L p regularity for elliptic operators with unbounded coefficients

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1 L regularity for ellitic oerators with unbounded coefficients G. Metafune J. Prüss A. Rhandi R. Schnaubelt Abstract Under suitable conditions on the functions a C 1, ), F C 1, ), and V : [0, ), we show that the oerator Au = a u) + F u V u with domain W, V RN ) = {u W, ) : V u L )} generates a ositive analytic semigrou on L ), 1 < <. Analogous results are also established in the saces L 1 ) and C 0 ). As an alication we show that the generalized Ornstein Uhlenbeck oerator A Φ,Gu = u Φ u + G u with domain W,, µ) generates an analytic semigrou on the weighted sace L, µ), where 1 < < and µdx) = e Φx) dx. 1 Introduction In recent years there has been an increasing interest in differential oerators with unbounded coefficients on arising in the analytic treatment of stochastic differential equations; see [6], [7], [10], [11], [1], [13], [14], [1], [3], [4], [5], [7], and the references therein. An imortant examle of such oerators is the generalized Ornstein Uhlenbeck oerator A Φ,G u = u Φ u + G u. This oerator has good roerties in the sace L, µ) with weighted measure µdx) = e Φx) dx. We show in Theorem 7.4 that A Φ,G with the natural domain W,, µ) generates an analytic C 0 semigrou on L, µ) if 1 < <, where in the simler case G = 0 we only require that e Φ is integrable and that D Φ is small comared to Φ Corollary 7.8). Our theorem extends and simlifies recent results by Da Prato and Vesri [13] for G = 0) and the authors [7] for quadratic Φ and linear G); see also [7], [11], [1], [1], and in articular [10] where a descrition of the domain is given if G = 0 and Φ is convex). Via the transformation v = e Φ/ u, the oerator A Φ,G on L, µ) is similar to an oerator of the form Av = v + F v V v, or more generally, Av = a v) + F v V v in the unweighted sace L ), cf. [13]. In fact, most of our aer deals with the oerator A aiming at a recise descrition of the domain on which A generates an analytic semigrou. We make the following assumtions on the coefficients of A, where [1, ) is given. H1) a hk C 1 b RN ) are real-valued functions with a hk = a kh and N h,k=1 a hk x)ξ h ξ k ν ξ 1

2 for all x, ξ and some constant ν > 0. H) U C 1 ) is a function such that U c 0 > 0 and U γu 3 c 0, γ > 0 and C γ 0. + C γ for some constants H3) V : R is measurable and U V c 1 U for some constant c 1 1. H4) The function F C 1, ) satisfies F κu 1 for some constant κ > 0. H5) There is a constant θ < such that θu + div F 0. We will also require below that γ > 0 is sufficiently small. The auxiliary otential U is mostly used to simlify the resulting hyotheses on the coefficients of A Φ,G. In addition, it allows to avoid assumtions on the oscillation of V itself. There are several aroaches to construct semigrous whose generator extends the closure of A defined on test functions; see also the references in [6], [11], [13], [3], [5], [33]. Assuming essentially H1) and H5), one can imitate the arguments in [5, 4] yielding a contractive C 0 semigrou on L ) for 1, cf. Remark.7. If one adds hyothesis H4), then the quadratic form corresonding to A on the natural form domain is closed, ositive, and sectorial, see Section 6. Due to [33] the resulting analytic C 0 semigrou on L ) induces contractive C 0 semigrous on L ) for 1, which are analytic for > 1; see also [15]. It is further ossible to solve first a related stochastic equation assuming only a dissiativity condition and use this solution to define a semigrou acting on, say, bounded Borel functions. This aroach is exosed in, e.g., [6] for oerators such as the generalized Ornstein Uhlenbeck oerator A Φ,G. One can then extend the semigrou to weighted L saces under suitable hyotheses as it is done in [11]. In su norm context the semigrou can also be constructed using ariori estimates of Schauder tye and an aroximation rocedure, [3]. The assumtions in [6], [11], [3] differ from ours. However, we oint out that the aroaches mentioned in the above aragrah do not directly) allow to comute the domain of A or A Φ,G ) on L. In [11] the case = is treated by additional arguments.) In our main results Theorems 3.4, 4.4, and 5. we establish that A with the natural domain D ) DV ) generates an analytic and contractive C 0 semigrou on L ), 1 <, and C 0 ) where a hk = δ hk if = 1, ). The recise descrition of the domain corresonds to good ariori estimates for the ellitic roblem λu Au = f stated in Corollaries 4.5 and 4.6. This semigrou then solves the arabolic artial differential equation corresonding to the ellitic oerator A. Here we imrove results due to Okazawa [8], who studied Schrödinger oerators where F = 0 in L ) with 1 < <, and results due to Cannarsa and Vesri [4], [5], who relaced our condition H5) by an additional bound on the constant κ in H4). We note that the results in [13] on A Φ,G rely on [5] and that the additional restriction on κ in [5] leads to difficulties in [13] cf. Examle.4 and Theorem.7 in [13]). Cannarsa and Vesri first show generation in L ), then in Morrey Camanato saces, then in C 0 ) and L 1 ), and finally in L ). Our aroach is much more direct. In Proositions 3.1 and 3. we rove the crucial ariori estimates in L emloying only integration by arts and related elementary techniques besides standard regularity roerties of the diffusion art, cf..6)). We combine these ariori estimates with known results and methods in semigrou theory to obtain the semigrous on L ), 1 <, which are analytic for > 1. The Stewart Masuda technique then gives the generation result in C 0 ). A duality argument yields analyticity for = 1 if also div F c U). As can be seen from the roofs of Proositions.3 and 3., hyothesis H) is the essential ingredient to determine the domain. In fact, in Examle 3.7 we resent a Schrödinger oerator A = V on

3 L R 3 ) such that H) holds with a too large constant γ and DA) DV ). Condition H4) combined with H) and H3) allows to control the drift term by the diffusion term and the otential, emloying Proosition.3. Assumtions H4) and H5) further lead to the sectoriality of A, while H5) already guarantees that A is dissiative; see Proosition 3. and Lemma.6. In the next section we give some reliminary results. The following three sections deal with generation in L, C 0, and L 1, resectively. In Section 6 we show additional qualitative roerties of the semigrous, such as ositivity, comactness, and maximal regularity of tye L q for the inhomogeneous arabolic roblem. Section 7 is devoted to the Ornstein Uhlenbeck oerator A Φ,G. Notation. C0 ) means the sace of C -functions with comact suort and C b ) C 0 )) the sace of bounded continuous functions vanishing at infinity). For 1 and k N, W k, ) denotes the usual Sobolev sace. Bx, r) designates the oen ball in with center x and radius r. The norm of L ) is denoted by and that of L Bx, r)) by,r, thus without indicating the center of the ball. We set A 0 u = N h,k=1 D ha hk D k u) and aξ, η) = N h,k=1 a hk )ξ h η k for ξ, η. Preliminary results First of all, let us observe that if a function U satisfies assumtion H), then U + λ satisfies H) with C γ = 0 for λ large enough. In this case, H) is equivalent to the inequality U 1 γ/. From the mean value theorem it follows that 1 δγ )Ux) 1 Ux0 ) δγ )Ux) 1.1) if x x 0 r and r = δux 0 ) 1. Some of the following results are valid assuming only.1) but we refer to kee H) in order to simlify the exosition. If N = 1 or N > 1 and U is rotational symmetric), H) holds whenever U does not oscillate too fast. For examle, it is satisfied by all olynomials and functions like e P, with P a olynomial. The icture is more comlicated in several variables, if U is not rotational symmetric. For examle, the function 1 + x y fails H). However, H) is valid if U is a olynomial whose homogeneous art of maximal degree is ositive definite or if, more generally, U is hyoellitic, see [19, Chater 11]. The interolation result stated in Proosition.3 below will be crucial throughout the aer. To establish this result, we need the following well known fact; its roof is given for the sake of comleteness. Lemma.1 For every u C 0 ) and 1, we have with a constant C > 0 deending only on N. u C u 1 u 1.) Proof. For λ > 0 and u C 0 ) we set f = λu u. Then ux) = 0 e λt Gt)f ) x) dt, where Gt)f ) x) = 4πt) N e x y 4t fy) dy for t > 0 and x. Differentiating under the integral sign and using Young s inequality, we obtain Gt)f c t f, 3

4 with c = cn), whence u c λ f c λ λ u + u ) for each λ > 0. The assertion follows if we take λ = u u 1. We next show a variant of Besicovitch Covering Theorem, needed in the roof of Proosition.3. Lemma. There exists a natural number KN) with the following roerty. Let {Bx, r x ) : x } be a collection of balls such that 0 < r x C and r x r y 1 x y for every x, y RN. Then there exists a countable subcovering {Bx n, rxn )} of RN such that at most KN) among the balls Bx n, r xn ) overla. Proof. Due to the Besicovitch Covering Theorem, [36, Theorem 1.3.5], there exists a countable subcovering {Bx n, rn )} of RN such that at most ξn) among the balls Bx n, rn ) overla. Here we write r n for r xn and remark that the number ξn) deends only on the dimension N. We now have to estimate the number of overlaing doubled balls Bx n, r n ), where we follow the roof of [36, Theorem 1.3.4]. Fix x = x k for some k and assume that Bx, r x ) and Bx j, r j ) overla for j J. Then x x j r x + r j, hence r x r j 1 r x + r j ) by the assumtion. This yields r x 3r j and r j 3r x. Thus the balls Bx j, rj ), j J, are contained in Bx, 5r x). Since at most ξn) of the balls Bx j, rj ) overla, we obtain 6 N J r N x j J N r N j ξn)5 N r N x and hence J 30 N ξn). This yields KN) N ξn). Proosition.3 Let U be a function satisfying H). Then there are two constants α, ε 0 > 0 deending only on γ, C γ, c 0 ) such that for all 0 < ε ε 0, 1, and v C 0 ), we have U 1 v ε v + α ε Uv..3) Proof. We relace U by U + λ for λ 0 such that H) holds for U + λ with C γ = 0. Since U c 0 > 0, estimate.3) for U + λ imlies.3) for U with a different α). So we may assume that C γ = 0 in H). Fix x 0 and choose η C0 ) such that 0 η 1, η = 1 in Bx 0, r/), su η Bx 0, r), ηx) c/r, and D ηx) c/r for a constant c indeendent of x 0 and r. Using.) for ηv and Young s inequality, we estimate Ux 0 ) 1 v, r Ux 0 ) 1 ηv) C ηv) 1 Ux 0 )ηv 1 ε v,r + 1 r v,r + 1 ) r v,r + C 1 ε Ux 0)v,r.4) for a constant C 1 and each ε > 0. We fix r = δux 0 ) 1 and δ = 1/γ. Then.1) shows that 1 Ux) 1 Ux 0 ) 1 3 Ux) 1 if x x 0 r. Thus.4) yields 1 U 1 v, r ε v,r + εc U 1 v,r + C ε Uv,r.5) for a constant C > 0. Let us first deal with the simler situation where =. In this case, varying x 0 in, equation.5) imlies 1 U 1 v ε v + C ε U 1 v + C ε Uv. 4

5 Thus the asserted estimate holds with ε 0 = 4C ) 1. Suose now that 1 <. Alying Lemma., we find a countable covering {Bx n, rn )} of RN with r n = δux n ) 1 such that at most KN) among the balls Bx n, r n ) overla. Taking the -th ower of.5) relative to each ball Bx n, rn ) and summing u, we arrive at 1 U 1 v 3 1 KN) 1 ε v + C ε U 1 v + C ) ε Uv, where 1/ + 1/ = 1. The asserted estimate follows with ε 0 = 1C KN)) 1. Remark.4 Let 1 < <. Thanks to the ellitic estimate v W, ) C A 0 v + v ).6) for v L ), see e.g. [18, Chater 9], for uniformly ellitic oerators and U c 0 > 0) one can substitute v by A 0 v in Proosition.3. This will be used systematically in the next section. If 1 < <, there is also a simler roof of Proosition.3 with D v instead of v involving Taylor s formula. We now introduce the saces D 1 = {v L 1 ) : v L 1 ), Uv L 1 )}. D = W, U RN ) = {v W, ) : Uv L )}, 1 < <, D = {v C 0 ) : v W,q loc RN ) for all q <, v C 0 ), Uv C 0 )}. Here is understood in the sense of distributions. These saces are Banach saces endowed with the norms v D = v W, ) + Uv L ), 1 < <, v D = v L ) + v L ) + Uv L ), = 1,. C 0 ) is a dense subsace of D as we show in the next lemma. Therefore.) and.3) also hold for v D. In articular U 1 v L ) for v D, 1 <, and U 1 v C0 ) for v D. Lemma.5 The sace C 0 ) is dense in D for 1. Proof. Let η be a cutoff function such that 0 η 1, η = 1 in B0, 1), su η B0, ). Define η n x) = ηx/n). For f D it is easy to see that η n f f as n in D. This shows that the set of all functions in D having comact suort is dense. On the other hand, if f D has comact suort, a standard convolution argument shows the existence of a sequence of smooth functions with comact suort converging to f in D. In the following lemma we establish the dissiativity of A under rather weak assumtions. Lemma.6 Assume that assumtion H1) is satisfied. a) If F C 1, ), V L loc RN ), and V + 1 div F 0 for some 1 <, then A, C 0 )) is dissiative in L ). b) If F C, ), V C, R), and V 0, then A, C 0 )) is dissiative in C 0 ). 5

6 Proof. a) Let 1 < and u C0 ). It is known that A 0 is dissiative in L ), that is, ) Re A 0 u)u u dx 0. Moreover, from u)u u = 1 u ) + iimu u) u it follows that As a result, ) Re F u)u u dx = 1 u div F dx. R N ) Re Au)u u dx V + 1 R div F ) u dx 0. N b) The dissiativity of A in C 0 ) is a standard consequence of the maximum rincile. Remark.7 Suose that the assumtions of Lemma.6 hold for some [1, ) and that V is locally Hölder continuous. Then there exists a contractive C 0 semigrou on L ) whose generator extends the closure of) A defined on test functions. This semigrou can be constructed as the limit of the contraction semigrous on L B0, n)) which are generated by the oerator on L B0, n)) induced by A with Dirichlet boundary conditions, comare [5, 4]. 3 Generation of analytic semigrous on L ), 1 < <. In this section we establish that, under the assumtions H1) H5), the oerator A with domain D = W, U RN ) generates an analytic semigrou on L ) if 1 < <. We start by showing that A is regularly dissiative, that is, for some φ 0, π/) the oerator e ±iφ A is dissiative. This roerty is clearly equivalent to the estimate 3.4) below with δ = cot φ). Proosition 3.1 Let 1 < < and assume that H1), H4), and H5) are satisfied with U relaced by V L loc RN ). Then the oerator A defined on C 0 ) is regularly dissiative in L ) with angle φ > 0 only deending on and the constants in H1), H4), and H5). Proof. We first assume that. Let u C0 ) and set u := u u. Integrating by arts, one comutes ) Re Au)u dx = 1) u 4 areu u), Reu u)) dx 3.1) R N + u 4 aimu u), Imu u)) dx + V + 1 ) div F u dx. If we ut B := u 4 areu u), Reu u)) dx, C := u 4 aimu u), Imu u)) dx, and D := V u dx, then we deduce from H5) ) Re Au)u dx 1)B + C + D 1 θ R ) 0. N On the other hand, roceeding in a similar way for the imaginary art, we have R Im Au)u dx R u 4 areu u), Imu u)) dx+ F u Imu u) dx. 3.) N N 6

7 Conditions H1) and H4) imly F u Imu u) dx κ V 1 Imu u) u u 4 dx κ V u dx κ ν V u dx ) 1 RN ) 1 RN u 4 Imu u) ) 1 ) 1 u 4 aimu u), Imu u)) dx. 3.3) Therefore, Im Au)u dx) BC + κ ν CD Taking δ = δ such that δ = 4 1) + κ 4ν1 θ/), we see that [ )] R Im Au)u dx) δ Re Au)u dx. 3.4) N This shows the assertion for. If 1, ), we relace u by u ε = u + ε for ε > 0 in the calculations involving A 0. Passing to the limit ε 0, one then establishes 3.1) and 3.) in articular, all integrands are integrable). Thus estimate 3.3) is also valid and one can deduce 3.4) as above. Next we rove the closedness of the oerator A, W, U RN )) for 1 < <. Proosition 3. Let 1 < <. Assume that the assumtions H1), H3), H4), and H5) are satisfied. If H) holds with γ satisfying [ θ κ + 1)γ + Mγ ] < 1, 3.5) 4 where M := su x max ξ =1 aξ, ξ), then A, W, U RN )) is closed. Moreover, there exist constants C, C 0 deending only on M,, on the constants in H1) H5), and on C from.6) such that u W, U RN ) C u Au C u W, U RN ) for u W, U RN ). Proof. We assume reliminarily that H) is satisfied with C γ = 0. Proosition.3, H3), and H4) imly that A : W, U RN ) L ) is continuous. Since C0 ) is dense in W, U RN ) and A is dissiative, it remains to show the estimates Uu, D u C Au for u C 0 ). We consider first the case. For a fixed real u C 0 ) we set f := Au = A 0 u F u + V u. 3.6) 7

8 Integrating by arts, one deduces U 1 F u)u u dx = 1 U 1 F u ) dx R N = 1 div F )U 1 u dx 1 1 ) U u F U dx, R N N A 0 u)u 1 u u dx = a hk x)d h ud k U 1 u u ) dx h,k=1 = 1) 1) N h,k=1 N h,k=1 If we multily 3.6) by U 1 u u and integrate, we thus obtain V U 1 u dx = 1) Conditions H3) and H5) yield so that 1) N h,k=1 N h,k=1 1 div F )U 1 u dx R N + fu 1 u u dx. a hk x)u 1 u D h ud k u dx a hk x)u u u D h ud k U dx. 3.7) a hk x)u 1 u D h ud k u dx a hk x)u u U D h ud k U V + 1 div F 1 θ ) U, 1 1 ) U u F U dx 1 θ ) U u dx + 1) u U 1 a u, u) dx R N 1) U u 1 a u, U) dx ) U u F U dx R N R N + fu 1 u 1 dx. Using H) and H4), we estimate U u F U dx κγ U u dx. 8

9 Moreover, H) imlies that U u 1 a u, U) dx U u 1 a u, u) 1 1 a U, U) dx γ M = γ M U 1 u 1 a u, u) 1 dx U 1 u a u, u) 1 u U dx γ M u U 1 a u, u) dx ) 1 RN ) 1 u U dx. 3.8) From Hölder s and Young s inequalities it follows that for each ε > 0 there is c ε > 0 such that [1 θ κγ1 1 ] ) U u dx + 1) u U 1 a u, u) dx 1)γ ) 1 ) 1 M u U 1 a u, u) dx U u dx RN + ε U u dx + c ε f dx. Setting B := U u dx and D := u U 1 a u, u) dx, we arrive at [1 θ κγ1 1 ) ε ] B + 1)D 1)γ MBD + c ε f. Thanks to 3.5) we may choose a small ε to deduce Uu C f. Then H4), H), Proosition.3,.6), and H3) imly that F u κε u + α ε Uu ) κc ε A 0 u + c ε Uu for ε ε 0. Thus, for sufficiently small ε, we have Finally, κc ε f + F u ) + c ε Uu. F u C Uu + f ) C f. D u C A0 u + u ) C f + F u + V u + u ) C f. If <, then one can verify as in the roof of Proosition 3.1 that 3.7) holds and, in articular, that D is a finite number. Thus estimate 3.8) is also valid and one can conclude the roof as above. In order to remove the assumtion C γ = 0 we fix a large λ such that U + λ satisfies H) with C γ = 0 and aly the revious estimates to the oerator A λ. Then u W, U RN ) C λ + 1)u Au C u Au + λ u ) C1 + λ) u Au, by the dissiativity of A. 9

10 Remark 3.3 If F 0 and a hk = δ hk, then condition 3.5) becomes γ < 4 1, which was already used by N. Okazawa in [8, Theorem.5]. We now come to the main result of this section. Theorem 3.4 Assume that the assumtions H1), H3), H4), and H5) are satisfied. If H) holds with γ satisfying [ θ κ + 1)γ + Mγ ] < 1, 4 where M := su x max ξ =1 aξ, ξ), then A, W, U RN )) generates an analytic C 0 semigrou T ) in L ), 1 < <, such that T z) 1 for arg z φ and some φ > 0. Proof. It suffices to show that 1 belongs to the resolvent set ρa) of A. In fact, Proositions 3.1, 3. and the Lumer Phillis theorem then imly that the oerators e ±iφ A generate contractive C 0 semigrous for some φ > 0. This yields the assertion by [17, Theorem II.4.9]. To verify 1 ρa), we emloy the continuity method, cf. [18, Theorem 5.]. For t [0, 1] and u W, U RN ) we set L t u := A 0 u + tf u V u. Note that these oerators satisfy the assertions of Proositions 3.1 and 3. with uniform constants. In articular u W, U RN ) C u L tu for every u W, U RN ), with C indeendent of t [0, 1]. Then 1 ρl 0 ) if and only if 1 ρl 1 ) = ρa). To establish that 1 ρl 0 ), we use as in [8] the aroximating otentials U ε := U 1+εU and V ε := V 1+εV, where ε > 0. Then we have c 0 U ε V ε c 1 U ε c εc 0 ε and U ε γu 3 ε + C γ. 3.9) Let f L ). Observe that A 0 V ε with domain DA 0 V ε ) = W, ) generates a ositive contraction C 0 -semigrou on L ). Thus the equation has a unique solution u ε W, ) satisfying u A 0 u + V ε u = f u ε f and U ε u ε C f, where the last inequality follows from the Proosition 3.. We remark that the constant C does not deend on ε due to 3.9). From.6) we thus deduce u ε W, ) C A 0 u ε + u ε ) C f. Therefore, there exists a sequence u εn ) converging weakly to a function u in W, ) as ε n 0. The Rellich Kondrachov theorem imlies that a subsequence of u εn ) tends strongly to u in W 1, loc RN ) and therefore we may assume that u εn x) ux) a.e. in. This shows that Uu C f, and hence u W, U RN ). Passing to the limit in L loc RN ) in the equality u εn A 0 u εn + V εn u εn = f, we derive u A 0 u + V u = f. This concludes the roof. 10

11 Remark 3.5 Let c be a ositive constant. By considering V + c instead of V, Theorem 3.4 shows that A, D ) generates an analytic semigrou on L ), 1 < <, if we relace H3) by U V + c c 1U. Remark 3.6 As already ointed out in the introduction, condition H4) is crucial to treat the drift term F as a erturbation of the Lalacian and of the multilication oerator u V u. This is done through Proosition.3. We remark, however, that the oerator F is not a small erturbation of the Schrödinger oerator V. If this were true, the generated semigrou T ) would be analytic of angle π/ and the same would hold for the Ornstein-Uhlenbeck semigrous in L, µ), treated in Section 7 see the roof of Theorem 7.4). However, this is false even for =, see [4]. It is clear that H4) is, in general, necessary for the domain characterization given in Theorem 3.4. In the following examle we show that the closedness of A, W, U ) may fail even if F = 0 and U fulfills H) with a too) large constant γ. Examle 3.7 We consider the Schrödinger oerator B = V with V x) = 3 4 x in L R 3 ). Since 0 V L 1 loc R3 ), B can be defined by the quadratic from bu) = u + V u ) dx R 3 for u Db) = {u W 1, R 3 ) : V u L 1 R 3 )}. Observe that the function wx) = x 1/ satisfies w V w = 0 on R 3 \ {0}. Let u = ηw, where η C0 R 3 ) is equal to 1 near the origin. It is easy to see that u Db) W,1 R 3 ) and that u V u = f C0 R 3 ). It follows that u DB) and Bu = f. Observe, however, that neither u nor V u belong to L R 3 ). Since V x) = γv x) 3/ with γ = 4/ 3, it is not difficult to construct bounded ositive otentials V n C 1 ) such that 0 V n V n+1, V n x) V x) as n, and V n x) γv n x) 3/ it is sufficient to regularize r for r 1/n). Since V n converges monotonically to V, we have 1 + V n ) 1 g 1 + V ) 1 g for every g L R 3 ), see [3, Theorem S. 14]. Taking g = u f and ossibly considering a subsequence n k ), we obtain from Fatou s lemma = V u lim inf k V n k 1 + V nk ) 1 u f) and therefore c k := V nk 1 + V nk ) 1 as k. Since C 0 R 3 ) is a core for V nk, we find u k C 0 B0, R k )), R k 1, such that V nk u k c k u k + u k V nk u k. 3.10) Take now oints x k R 3 and define W k x) = V nk x x k ), v k x) = u k x x k ), η k x) = ηx x k )R 1 k ), where η is a cutoff function such that ηx) = 1 for x 1 and ηx) = 0 for x. One can choose the oints x k with large distances x k x h in such a way that the balls Bx k, R k ) do not overla. Hence, the ositive otential W = k η kw k C 1 ) satisfies W γw 3/ + c. On the other hand, 3.10) yields W v k c k v k + v k W v k for every k. The oerator W 1 + W ) 1 is thus unbounded in L R 3 ), and W, R 3 ) DW ) is not the domain of W. We remark that the constant c = 3/4 in the definition of V hence γ = 4/ 3) is the best ossible constant for such a counterexamle. In fact c x is closed on W, R 3 ) D x ) for every c > 3/4, see [9, Theorem 3.6] where a very detailed analysis of the otential x is carried out in L ). However, the hyotheses of Theorem 3.4 hold for γ < for Schrödinger oerators in L ). 11

12 4 Generation of an analytic semigrou on C 0 ) In this section we use the Stewart Masuda localization technique to rove that the oerator A with the domain D = {v C 0 ) : v W, loc RN ) for all <, v C 0 ), Uv C 0 )} generates an analytic semigrou in C 0 ). For simlicity, we assume that the rincial art of A is the Lalacian and that H) holds for all γ > 0. We first show in two stes that A generates a contraction semigrou in C 0 ). Proosition 4.1 Assume that H) holds for all γ > 0 and that H4) is satisfied, where U = V. Then A = + F V with DA) = D is closed in C 0 ). Moreover, u + u + V u C u Au C u + u + V u ) for every u D, where C and C only deend on γ, C γ, c 0, κ. Proof. Proosition.3 shows the second estimate in the assertion. Considering V + λ instead of V for a sufficiently large λ = λγ) 0, and using the dissiativity of A, we may assume that V γv 3, as in the roof of Proosition 3.. The arameter γ will be fixed below. Let u C0 ), f = Au, and fix x 0. We take a smooth cutoff function η such that 0 η 1, η = 1 in Bx 0, r/), su η Bx 0, r), ηx) c/r, and D ηx) c/r for some constant c > 0. Observe that ηu) + F ηu) V x 0 )ηu = ηf + u η + u η + uf η + V V x 0 ))ηu. Since V x 0 ) > 0, the dissiativity of + F on C 0 ) yields V x 0 )ηu,r f,r + c r u,r + c r u,r + cκ r V 1 u,r + V V x 0 ))u,r, where we have also used H4). Let γ 1. We choose now r = 3γ) 1 V x 0 ) 1 so that.1) yields 5 6 V x) 1 V x 0 ) V x) 1 for x x 0 r. Hence, V x) V x 0 ) V x) for x x 0 r and 5 36 V u, r f + γc κ) V u,r + γc 1 V 1 u,r V u,r, where C 1 := 13c. Letting x 0 vary in and then taking γ min{1, 61 + κ)c 1 ) 1 }, we obtain Now the equation Au = f and H4) imly V u 6 f + 6γC 1 V 1 u. u C f + V 1 u ) for C := max{7, κ + 6C 1 }. At this oint we use Proosition.3 for 0 < ε ε 0 and estimate V 1 u ε u + α ε V u c ε f + εc + 6αC ) 1γ V 1 u. ε Setting ε := γ 1 and choosing γ small enough, we arrive at V 1 u C f and, by the revious inequalities, u, V u C f, with C indeendent of f. These facts show that u D C Au C u Au, for u C 0 ), using the dissiativity of A. Since C 0 ) is dense in D, the roof is comlete. 1

13 Proosition 4. Assume that H3) and H4) hold, that V is continuous, and that H) is satisfied for all γ > 0. Then A = + F V with DA) = D generates a contraction semigrou T ) on C 0 ). Proof. First we assume that U = V and we show that the oerator I V ) : D C 0 ) has dense range. Therefore V generates a contraction semigrou by Lemma.6, Proosition 4.1, and the Lumer Phillis theorem. The result for F 0 is then deduced alying the continuity method to the oerators + tf V, cf. [18, Theorem 5.], using that the ariori estimate from Proosition 4.1 is indeendent of t [0, 1]. Let > N and γ be so small such that the assumtions of Theorem 3.4 and Proosition 4.1 are satisfied. We may assume that C γ = 0 relacing V by V + λ. We define V ε = V 1+εV C b ) for ε 0, 1] and fix f C0 ). Note that V ε with D V ε ) = D ) generates a contraction semigrou on L ) and C 0 ). Thus there exists u ε W, ) C 0 ) such that u ε u ε + V ε u ε = f 4.1) and u ε r f r for r =,. Moreover, u ε C 0 ). Since V ε fulfills H) and H4) with uniform constants, Proositions 3. and 4.1 yield u ε r, V ε u ε r C 1 f r for r =,. Here and below the constants do not deend on ε.) Lemma.1 then gives u ε C f. For a suitable sequence ε n ), u εn thus converges uniformly on comact sets to a continuous function u. Combined with Lemma.1, this fact yields that u C 1 ). Due to 4.1), also u εn converges uniformly on comact sets. Local ellitic regularity see e.g. Theorem 8.8 and Lemma 9.16 of [18]) now imlies that u W,q loc RN ) for every q <. Moreover, u u + V u = f and V u r, u r C 1 f r for r =,. Therefore u W, ), and hence u, u C 0 ) by Sobolev s embedding theorem. We next show that V u belongs to C 0 ). Take η C ) such that 0 η 1, η = 1 in Bx 0, R), η = 0 outside Bx 0, R), η c/r, and D η c/r. Then ηu ηu) + V ηu = ηf η u u η and Proosition 4.1 alied on ηu D ) shows that V x 0 )ux 0 ) V ηu C 3 [ ηf + 1 R u + 1 ] R u. Since u, f C 0 ), the above inequality imlies that V u C 0 ). Thus u = u + V u f C 0 ) and u D. This roves that I A has dense range and concludes the roof in the case where U = V. Finally, we deal with the general case U V c 1 U. Let V t = U + tv U), A t = + F V t, DA t ) = D, 0 t 1, and observe that A t = A 0 + V t U). Since U1 A 0 ) 1 C by Proosition 4.1, we have V t U)1 A 0 ) 1 tc 1 1)C. Thus standard erturbation theory for contraction semigrous shows that A t generates a contraction semigrou C 0 ) if tc 1 1)C < 1. Due to V t U, the maximum rincile [, Proosition ] yields 0 1 A t ) 1 1 A 0 ) 1 whenever 1 A t ) 1 exists. Then 0 V s V t )1 A t ) 1 c 1 1)s t)u1 A 0 ) 1 if s t, and hence V s V t )1 A t ) 1 s t)c 1 1)C. The same argument as before alies and roves that A t generates a contraction semigrou in C 0 ) for tc 1 1)C <. Iterating this rocedure a finite number of times, the roof is comlete. 13

14 In the following theorem we show that the semigrou is analytic in C 0 ) if one more condition holds. In the roof we need the next lemma, which is of interest in itself. Lemma 4.3 Let 1 < <, 1 < q, A = A, D ), A q = A, D q ) and assume that the hyotheses of Theorem 3.4 hold for A if 1 < < and that those of Theorem 3.4, res. Proosition 4., hold for A q if q <, res. q =. If q < and f L ) L q ) or if q = and f L ) C 0 ), then λ A ) 1 f = λ A q ) 1 f for Re λ > 0, and hence T t) = T q t). Proof. First let q <. Then the roof of Theorem 3.4 alies and shows that the oerator λ A, Re λ > 0, with domain W, U RN ) W,q U RN ) is invertible in L ) L q ). Hence the equation λu Au = f with f L ) L q ) has a unique solution u W, U RN ) W,q U RN ) and the assertions easily follow. If q =, the roof is similar using Theorem 3.4 and Proosition 4.. Theorem 4.4 Assume that H3) and H4) hold, that V is continuous, and that H) is satisfied for all γ > 0. If θu + div F 0 for some θ 0 then A = + F V with DA) = D generates a bounded analytic C 0 semigrou T ) on C 0 ). Proof. Again we assume that C γ = 0, relacing A by A w for some w = wγ) 0, where we fix γ > 0 such that the hyotheses of Theorem 3.4 and Proosition 4. hold. We fix N, ) with > θ such that Theorem 3.4 is valid. Thus A, W, U RN )) generates an analytic semigrou in L ). Let f C ) have comact suort and Re λ > 0. Due to Proosition 4. there exists u D such that λu Au = f. The above lemma shows that u W, U RN ) and hence λ u + Au C f by Theorem 3.4. Proosition 3. further yields D u + Uu C f. Using these estimates as well as Hölder s and the Gagliardo Nirenberg inequalities, we conclude that λ 1 U 1 u C f and λ 1 u C f. Taking into account Proosition.3, we have shown that D u + Uu + U 1 u + λ 1 u + λ 1 U 1 u + λ u C f. 4.) Here and below C is a generic constant indeendent of λ and f.) Fix x 0 and a smooth function η such that 0 η 1, η = 1 in Bx 0, r/), su η Bx 0, r), ηx) c/r, and D ηx) c/r. Observe that ληu) Aηu) = ηf u η u η uf η. Alying 4.) to ηu and emloying H4), we obtain D u, r + Uu, r + U 1 u, r + λ 1 u, r + λ 1 1 U u, r + λ u, r C f,r + 1 r u,r + 1 r u,r + 1 ) r U 1 u,r Cr N/ f + 1 r u,r + 1 r u,r + 1 r U 1 u,r ). From the comactness of the embedding W 1, B 1 ) C B 1 ) and a rescaling argument it follows that for every ε > 0 there is C ε > 0 such that v, r C 1r N/ ) εr v, r + C ε v, r v, r C 1r N/ ) εr D v, r + C ε v, r for v belonging to the resective saces, where C 1 does not deend on ε, r. Hyothesis H) further yields U 1 u), r U 1 u, r + C Uu, r 14

15 so that U 1 u, r C 1r N/ ) εr U 1 u, r + C εr Uu, r + C ε U 1 u, r. Using the above inequalities, one deduces λ 1 u, r + λ 1 1 U u, r + λ u, r C 3 εr λ 1 + C ε ) f + 1 r u + 1 r u + 1 ) r U 1 u. We set r = M λ 1 and let x 0 vary in. Then we have λ 1 u + λ 1 U 1 u + λ u C 3 εm + C ε) f + λ Taking first a small ε and then a large M, we arrive at M u + λ 1 M u + λ 1 ) M U 1 u. λ 1 u + λ 1 U 1 u + λ u C f. 4.3) In view of the rescaling made at the beginning, this establishes the theorem for A w by density, that is, with f = λu + ωu Au in 4.3). Since V c 0 > 0, the oerator A + c 0 / satisfies the assumtions of Proosition 4. and thus generates a contraction semigrou on C 0 ). It is then easy to verify 4.3) also for A, that is, with f = λu Au. We state two useful consequence of the above roof refining our revious ariori estimates. Corollary 4.5 Under the hyotheses of Theorem 4.4, if Re λ > 0 and λu Au = f, then u + U 1 u + Uu + λ 1 u + λ 1 U 1 u + λ u C f. Proof. Since V c 0 > 0, the oerator A is invertible on C 0 ) and the closedness of A, D ) yields u + Uu C Au C ) λu Au + λ u. It now suffices to combine 4.3) for A with Proosition.3 to conclude the roof. Corollary 4.6 Under the hyotheses of Theorem 3.4, if Re λ > 0 and λu Au = f, then u + U 1 u + Uu + λ 1 u + λ 1 U 1 u + λ u C f. Proof. In 4.) we have roved the statement for A w, continuous f with comact suort and > N. The roof given in Theorem 4.4 extends to all > 1 and all f L ). Moreover, as above the initial rescaling does not matter since A + θ)c0 is dissiative. 5 Generation of an analytic semigrou on L 1 ) Also in this section we assume that A 0 =. We observe that Proosition.3 shows that A = + F V with domain D 1 = {v L 1 ) : v L 1 ), Uv L 1 )} is well defined in L 1 ). Recall that C0 ) is dense in D 1 by Lemma.5. It is known that D 1 embeds continuously into W 1, ) for every <, see e.g. [34, Theorem 5.8]. 15 N N 1

16 Proosition 5.1 Assume that H) H5) hold for some γ > 0 and θ < 1. Then A, D 1 ) generates a contraction semigrou T 1 t) in L 1 ). If also the assumtions of Theorem 3.4 for some 1, ), res. of Proosition 4. if =, are satisfied, then T 1 t)f = T t)f for f L 1 ) L ), res. f L 1 ) C 0 ). Proof. Let u C0 ) be real and set f = u F u + V u. If we multily this equation by sign u, integrate by arts, and use the dissiativity of the Lalacian on L 1 ), we estimate V + divf ) u dx u) signu dx + V u F u) signu dx = f signu dx f 1 and therefore 1 θ)uu 1 f 1. Taking into account Proosition.3 and roceeding as in the roof of Proosition 3., we obtain the closedness of A, D 1 ). Moreover, A, D 1 ) is dissiative by Lemmas.5 and.6. As before we may assume that C γ = 0 to show the surjectivity of I A. We first assume that F = 0. The general case is then deduced by means of the continuity method, cf. [18, Theorem 5.]. We kee the notation introduced in the roof of Theorem 3.4. For f L 1 ) there is u ε L 1 ) such that u ε u ε + V ε u ε = f. Then u ε 1 f 1, U ε u ε 1 C f 1, 5.1) with C indeendent of ε since V ε and U ε satisfy the assumtions with uniform constants. It follows that u ε 1 + C) f 1. We fix 1 < < N N 1. Then u ε W 1, ) C 1 f 1 and so there is a sequence ε n ) converging to 0 such that u n := u εn tends weakly in W 1, ) to a function u. By the Rellich-Kondrachov theorem, a subsequence u nk ) converges to u in L 1 loc RN ) and after relabeling we have u n u almost everywhere. We infer from 5.1) that u 1 f 1 and Uu 1 C f 1. Moreover, u n u in L 1 loc RN ) and therefore u u + V u = f and u D 1. The last assertion can be shown as in Lemma 4.3. In order to establish the analyticity of T ) on L 1 ) we emloy a duality argument taken from [31, Theorem ], see also [1]. Under the assumtions of the revious roosition, we define the formal adjoint of A by A v = v F v V + divf )v. This oerator has the same structure as A with the new drift F = F and the new otential Ṽ = V + divf. Set Ũ = 1 θ)u. Hyotheses H) H5) yield Ṽ Ũ and that F and Ũ satisfy H) and H4). In addition, we suose that divf c U for some constant c 0 and that V is continuous. Then H3) holds for Ũ and Ṽ and c 1 + c 1 θ Ũ + div F Ṽ + div F = V 0. We can therefore aly Theorem 4.4 and thus A induces as in a generator A of a bounded analytic semigrou on C 0 ). Observe further that Au)v dx = ua v) dx for u C0 ) and v W,1 loc RN ). Due to [31, Lem.7.3.8] we have { } u 1 = su uϕ dx : ϕ C0 ), ϕ 1 for u L 1 ). 5.) 16

17 Theorem 5. Assume that H) H5) hold for all γ > 0 and some θ < 1, that V is continuous, and that divf c U for some constant c 0. Then A, D 1 ) generates a bounded analytic semigrou on L 1 ). Proof. Let u C0 ) and Re λ > 0. As observed above, the formal adjoint A induces a generator A of a bounded analytic semigrou on C 0 ) by Theorem 4.4. In 5.) we relace ϕ by ϕ = λ A )v with v DA ) and ϕ 1. Since v M/ λ, we obtain { u 1 su uλ A )v dx : v DA ), v M } R λ { N = su vλ A)u dx : v DA ), v M } R λ N M λ λ A)u 1. This inequality imlies the assertion because of Proosition 5.1 and an aroximation argument. 6 Consequences In this section we establish several regularity roerties of the semigrous obtained above. In order to treat the scale of L -saces simultaneously, and for the sake of simlicity, we assume here that Au = u + F u V u, that H) H5) are satisfied for every γ > 0 and some θ < 1, and that V is continuous. When we need the analyticity in L 1 ), we also have to suose that divf c U for some c 0. Theorem 3.4 and Proosition 5.1 show that A with domain DA ) = D generates a contraction semigrou T t)) t 0 on L ) for 1 < and on C 0 ) for =. Moreover, T t)f = T q t)f if f L ) L q ), by Lemma 4.3 and Proosition 5.1. Therefore we dro the index and write simly T t). Observe that the analyticity estimate AT t)f Ct 1 f for t > 0 and the domain characterization of the revious sections yield T t)f Ct 1 f. Hence the gradient estimate T t)f Ct 1/ f holds which is well known for uniformly ellitic oerators and is also satisfied by certain oerators with unbounded coefficients, see e.g. [3]. We next associate a sesquilinear form a in L ) with the oerator A, D ) setting au, v) = u v + V uv vf u) dx 6.1) for u, v Da) = {f W 1, ) : U 1 f L )}. The form is well defined thanks to H4). Moreover, as in Lemma.5 it can be verified that C0 ) is dense in Da). Integrating by arts and using H5), with =, we deduce Re au, u) = u + V u + 1 divf u ) dx u + 1 θ ) U 1 u. 6.) This shows that a is a closed ositive form. Since au, v) = Au, v) for u D and v Da), a is sectorial by Proosition 3.1 and the m sectorial oerator induced by a coincides with A, since we know that D Da). If divf c U, the adjoint form a u, v) = av, u) = u v + V + divf)uv + vf u) dx, 17

18 with Da ) = Da) has the same roerties as a and defines the adjoint oerator A, D ), where A v = v F v V + divf )v. We oint out that this aroach yields the analyticity of our semigrou T ) in L ) for 1 < < see [33, Theorem 1.1]) but not the descrition of the domain of the generator. We refer to [15], [33], and the references therein for recent develoments of these methods using rather weak assumtions. In the roof of Proosition 4. we have already roved the ositivity of T ), emloying the maximum rincile for functions in D, [, Proosition ], to show the ositivity of the resolvent on C 0 ). Here we give an alternative roof, based on form methods. Proosition 6.1 The semigrou T ) is ositive, i.e., f 0 imlies T t)f 0. Proof. It suffices to treat the case =, due to Lemma 4.3 and Proosition 5.1. Observe that the form a defined in 6.1) is real and if u Da) is real valued, then u +, u Da) and au +, u ) = 0. The ositivity of T t) on L ) thus follows from the non symmetric first Beurling Deny criterion, see [30, Theorem.4], since its generator A, D ) is the oerator associated to a. The semigrou further imroves the integrability of the initial datum. Proosition 6. For every t > 0 and 1 < <, T t) mas L ) into C 0 ) and thus also into L q ) for q, ). Proof. It suffices to show that T t) mas L ) into C 0 ). Let > 1 and f L ). Since T ) is analytic, T t)f W, U RN ) W, ). If > N/ we are done due to Sobolev s embedding theorem. If N/ we use Sobolev s embedding theorem and the semigrou law) finitely many times to deduce that T t)f L r ) for some r > N/ and conclude as above. In order to deal with the case = 1 and to give an estimate of the oerator norm of T t) acting from L ) into L 1 ), we use form methods as in the roof of Proosition 6.1. The oerator norm of T t) : L ) L q ) can then easily be estimated by interolation. Proosition 6.3 There is a ositive constant M such that T t)f Mt N 4 f 1 for t > 0 and f L 1 ). If divf c U, then T t)f Mt N f 1 for t > 0 and f L 1 ). Proof. Let us first show that T t)f Mt N/4 f 1, 6.3) without assuming that divf c U. To this aim we emloy the sesquilinear form a introduced in 6.1) and observe that Re au, u) u for every u W, U RN ) due to 6.). Therefore, Sobolev s inequality yields u +4/N Cau, u) u 4/N 1 for every u W, U RN ). Now 6.3) follows as in [15, Theorem.4.6] since T ) is analytic in L ). Next we assume that divf c U and consider the adjoint A v = v F v V + divf )v with domain D. It is associated with the adjoint form a and therefore it is the generator of the adjoint semigrou T t) ) t 0. Therefore 6.3) holds with T t) instead of T t) and yields, by duality, By the semigrou law, the roof is comlete. T t)f Mt N/4 f. 18

19 It is well-known that the estimate T t)f Ct N/ f 1 imlies the existence of a bounded kernel k t x, y) such that T t)f ) x) = k t x, y)fy) dy and, moreover, su x,y k t x, y) Ct N/. This kernel is ositive by Proosition 6.1. In the following roosition we show the comactness of T ) assuming that the otential V tends to infinity as x. Proosition 6.4 Assume that lim x V x) =. Then D is comactly embedded in L ) if 1 < and in C 0 ) if =. The sectrum of A, D ) is indeendent of [1, ]. Moreover, T t), t > 0, is comact in L ) for 1 < < and in C 0 ). T t), t > 0, is comact in L 1 ) if divf c U. Proof. Let F be a bounded set in D and assume that 1 <. By the assumtion, given ε > 0, we can find R > 0 such that fx) dx ε x >R for every f F. At this oint the comactness of F in L ) easily follows from the comactness of the embedding of W, B R ) into L B R ). A similar argument works for =. The -indeendence of the sectrum is now a consequence of e.g. [3, Proosition.6]. Since T ) is analytic, its comactness is equivalent to the comactness of the embedding of D into L ), res. C 0 ). An analytic semigrou S ) on a Banach sace X with generator B has maximal regularity of tye L q 1 < q < ) if for each f L q [0, T ], X) the function t ut) = t St s)fs) ds belongs to 0 W 1,q [0, T ], X) L q [0, T ], DB)). This means that the mild solution of the evolution equation u t) = But) + ft), t > 0, u0) = 0, is in fact a strong solution and has the best regularity one can exect. It is known that this roerty does not deend on 1 < q < and T > 0. In recent years this concet has thoroughly been studied and alied in various directions, see e.g. [], [16], [35], and the references therein. For our uroses we only need the following facts. Let X = L ) for some 1 < <. Then the oerator B has maximal regularity of tye L q if its imaginary owers satisfy B) is Me a s for some a [0, π/) and all s R thanks to the Dore Venni theorem, see e.g. [, Theorem II ]. If = and B is maximal dissiative and invertible, then B) is Me π s / by a result due to Kato, [0, Theorem 5]. Hence, if = and B is regularly dissiative, then B) is Me a s for a = π/ φ and some φ 0, π/]. Moreover, if B generates a ositive contraction semigrou on L ), then B) is M ε exε + π/) s ) for each ε > 0 and s R because of the transference rincile [9, 4], see [8, Theorem 5.8]. If we combine these facts with the Riesz-Thorin interolation theorem, Proosition 3.1, Theorem 3.4, and Proosition 6.1, we obtain the following result. See also [5, Theorem 7.1].) Proosition 6.5 A has maximal regularity of tye L q on L ) for all 1 <, q <. 7 Generalized Ornstein Uhlenbeck oerators In this section we investigate the oerator A Φ,G u = u Φ u + G u 19

20 on L, µ), 1 < <, where µdx) = e Φ dx. Under the following assumtions on Φ : [0, ) and G : we show that A Φ,G with domain W,, µ) generates a ositive analytic semigrou T ) on L, µ) emloying our revious results. A1) Φ C, R), G C 1, ), and e Φx) dx <. A) For each ε > 0 there is c ε > 0 with div G + D Φ ε Φ + c ε. A3) There is a constant w R such that G Φ div G w. A3 ) div G = G Φ. A4) There is a constant d > 0 such that G d Φ + 1). Below we will see that A1) and A3 ) say that the finite measure µdx) is, u to a multilicative constant, the unique invariant measure of T ). In fact, A3 ) is only needed for this urose. Moreover, A) with ε = 1 and A3) imly that Φ Φ + G) w + c 1 =: w, i.e., Φ is a Liaunov function for the ODE corresonding to the shifted vector field Φ + G w. Lemma 7.3 also shows that A3) imlies the dissiativity of A Φ,G w. We can omit A3) if the constant d in A4) is small enough, see Remark 7.5. The growth assumtions A) and A4) allow us to use Theorem 3.4 and to comute the domain. Notice that the above hyotheses simlify considerably if G = 0, i.e., for the symmetric) oerator A Φ := A Φ,0, see Corollary 7.8. It can be roved that the analyticity of the semigrou fails in some cases where A4) is not satisfied, see [6]. We need some roerties of the weighted Sobolev saces { } W k,, µ) = u W k, loc RN ) : D α u L, µ) if α k. First of all, we observe that C 0 ) is dense in W k,, µ). This can be seen as in Lemma.5. In the following lemma we do not need the full strength of hyothesis H) but only a weaker, one sided estimate. Lemma 7.1 Let 1 < <, assume that A1) holds, and that Φ + )1 + Φ ) 1 D Φ Φ, Φ ε Φ + c ε for some ε < 1. Then the ma u Φ u is bounded from W 1,, µ) to L, µ) and the ma u Φ u is bounded from W,, µ) to L, µ). In articular, A Φ,G : W,, µ) L, µ) is bounded in view of A4). Proof. Fix 1, ). Since C 0 ) is dense in W,, µ), it suffices to rove that Φ u L µ c u L µ + u L µ ) for u C 0 ) and some constant c > 0. Observe that there are two constants a, b > 0 such that Φ a1 + Φ ) 1 Φ + b, 0

21 so that we only have to estimate 1 + Φ ) 1 Φ u e Φ dx. An integration by arts yields 1 + Φ ) 1 Φ u e Φ dx = u 1 + Φ ) 1 Φ e Φ ) dx R N = u 1 + Φ ) 1 Φ µdx) + ) + ε + u u 1 + Φ ) 1 Φ u µdx) 1 + Φ ) 1 Φ u µdx) + c ε 1 + Φ ) 1 Φ u 1 u µdx). Using the inequality 1 + t ) 1 η1 + t ) 1 t + c with η = 1 ε c ε 1 ε 1 + Φ ) 1 Φ u µdx) u 1 + Φ ) D Φ Φ, Φ µdx) 1 + Φ ) 1 u µdx) 1 + Φ ) u µdx) and Hölder s inequality, we obtain ) 1 1) Φ u µdx) ) 1 + C u µdx) is the conjugate of ). Now, since 1 + t ) 1) t c t ) 1 t + c for certain constants c 1, c > 0, the conclusion follows from Young s inequality. The following lemma can be roved suosing only the one sided estimate Φ + 1)1 + Φ ) 1 D Φ Φ, Φ ε Φ + c ε. for some ε < 1. This can be done imitating the roof of Lemma 7.1. Since we need hyothesis A) in the sequel, we refer to assume it to shorten the roof. Lemma 7. Suose that 1 < < and that hyotheses A1) and A) hold. u Φ u is bounded from W,, µ) to L, µ). Then the ma Proof. We aly Lemma 7.1 to the vector) function u Φ and we obtain ) u Φ L µ C u Φ W 1, C u Φ µ L + ud µ Φ L + u Φ µ L µ ) C u W, + ε u Φ µ L µ + c ε u L µ. In the second inequality we have used Lemma 7.1 twice and A). Taking a small ε, we establish the assertion. Lemma 7.3 Assume that A1) and A3) hold. For u C0 ) and v W 1,1 loc RN ) we have A Φ,G u)v µdx) = u v µdx)+ G Φ divg) uv µdx) ug v µdx). 7.1) In articular, A Φ,G w defined on C 0 )) is dissiative in L, µ) for 1 < <. Moreover A Φ is symmetric in L, µ). 1

22 Proof. The first assertion is straightforward to check and imlies the last one. Let. Take a real u C0 ) and set u = u u. Then u = 1) u u and therefore A Φ,G u)u µdx) = 1) u u µdx) + G Φ divg) u µdx) R N 1) u G u µdx). 7.) This equality and A3) imly that A Φ,G u)u µdx) w u, as claimed. If 1, ), we relace u by η + u ) 1/. By similar calculations and letting η 0, one also obtains 7.), and thus the dissiativity of A Φ,G w/. We now come to the main result of this section. We say that a finite Borel measure ν on is an invariant measure for a semigrou T ) on C b ) if T t)f νdx) = f νdx) for f C b ), t ) Theorem 7.4 Assume that assumtions A1), A), A3), and A4) are satisfied. Then the oerator A Φ,G = Φ + G with domain DA Φ,G ) = W,, µ) generates a ositive analytic C 0 semigrou T ) on L, µ), 1 < <, such that T t) e tw/, where µdx) = e Φx) dx. Further, µ is an invariant measure of T ) if and only if A3 ) holds in addition. Moreover, T t) is symmetric if = and G = 0. Finally, if Φx) as x, then T t), t > 0, is comact in L, µ) and the sectrum of A Φ,G is indeendent of for 1 < <. Proof. Fix 1, ). We make a change of variable in order to work with an oerator on L, dx) instead of L, µ). Namely we define the isometry J : L, µ) L ), Ju = e Φ u. A straightforward comutation shows that A Φ,G u = J 1 AJu for u C 0 ), where [ ) ] Av = v + 1 Φ + G v 1 [ 1 1 ) ] Φ Φ G Φ v. We define F = 1) Φ + G and V = 1 [ 1 1 ) ] Φ Φ G Φ. 7.4) We now want to show that A satisfies assumtions H)-H5) for a suitable function U. Taking ε = ) in A) and using A3), we see that V ) Φ 1 c ε + w). 7.5)

23 We set therefore U = c ) Φ for a number c 0 > 0 to be determined below. Assumtion A) imlies that H) holds for every γ > 0. Set c = c c ε + w). Due to A) and A4), there is a ositive number c 1 0 such that 0 < c 0 U V + c c 1 U, i.e., H3) holds for V + c. It is clear that H4) follows from A4). It remains to choose c 0 so that H5) is satisfied. Using A) with ε = 1 8, we obtain if we set c 0 = 4c ε ) div F = div G 1 Φ ε 1 + ε Φ + c ε U ) Φ 1 + c ε 1 + ) 1,. Theorem 3.4 and Remark 3.5) thus shows that A, WU RN )) generates an analytic C 0 semigrou S ) on L ) for 1 < <, which is ositive and comact if Φx) as x ) due to Proositions 6.1 and 6.4, resectively. As a result, A Φ,G = J 1 AJ with domain DA Φ,G ) := {u L, µ) : Ju W, U RN )} generates a ositive analytic C 0 semigrou on L, µ), 1 < <, which is comact if Φx) as x. The indeendence of the sectrum of 1 < < is a consequence of the comactness of T ) as in Proosition 6.4. Suose that u DA Φ,G ). Then v := Ju W, ) and Φ v L ). Hence e Φ Dj u = 1 vd jφ + D j v L ), since Φ 1 + Φ. Moreover, from A) and Proosition.3 we deduce e Φ Dij u = 1 vd ijφ + D ij v + 1 D jvd i Φ + 1 D ivd j Φ + 1 vd iφd j Φ L ), i.e., u W,, µ). Conversely, take u W,, µ) and set v := Ju. Then, by Lemma 7.1, D j v = e Φ 1 ) ud jφ + D j u L ). Lemma 7. further imlies that Φ v L ). Using Lemma 7.1 and A), we obtain D ij v = e Φ D ij u 1 ud ijφ + 1 ud jφd i Φ 1 D jud i Φ 1 ) D iud j Φ L ). Thus u DA Φ,G ) and we have established DA Φ,G ) = W,, µ). In articular, C 0 ) is a core of A Φ,G. Therefore Lemma 7.3 yields the asserted estimate and, if G = 0, the symmetry of T t). Moreover, µ is an invariant measure of T ) if and only if A Φ,G u dµ = 0 for each u C 0 ). This roerty is equivalent to A3 ) by 7.1). Remark 7.5 In the above roof we have used A3) only to deduce 7.5) and to obtain the asserted estimate. Therefore A Φ,G with domain W,, µ) also generates a ositive analytic C 0 semigrou on L, µ) if we assume that A1) and A) hold and that G d Φ + ĉ for some constants 0 < d < 1 1 and ĉ > 0. 3

24 Remark 7.6 Observe that in the above theorem the domains of generators A Φ,G become smaller if we increase 1, ). Thus the semigrous T ) generated by A Φ,G on L, µ) and L q, µ) coincide on L, µ) if > q. As in Remark.7 one further sees that these semigrous also extend the semigrou on C b ) constructed in [5, 4]. Corollary 4.7 in [5] thus shows that the semigrous T ) in Theorem 7.4 are irreducible and have the strong Feller roerty, As a result, µ is the u to a multilicative constant) unique invariant measure of T ) if A3 ) holds due to [14, Theorem 4..1]. Remark 7.7 Observe that the change of variable used in the roof of Theorem 7.4 and the resulting oerator A and the auxiliary otential U deend on. Moreover, in the limiting cases = 1, hyotheses H) H5) may fail. This is not only a technical roblem, but some of the conclusions of Theorem 7.4 can in fact be wrong for = 1,. For examle, T ) is, in general, not analytic and even the inclusion DA Φ,G ) W 1,1, µ) may be false, see e.g. [4]. Corollary 7.8 Let Φ : R be a C function satisfying e Φx) dx <. Assume that for each ε > 0 there is c ε > 0 such that D Φ ε Φ + c ε. Then the oerator A Φ u = u Φ u with domain DA Φ ) = W,, µ) generates an analytic semigrou in L, µ) for 1 < <. Corollary 7.8 imroves Theorems. and.7 in [13]. These results require condition H) for all γ > 0 for the otential V with G = 0) defined in 7.4) and additional growth assumtions on Φ and D Φ. See also [11] for further develoments in the case =. Observe that if Φx) = ϕ x ) for x r 0 > 0 and G = 0, then H) says that ϕ ε ϕ + C ε on [r 0, ). This holds e.g. if ϕr) = r α, α > 0. As a articular case of the Theorem 7.4 we consider the Ornstein Uhlenbeck oerator Lux) = T rqd ux)) + Bx, ux), x, u C 0 ), where Q is a real, symmetric, ositive definite N N matrix and B is a real N N matrix whose eigenvalues are contained in the oen left half lane. The Ornstein Uhlenbeck semigrou is given by T t)φ)x) = 4π) N det Qt ) 1 φe tb x y)e 1 4 Q 1 t y,y dy, x, t > 0, φ C b ), where Q t := t 0 esb Qe sb ds, t > 0. We know from [7] that, by a change of variables, L is similar to the oerator A Φ,G u = u Φ u + G u, where Φx) := N j=1 x j 4λ j and Gx) := B 1 x for x. Here B 1 is a real N N matrix satisfying B 1 D λ = D λ B 1, 7.6) where D λ := diagλ 1,..., λ N ) and λ 1,..., λ N > 0. Condition 7.6) imlies that div G = G Φ = 0. It is easy to see that Φ and G satisfy the assumtions A1), A), and A4). Thus we obtain the following result, which was roved in [7, Theorem 3.4] using comletely different arguments. We also refer to [1], where the case = was treated, and to [7], where the symmetric case B 1 = 0 was studied with relaced by an searable Hilbert sace. Corollary 7.9 Suose that Q is a real, symmetric, ositive definite N N matrix and that B is a real N N matrix with eigenvalues in the oen left half lane. Then the Ornstein-Uhlenbeck oerator L with domain W,, µ), where µdx) = e Φx) dx, generates the Ornstein-Uhlenbeck semigrou T ) on L, µ) for 1 < <. Moreover, µ is the invariant measure of T ). 4