND ADAM SIKORA It is well-known that, if f Lip(M), then f is differentiable almost everywhere and f Lip = f. By the reiteration lemma (see for instance [4, Proposition 2., p.36]), ineuality (3.7) is euivalent to the embedding [L α, L ] θ, Lip, where [X, Y ] θ,r denotes the real interpolation space between X and Y with parameters θ and r, θ = ν+ α, and L α is the completion of (M) with respect to the norm α/2 f. Then it is a well-known fact (see for instance [5, Proposition 3.5.3] and modify it to obtain a version for homogeneous spaces) that [L α, L ] θ, = Λ, θα, where the Besov space Λ p, α Λ, α (f) = is defined via the norm + for k > α/2. Finally (3.7) is euivalent to Λ, ν t k α 2 k e t f dt t, + Lip. Let us finally consider the limit case = in Theorem 3.3. Here, no upper volume bound is needed. Theorem 3.5. Let M satisfy (D) and (DUE). Assume (3.9) f f α α/2 f α, f (M), for some α >. Then (E p ) holds for all p (, ). Let us emphasize the particular case α = 2 of ineuality (3.9): (3.) f 2 f f, f (M). Proof. Substituting exp( t )f in (3.9) yields exp( t )f exp( t )f α α/2 exp( t )f α. Recall that it follows from (DUE) that the heat semigroup is analytic on L (M, µ), hence by duality α/2 exp( t )f t α/2 f. The heat semigroup being uniformly bounded on L (M, µ), one obtains exp( t )f t /2 f, that is, (G ), or sup t x p t (x, y) dµ(y) <. x M,t> M

11 RIESZ MEETS SOBOLEV It is well-known and easy to see that (G ) together with (D) and (UE) implies (G) (in fact, these conditions are euivalent, because of the alreadymentionned self-improvement of (G)). One concludes again by applying [2], namely Theorem. above. Next we discuss a result which does not reuire any assumption on the volume growth and which is motivated by (3.). This result is contained in [6], with a similar argument, in a discrete setting. For another approach to ineuality (3.) below, see [, Section 4]. Proposition 3.6. For any p, condition (G p ) is euivalent to : (3.) f 2 p f p f p, f (M). Proof. To prove that condition (3.) implies (G p ) we modify slightly the argument of the proof of Theorem 3.5. Namely, we put α = 2 and replace L norm by L p norm. To prove the opposite direction, write Hence, for suitable f, (I + t ) = (I + t ) f p exp( s( + t )) ds. Assuming (G p ), one obtains, for f L (M, µ), Hence (I + t ) f p e s exp( st )f p ds. e s (ts) /2 f p ds = t /2 f p s /2 e s ds = t /2 f p. f p t /2 (I + t )f p t /2 ( f p + (t )f p ) = t /2 ( f p + t f p ). Taking t = f p f p yields (3.). 4. Polynomial volume growth Theorem 4.. Let n >. Suppose that M satisfies upper and lower n- dimensional Gaussian estimates ct n/2 exp d2 (x, y) p t (x, y) t n/2 exp d2 (x, y), ct t for some, c >, all x, y M and t >.

12 2 THIERRY OULHON AND ADAM SIKORA Let p (2, ]. Then the following are euivalent: i) (4.) f p p, n 2 ( p)+ 2 f, f (M), for some (, p), and all p (2, p ). ii) (R p ) holds for all p (2, p ). Proof. Let and p be such that < < p < and p > 2. According to [26], the following Sobolev ineuality is a conseuence of the upper heat kernel estimate : and in particular f p n 2 ( p) f, f (M), /2 f p n 2 ( p)+ 2 f, f (M). Thus (R p ) for some p > 2 implies (4.) for all such that < < p, and in particular ii) implies i). onversely, observe that the heat kernel estimates imply V (x, r) r n, r >, x M (see [9, Theorem 3.2]). Therefore Theorem 3. applies with ν = n and shows that i) implies ii). Remarks similar to those after Theorem 3. are in order. We add one more. Remark: According to [, Theorem.4], under the assumptions of Theorem 4., there always exists a p such that ii) holds. It would be nice to have a proof of this fact using i). Again, we shall now consider the limit case p = of ineuality (4.). We shall have to make local assumptions in order to ensure that the uantity θ(t) := sup u n/2 p u (x, x) = sup u n/2 exp( u ) <u t, x M <u t is finite for some (all) t >. For instance, a local Sobolev ineuality of dimension n is enough, since then sup M p t (x, x) t n/2, < t. This holds for instance if dim M n, M has Ricci curvature bounded from below and satisfies the matching condition to (3.6) : inf x M V (x, ) >. Theorem 4.2. Assume that M has Ricci curvature bounded from below. Let n N. Assume that (4.2) V (x, r) r n, r >, x M.

13 RIESZ MEETS SOBOLEV 3 Then M satisfies the heat kernel gradient estimate (G), that is (4.3) x p t (x, y) t n+ 2 exp d2 (x, y), t for some >, all x, y M and t >, if and only if (4.4) f f n+ α α/2 n+ α f, f (M), for some (all) (, ) and some (all) α > n +. Moreover if (4.3) or (4.4) holds, then M satisfies (LY ), that is the upper and lower n- dimensional Gaussian estimates (4.5) ct n/2 exp d2 (x, y) p t (x, y) t n/2 exp c d2 (x, y), t t for some, c >, all x, y M and t >. The following result is a direct conseuence of Theorem 4.2 together with [2, Theorem.4], that is, Theorem. above. orollary 4.3. Assume that M has Ricci curvature bounded from below, and satisfies (4.2) and (4.4). Then (E p ) holds for all p (, ). Let us prepare the proof of Theorem 4.2 with two lemmas. The first one is reminiscent of Proposition 2. : it shows that a certain gradient estimate implies an upper bound of the heat kernel. Lemma 4.4. Assume that M has Ricci curvature bounded from below and let n >. Assume that, for some c >, (4.6) V (x, r) cr n, for all x M and r >. Next suppose that, for some [, ], (G n, ) exp( t ) t n+ 2 for all t >. Then there exists a constant such that for all t >. sup p t (x, x) t n/2 x M Proof. Set θ(t) = sup <u t, x M u n/2 p u (x, x) = sup <u t u n/2 exp( u ). Remember that the curvature assumption together with the volume lower bound ensures the finiteness of θ(t) for all t >. Using (G n, ) and interpolation, we can write exp( 2s ) exp( s ) exp( s ) s n+ 2 θ(s)s n 2 = s n+ 2 θ(s).

14 4 THIERRY OULHON AND ADAM SIKORA For x M and s >, define For all y B therefore Hence ( x, K s θ(s) K = K(s, x) = sn/2 p 2s (x, x). 2 ), p 2s (y, x) p 2s (x, x) d(y, x) sup p 2s (z, x) z M d(y, x) exp( 2s ) K s θ(s) θ(s) s n+ 2 = K s n/2, p 2s (y, x) p 2s (x, x) K 2K = sn/2 s K n/2 s = K n/2 s. n/2 M p 2s (y, x) dµ(y) K s B x, θ(s) K s B x, θ(s)! p 2s (y, x) dµ(y)! K s n/2 dµ(y) c K n+ n θ(s) n( ), using (4.6) in the last ineuality. Since θ is obviously non-decreasing, we also have K n+ c n θ(2s), n( ) that is n K(s, x) c θ(2s)n( ) n+, hence n K(s, x) c θ(2t)n( ) n+ for < s t. Taking supremum in x and s yields 2 n 2 n θ(2t) c θ(2t)n( ) n+. Since n( ) <, it follows that θ is bounded from above, which proves n+ the claim. Remark: One can write a version of the above lemma in the case where V (x, r) v(r), for some doubling function v. The lemma below yields as a by-product a new proof of ineuality (4.4) in R n. It does not reuire any volume growth assumption.

15 RIESZ MEETS SOBOLEV 5 Lemma 4.5. Let < < and n >. euivalent: i) (G n, ) exp( t ) t n+ 2, t >. ii) (I + t ) α/2 α t n+ 2, The following estimates are for some (all) α > n + and all t >. iii) n+ α f f α/2 n+ α f, f (M), for some (all) α > n +, that is, (4.4). Proof. We shall show that i) ii) iii) i). Write (I + t ) α/2 = Hence, for suitable f, (I + t ) α/2 f s (α/2) exp( s( + t )) ds. Assuming (G n, ), one obtains, for f L (M, µ), (I + t ) α/2 f since α > n +. Assume ii), and write s (α/2) e s exp( st )f ds. = t n+ 2 = α t n+ 2 f, s (α/2) e s (ts) n+ 2 f t n+ 2 (I + t ) α/2 f f ds f s (α/2) n 2 (3/2) e s ds α t n+ 2 ( f + (t ) α/2 f ) = t n+ 2 ( f + t α/2 α/2 f ). The second ineuality relies on the L p -boundedness of the operator (I + t ) α/2 (I + (t ) α/2 ) (see [25], or use analyticity). Taking t = f 2/α α/2 f 2/α yields iii). Finally, assume iii). Replacing f by exp( t )f, one obtains, by contractivity and analyticity of the heat semigroup on L (M, µ), exp( t )f exp( t )f that is, i). n+ α n+ α t 2α f = t n+ 2 f, α/2 n+ α exp( t )f

16 6 THIERRY OULHON AND ADAM SIKORA Remark: The above lemma also holds for =,, provided the heat semigroup is analytic on L (M, µ), which is the case, as we already said, if it satisfies Gaussian estimates and (D) holds. Note that, according to Lemma 4.4, this is automatic from i) under the boundedness from below of the Ricci curvature and (4.2). Proof of Theorem 4.2. Assume (4.4). By Lemma 4.5, (G n, ) follows, and by Lemma 4.4, (4.7) sup p t (x, x) = exp( t ) t n/2 x M for all t >. The Gaussian upper bound follows: p t (x, y) t n/2 exp d2 (x, y), t for some, c >, all x, y M and t >. By interpolation, (4.7) yields (4.8) exp( t ) t n( )/2. ombining (G n, ) with (4.8) yields (4.9) sup x p t (x, y) = exp( t ) t n+ 2, x,y M which together with the upper bound yields the Gaussian lower bound ct n/2 exp d2 (x, y) p t (x, y). ct Finally, (4.9) self-improves into (4.3). onversely, (4.3) obviously implies (G n, ) and, together with the volume upper bound, (G n, ) or, in other words, (G ): sup x M M x p t (x, y) dµ(y) = exp( t ) t /2. By interpolation, one obtains (G n, ), therefore (4.4), thanks to Lemma 4.5. Remarks: -As a conseuence of Theorem 4.2, (4.4) implies, using the results in [7], f f n α α/2 f n α, f (M), for α > n/, [, ), and, using the results in [8], f(x) f(y) [d(x, y)] α (n/) α/2 f, x, y M, f (M), for α > n/, [, ). It would be interesting to have a direct proof of these two implications. -According to known results on Riesz transforms (see [2] for references), (4.4) is true for manifolds with non-negative Ricci curvature, Lie groups with

17 RIESZ MEETS SOBOLEV 7 polynomial volume growth, cocompact coverings with polynomial volume growth. Again, it would be interesting to have direct proofs. -It would interesting to study the stability under perturbation of ineualities (4.4) or (4.), in the light of the result in []. 5. Applications Now we consider a uniformly elliptic operator H in divergence form acting on R n, n N, that is n Hf = i (a ij j f) = where a ij L for all i, j n, and the matrix (a ij (x)) n is a symmetric matrix with real coefficients, such that a ij (x)ξ j ξ i c ξ 2, for a.e. x, ξ R n, for some c >. Next let denote the standard non-negative Laplace operator acting on R n. It follows from the above uniform ellipticity assumption and the boundedness of the coefficients that H f(x) 2 = a ij (x) j f(x) i f(x) f(x) 2. We say that H satisfies (R p ) for some p (, )) if H f p p H /2 f p, f (R n ), which according to the above remark is euivalent to f p p H /2 f p, f (R n ). To avoid technicalities we assume in what follows that all coefficients a ij, b ij discussed below are smooth. However we point out that this assumption can be substantially relaxed. Recall that the Gaussian estimates do hold for e th and that the above framework applies. The assumption in our first application may be seen as some boundedness for the higher order Riesz transform associated with H. Theorem 5.. Suppose that (5.) α/2 f H α/2 f, f (R n ), for some α > and < <. Then, if α < n +, H satisfies (R p ) for n all p (, p ), where p =, and if α n n + α +, H satisfies (R p ) for all p (, ).

18 8 THIERRY OULHON AND ADAM SIKORA Proof. The boundedness of the classical Riesz transform on L p (R n, dx) together with the Sobolev ineuality in R n imply, for < < p <, f p n + 2 p 2 f α/2 f θ f θ, f (R n ), as soon as α n( p ) +, θ (, ] being such that αθ = n( p ) +. Now let α and be such that (5.) holds. If α n +, choose any p >. If α < n +, choose p in (, ) so that α = n( p ) +. In both cases, f p H α/2 f θ f θ, f (R n ), and (R p ) for < p < p follows from Theorem 3. and the remarks afterwards. Our next application says that (R p ) also holds for small L W,n perturbations of operators with bounded second order Riesz transform. To state this result we set H ε f = Hf + ε i b ij (x) j f, where H = H is as above. We do not assume here that the matrix (b ij (x)) n is positive definite. However we assume that b ij L (R n, dx) and that ε is small enough so that the operator H ε is uniformly elliptic. Theorem 5.2. Suppose that b ij L (R n, dx) for all i, j n and that j b ij L n (R n, dx) for all i, j n. Next assume that for some < n (5.2) f H f, f (R n ), Then there exists γ > such that (5.2) holds for H ε for ε < γ and so (R p ) holds for H ε for all ε < γ and < p < p where p + n =. Proof. Note that (5.2) is just condition (5.) for α = 2. We are going to prove that this ineuality extends from H to H ε for < ε < γ and apply Theorem 5.. To this purpose, it is enough to show that for some γ > and for all ε < γ (5.3) H ε f H f p 2 f p. Now Since H ε f H f ε i b ij j f. i (b ij j f) = b ij ( i j f) + ( i b ij )( j f), one may write i b ij j f max b ij i j f + ( i b ij )( j f),

19 RIESZ MEETS SOBOLEV 9 hence i b ij j f max b ij i j f + ( i b ij )( j f) n 2 max b ij i j f + i b ij n j f p. Here we have used the L boundedness of the second order Riesz transform in R n and the Hölder ineuality fg f n g p. Now recall that an ineuality similar to (4.) holds in R n, that is Therefore i b ij j f ( n 2 max This yields (5.3) with ( γ n 2 max j f p f. b ij i j + max i b ij i j + max i ) i p i b ij n f. ) i p i b ij n = 2. The second order Riesz transform bound (5.2) is known for various large classes of operators. We discuss one instance of such class next. Example: Assume the coefficients a ij of H are continuous and periodic with a common period and that n i= ia ij = for j n. Then Hf p f p, f (R n ), for all p (see [7, Theorem.3]), so that H satisfies the assumption of Theorem 5.2. In [7], it is proved that (R p ) holds for such H, but the above shows that is it also holds for small L W,n perturbations of H. Remark: It is interesting to compare Theorem 5., which proves that boundedness of second order Riesz transform implies boundedness of first order Riesz transform on a larger range on L p spaces, with the results obtained in [22]. See also [2, (.26)]. Acknowledgements: The authors would like to thank Professor Vladimir Maz ya for interesting discussions about the proof of ineuality (4.4) in the Euclidean space.

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21 RIESZ MEETS SOBOLEV 2 Thierry oulhon, Département de Mathématiues, Université de ergy- Pontoise, Site de Saint-Martin, 2, rue Adolphe hauvin, F 9532 ergy- Pontoise edex, France address: coulhon@u-cergy.fr Adam Sikora, Department of Mathematics, Macuarie University, NSW 29, Australia address: adam.sikora@anu.edu.au