RESOLVENT ESTIMATES FOR ELLIPTIC OPERATORS ON COMPACT MANIFOLDS

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1 ON p RESOVENT ESTIMATES FOR EIPTIC OPERATORS ON COMPACT MANIFODS KATSIARYNA KRUPCHYK AND GUNTHER UHMANN Abstract. We prove uniform p estimates for resolvents of higher order elliptic self-adjoint differential operators on compact manifolds without boundary, generalizing a corresponding result of [3] in the case of aplace Beltrami operators on Riemannian manifolds. In doing so, we follow the methods, developed in [] very closely. We also show that spectral regions in our p resolvent estimates are optimal.. Introduction and statement of results The purpose of this paper is to extend the result of [3], see also [], for the aplace-beltrami operator g on a compact Riemannian manifold (M, g) without boundary of dimension n 3, to the case of higher order elliptic self-adjoint differential operators, and specifically to show how the methods of [] apply in this context. In [3] it was established that given δ > 0 small, there exists a constant C = C(δ) > 0 such that for all u C (M) and all ζ R δ, the following p resolvent bound holds, u C ( g ζ)u, (.) n 2 (M) n+2 (M) where R δ = {ζ C : (Im ζ) 2 4δ 2 (Re ζ + δ 2 )}. Notice that R δ is the exterior of a parabolic region, containing the spectrum of g, see Figure. We observe that the bound (.) cannot hold if R δ intersects the spectrum of g, as the latter is discrete. The interesting question, posed in [3] and subsequently studied in [], is how close R δ can come to the spectrum of g near infinity, while still having the uniform estimate (.). Thans to the wor [], we now that the region R δ is in general the maximal possible for the uniform estimate (.) to hold. Indeed, in [] it is shown that the region cannot be improved when M is the standard sphere, or more generally, a Zoll manifold, due to a cluster structure of the spectrum of g on such manifolds, [7]. As explained in [], any sharpening in the spectral region is related to improvements in estimates for the remainder term in the sharp Weyl law for g, which measures how uniformly its spectrum is distributed. Consequently,

2 2 KRUPCHYK AND UHMANN Im ζ R δ Re ζ Figure. Spectral region R δ in the uniform resolvent bound (.). improvements in the spectral region R δ are available for manifolds of nonpositive curvature and in the case of the torus with a flat metric, see [], and also [3]. The corresponding uniform p resolvent estimates for the standard aplacian on R n, n 3, were obtained in [9]. Here in contrast to the case of a compact manifold, the estimates are valid for all values of the complex spectral parameter ζ. In [5] the results of [9] were generalized to the case of non-trapping asymptotically conic manifolds. To formulate our results let us begin by fixing some notation. et M be a compact connected C manifold without boundary of dimension n 2, equipped with a strictly positive C volume density dµ. et P be a differential operator on M of order m with C coefficients. We assume that P is elliptic and formally self-adjoint with respect to dµ, P uvdµ = up vdµ, u, v C (M). M M et p(x, ξ) C (T M) be the principal symbol of P, which is a real-valued homogeneous polynomial in ξ of degree m. Since p(x, ξ) 0 for ξ 0 and T M \ {0} is connected, without loss of generality we shall assume, as we may, that p(x, ξ) > 0 for ξ 0. The order m of the operator P is therefore even. If we equip the operator P with the domain C (M), P becomes an unbounded symmetric essentially self-adjoint operator on 2 (M), i.e. P has a unique selfadjoint extension, which we shall denote again by P. The domain of the selfadjoint extension is D(P ) = H m (M), the standard Sobolev space on M. An application of Gårding s inequality implies that there exists a constant C > 0 such that P CI in the sense of self-adjoint operators. Thus, after replacing P by P + CI, we assume, as we may, that P 0.

3 p RESOVENT ESTIMATES 3 The spectrum of P is discrete, consisting only of real eigenvalues, where each eigenvalue is isolated and of finite multiplicity. et 0 λ λ 2... be the eigenvalues of P repeated according to their multiplicity, and let e, e 2,... 2 (M) be the corresponding orthonormal basis of eigenfunctions. Seeing to generalize (.), our goal is to find a region R C, for which there holds a uniform p bound of the form, u q (M) C R (P ζ)u p (M), u C (M), ζ R, (.2) for suitable p and q. Motivated by the classical Sobolev inequalities, we shall be interested in the estimate (.2) for pairs (p, q) belonging to the Sobolev line p q = m n, (.3) assuming that p < n/m. Following [, 3], we shall also require the pairs (p, q) to be on the duality line, p + =. (.4) q The restrictions (.3) and (.4) imply that p = n + m, q = n m, n > m. It is clear that the estimate (.2) can only hold away from the spectrum of P. Similarly to the case of g, when establishing the estimate (.2), we shall in fact be concerned with the case of ζ away from all of [0, ). Given ζ C\[0, ), it will then be convenient to write ζ = z m with z Ξ, where This is due to that fact that the map Ξ = {z C : arg(z) (0, 2π/m)}. f = f m : Ξ C \ [0, ), z z m, is a conformal isomorphism. This map extends continuously to f : Ξ C with f( Ξ) = [0, ). Notice that the region R δ in the uniform bound (.) satisfies R δ = f 2 (Ξ δ ), Ξ δ = {z C : Im z δ}, By analogy with this, it is natural to try to establish the estimate (.2) for ζ = z m, where z Ξ δ = {z Ξ : dist(z, Ξ) δ}, with δ > 0 small but fixed. We have Ξ δ = {z C : arg(z) (0, 2π/m), Im z δ, Im (ze 2πi/m ) δ}. Associated with the principal symbol p(x, ξ) of the operator P is the cosphere Σ x = {ξ T x M : p(x, ξ) = }, x M.

4 4 KRUPCHYK AND UHMANN We may notice that for each x M, the cosphere Σ x is a C compact connected hypersurface in R n, see the discussion before emma 2.9 below. The cosphere Σ x is called strictly convex if the second fundamental form is definite at each point of Σ x. This is equivalent to the fact that the Gaussian curvature of Σ x is non-vanishing. The following theorem is the main result of this paper, which is a generalization of the uniform estimate (.), obtained in [3], to the case of higher order elliptic self-adjoint differential operators. Theorem.. Assume that n > m 2 and that for each x M, the cosphere Σ x is strictly convex. Then given δ > 0 small, there is a constant C = C(δ) > 0 such that for all u C (M) and all z Ξ δ, the following estimate holds u n m (M) C (P zm )u n+m (M). (.5) In the case of an elliptic operator P of order m 4, letting R δ = f(ξ δ ), a straightforward computation show that for R > 0 sufficiently large, we have where R + δ R δ {ζ C : ζ R} = (R + δ R δ ) {ζ C : ζ R}, m m 3 :={ζ C : Im ζ (Re ζ) m mδ + O((Re ζ) m ), Re ζ 0} {ζ C : Im ζ (Re ζ) m m 3 m mδ O((Re ζ) m ), Re ζ 0}, and R δ := {ζ C : Re ζ 0}. Thus, for ζ sufficiently large, similarly to the case of g, the region R δ is the exterior of a parabolic neighborhood of the spectrum of the operator P, see Figure 2. As an example of an operator P to which Theorem. applies, one can consider P = ( g ), 2 < n, where g is the aplace Beltrami operator on a compact Riemannian manifold (M, g). Our proof of Theorem. relies on the approach, developed in []. The main ingredients here are the spectral cluster estimates, obtained in [5] in the case of the aplace Beltrami operator on a compact Riemannian manifold, and in [] in the case of higher order elliptic operators, the method of stationary phase, as well as the Hörmander ax parametrix for the operator e it m P for small times. et us remar that the strict convexity of the cospheres Σ x in Theorem. guarantees that the Fourier transform of the surface measure on Σ x has essentially the same decay at infinity, as that of the surface measure on the sphere, thans to the method of stationary phase, see [4, Theorem.2., p. 50]. This assumption also plays a crucial role in the derivation of the spectral cluster estimates for higher order elliptic operators in [].

5 p RESOVENT ESTIMATES 5 Im z Im ζ R δ Ξ δ Re ζ Re z Figure 2. The spectral regions Ξ δ and R δ = f(ξ δ ) in the uniform estimate (.5). We may also notice that the a priori estimate (.5) implies that the 2 resolvent of P, (P ζ), ζ C \ [0, ), is a bounded operator: n+m (M) n m (M), see Proposition 2.0 below. Our next result shows that the region Ξ δ in (.5) is in general optimal for higher order elliptic operators, since it cannot be improved for an operator whose principal symbol has a periodic Hamilton flow. This is due to the fact that the spectrum of such an operator is distributed in a non-uniform fashion, displaying a cluster structure, see [2] and [7]. Theorem.2. Assume that n > m 2 and that for each x M, the cosphere Σ x is strictly convex. Assume furthermore that the subprincipal symbol of the operator P vanishes, and that the Hamilton flow of the principal symbol p is periodic, with a common minimal period on p (). Then there exist (i) a sequence z Ξ such that Re z, 0 < Im z 0 as, and and (P z m ) n+m (M) n m,, (M) (ii) a sequence z Ξ such that Re (z e 2πi/m ), 0 < Im (z e 2πi/m ) 0 as, and (P z m ) n+m (M) n m,. (M) As an example of the operator P in Theorem.2 we can tae P = ( g ), 2 < n, on a Zoll manifold M, similarly to the case when = in []. To prove Theorem.2 we shall also use the methods of [].

6 6 KRUPCHYK AND UHMANN The paper is organized as follows. Section 2 is devoted to the proof of Theorem. while Section 3 contains the proof of Theorem Proof of Theorem. 2.. Formula for the resolvent (P z m ) based on a half wave group for P /m. We shall denote by Ψ µ cl (M) the space of classical pseudodifferential operators of order µ on M. et Q = P /m be defined by the spectral theorem. According to Seeley s theorem, see [4, Theorem 3.3.], we have Q Ψ cl (M) with the principal symbol q = p /m. Furthermore, D(Q) = H (M), and the eigenvalues of Q are µ j = λ /m j, j =, 2,.... etting z Ξ and following [], let us derive a natural formula for the 2 resolvent (P z m ). To that end, we write (P z m ) = m z (Q), where the multiplier m z (Q) is given by m z (τ) = (τ m z m ). Using the inverse Fourier transform, we get m z (τ) = m z (t)e itτ dt, m z (t) = 2π τ m z m e itτ dτ. We shall need the following result. emma 2.. et z Ξ. Then for any t R, we have τ m z m e itτ dτ = 2πi mz m e 2πi/m+i t τ, (2.) where τ = ze 2πi/m, = 0,,...,. Here Im τ > 0, = 0,,...,. Proof. To show (2.) we shall use the residue calculus. To that end writing z = z e iϕ, 0 < ϕ < 2π/m, we obtain that the poles of the rational function C τ (τ m z m ) are given by τ = z e i(mϕ+2π)/m = ze 2πi/m, = 0,..., m. Notice that the poles are simple, none of them are on the real line, the poles τ, = 0,..., m/2, are in the upper half plane, and the poles τ, = m/2,..., m, are in the lower half plane. We have e itτ = e timτ. et first t 0. Deforming the contour of integration in the upper half plane, we get ( ) e itτ τ m z m e itτ dτ = 2πi Res τ m z ; τ e itτ m = 2πi mτ m = 2πi e 2πi/m itτ, t 0. mz m

7 p RESOVENT ESTIMATES 7 et now t > 0. Then by deforming the contour of integration in the lower half plane, we conclude that m ( ) e itτ m τ m z m e itτ dτ = 2πi Res τ m z ; τ e itτ m = 2πi mτ m = 2πi mz m m =m/2 =m/2 e 2πi/m itτ = = 2πi e 2πi/m+itτ, t > 0. mz m 2πi mz m Thus, (2.) follows. The proof of emma 2. is complete. et z Ξ. Then by (2.), we obtain that m z (τ) = i e 2πi/m mz m e i t τ +itτ dt. Therefore, we have the following formula for the resolvent of P, (P z m ) = m z (Q) = i e 2πi/m mz m Here τ = ze 2πi/m and Im τ > 0, = 0,,..., m/2. =m/2 e πi e 2πi/m itτ m/2+ e i t τ e itq dt. (2.2) 2.2. Consequences of the spectral projection estimates. Assume that, for each x M, the cosphere Σ x = {ξ T x M : q(x, ξ) = } is strictly convex. Consider the th spectral cluster of the operator Q, {µ j spec(q) : µ j [, )}, and denote by χ the spectral projection operator on the space, generated by the eigenfunctions, corresponding to the th spectral cluster, χ f = E j f, f C (M). µ j [,) Here E j : 2 (M) 2 (M) is the orthogonal projection onto the space, spanned by e j, i.e. ( ) E j f(x) = f(y)e j (y)dµ(y) e j (x). M It was shown in [], see also [4, Theorem 5..], that for p 2(n+), we have n ( χ 2 (M) p (M) C σ(p), σ(p) = n 2 ) p 2, (2.3)

8 8 KRUPCHYK AND UHMANN where C > 0 is a constant, and the dual estimate, χ p (M) 2 (M) Cσ(p), p = p p. (2.4) Similarly to [, emma 2.3], we have the following consequence of the spectral clusters estimates (2.3) and (2.4). emma 2.2. Assume that, for each x M, the cosphere Σ x = {ξ Tx M : q(x, ξ) = } is strictly convex. et α C([0, ), C) and define the operators α (Q) as follows, α (Q)f = α(µ j )E j f, f C (M), µ j [,) =, 2,.... Then if p 2(n+), we get n α (Q)f p (M) C 2σ(p) ( sup τ [ ;) α(τ) ) f p p (M), σ(p) = n ( 2 p where C > 0 is a constant independent of the function α. ) 2, (2.5) Proof. First notice that α (Q) = χ α (Q). et p 2(n+). Then using the n spectral clusters estimates (2.3) and (2.4), we obtain that α (Q)f p (M) C σ(p) α (Q)f 2 (M) ( = C σ(p) α(µ j ) 2 E j f 2 2 (M) C σ(p) ( µ j [,) sup τ [,) ( α(τ) ) µ j [,) = C σ(p) ( sup α(τ) ) χ f 2 (M) τ [,) C 2σ(p) ( sup τ [,) α(τ) ) f p p (M). ) /2 E j f 2 2 (M) ) /2 emma 2.3. Assume that for each x M, the cosphere Σ x = {ξ T x M : q(x, ξ) = } is strictly convex. et α C([0, ), C) be such that Then we have A = sup ( + τ m ) α(τ) <. (2.6) τ [0, ) α(q)f n m (M) CA f n+m (M), (2.7)

9 where α(q) is the operator defined by α(q)f = α(µ j )E j f, p RESOVENT ESTIMATES 9 j= f C (M), and C > 0 is a constant independent of the function α. Proof. To establish (2.7), we shall follow [, emma 2.3], see also [9], and use the one dimensional ittlewood Paley theory. To that end, let {, t [/2, ), χ(t) = 0, t / [/2, ), be the characteristic function of the interval [/2, ). Setting χ j (τ) = χ(2 j τ), we obtain the dyadic partition of unity in [0, ), χ 0 (τ) + j= χ j(τ) =, where χ 0 (τ) = when τ [0, ), and χ 0 (τ) = 0 otherwise. Define α j (τ) = α(τ)χ j (τ), j = 0,,.... Assume that we have proved that α j (Q)f S f n m (M) n+m, j = 0,,..., (2.8) (M) with some constant S > 0. By the ittlewood Paley theorem and Minowsi s inequality, we conclude from (2.8) that α(q)f n m (M) C q,ps f n+m (M), (2.9) where C q,p > 0 depends on q and p only, see [9] and [0]. et us present these arguments for the convenience of the reader. We shall write p = and q = n+m. Then < p < 2 < q. As q >, by ittlewood Paley theorem, we get n m ( ) /2 α(q)f q (M) C q α j (Q)f 2 q (M) j=0 /2 = C q α j (Q)f 2 j=0 q/2 (M) := I. As q/2, we may write from Minowsi s inequality that ( ) /2 ( /2 I C q α j (Q)f 2 q/2 (M) = C q α j (Q)f 2 (M)) := I q 2. j=0 j=0 As χ j = χ 2 j, j = 0,,..., it follows from (2.8) that ( ) /2 I 2 C q S χ j (Q)f 2 p (M) j=0 ( { = C q S M χ j (Q)f(x) p dµ(x)} l 2/p ) /p := I 3,

10 0 KRUPCHYK AND UHMANN where {a j } l 2/p denotes the l 2/p norm of the sequence {a j }. Since 2/p >, by Minowsi s inequality, ( /p ( ) /2 I 3 C q S { χ j (Q)f p } l 2/pdµ) = C q S χ j (Q)f 2 M p (M) C q C p S f p (M), which shows (2.9). Thus, we are left with proving (2.8). et f C (M). For j =, 2,..., we write α j (Q)f = α j (µ l )E l f = α j (µ l )E l f = l= 2 j 2 j r= µ l [2 j,2 j ) µ l [2 j +r,2 j +r) where the truncated operator α j, (Q) is given by α j, (Q)f = α j (µ l )E l f. Since get 2(n+) n m n α j (Q)f 2 j C r= µ l [,) j=0 2 j α j (µ l )E l f = α j,2 j +r(q)f,, by (2.5) and the fact that σ(/(n m)) = (m )/2, we n m (M) 2j r= α j,2 j +r(q)f n m (M) (2 j + r) m ( sup τ [2 j +r,2 j +r) Now using (2.6), we obtain that α j (Q)f n m (M) r= α(τ) ) f, j =, 2,.... n+m (M) 2j CA (2 j + r) m (2 j + r ) f m r= 2 j CA r= for j =, 2,.... We also have n+m (M) (2 j 2) m (2 j ) m f n+m (M) CA f n+m (M), α 0 (Q)f = and therefore, it follows from (2.5) that µ l [0,) α(µ l )E l f, (2.0) α 0 (Q)f C( sup α(τ) ) f CA f n m (M) τ [0,) n+m (M) n+m. (2.) (M)

11 p RESOVENT ESTIMATES We obtain (2.8) as a consequence of (2.0) and (2.). The proof of emma 2.3 is complete Derivation of the resolvent estimate with bounded z. et us first prove the resolvent estimate (.5) for all z Ξ δ when z is bounded by a fixed constant, i.e. z Ξ δ {z C : z D}. To that end, consider the multiplier m z (τ) = τ m z m, τ [0, ). First notice that τ m z m 0 for all τ 0 and all z C with arg(z) (0, 2π/m). Then by continuity of τ m z m on a compact set, we have that for any A, D, δ > 0, there exists a constant C > 0 such that τ m z m /C for τ [0, A] and z Ξ δ {z C : z D}. For τ large and z Ξ δ {z C : z D}, we have τ m z m τ m, and therefore, we conclude that m z (τ) C δ,d ( + τ m ) uniformly in z Ξ δ {z C : z D}. By appealing to emma 2.3, we obtain the resolvent estimate (.5) for z Ξ δ {z C : z D}. Remar 2.4. Notice that applying emma 2.3, we can immediately obtain the (non-uniform) estimate for all ζ C \ [0, ) and u C (M). u n m (M) C ζ (P ζ)u n+m (M), 2.4. Uniform bounds for a local term in the case of unbounded z. et z Ξ δ {z C : z }. Here it will be convenient to use the representation (2.2) for the multiplier m z (Q). To define the localized version of m z (Q), we fix a function ρ C (R) satisfying ρ(t) = {, t ε/2, 0, t ε, (2.2) where 0 < ε < /2 will be specified later. In view of (2.2), the localized version of m z (Q) is given by m loc z (Q)f = i mz m e 2πi/m ρ(t)e i t τ e itq fdt, f C (M). (2.3) Here τ = ze 2πi/m and Im τ > 0, = 0,,..., m/2. To prove the resolvent estimate (.5) for z Ξ δ {z C : z }, let us first establish this estimate for m loc z (Q), i.e. m loc z (Q)f n m (M) C f n+m (M). (2.4)

12 2 KRUPCHYK AND UHMANN When doing so we shall use a dyadic partition of the t interval in the definition (2.3) of m loc z (Q). To that end let ψ C0 (R) be such that supp (ψ) [ 2, 2], ψ = on [, ], and ψ is even. Define β(t) = ψ(t) ψ(2t). Thus, and β(t) = 0, t / [/2, 2], + j= It will be convenient to write, ρ(t) = Notice that ρ(t) = 0 when t. β(2 j t) =, t 0. + j=0 β(2 j t) C 0 (R). For a given z Ξ δ {z C : z }, we define the multipliers S z,j (τ) = and We have i e 2πi/m mz m S z (τ) = i e 2πi/m mz m β(2 j z t)ρ(t)e i t τ e itτ dt, j = 0,, 2,..., (2.5) ρ( z t)ρ(t)e i t τ e itτ dt. (2.6) S z,j = 0 if 2 j z. (2.7) Indeed, if t ε, then 2 j z t < /2 and therefore, β(2 j z t) = 0. The bound (2.4) follows once we show that there is a uniform constant C so that for all z Ξ δ {z C : z }, we have and S z,j (Q)f n m (M) m nm C2j f, j = 0,,..., (2.8) n+m (M) S z (Q)f n m (M) C f n+m (M). (2.9) et us start with establishing the estimate (2.9). When doing so, we shall follow [2] and obtain the following result. emma 2.5. The multiplier S z belongs to the symbol class S m (R) uniformly in z C, z, i.e. d j τ S z (τ) C j ( + τ ) m j, j = 0,, 2,..., (2.20) with the constants C j independent of z.

13 p RESOVENT ESTIMATES 3 Proof. Recall first that ρ( z t) = 0 when t / z. Furthermore, as Im τ > 0, = 0,,..., m/2, we conclude that e i t τ. et τ. Then for j = 0,,..., we have d j τ S z (τ) C z m / z / z t j dt C C, z m+j uniformly in z, z, which shows the estimate (2.20) in the case τ. Assume now that τ >. et us first prove the estimate (2.20) for j = 0. To that end we shall integrate by parts m times in the expression (2.6) for S z. et us first explain that all boundary terms vanish when we integrate by parts m times in (2.6). Indeed, integrating by parts once in (2.6), we obtain the following boundary terms, i ( ) e 2πi/m ρ( z t)ρ(t)e itτ e itτ t=0 iτmz m t= + ρ( z t)ρ(t)e itτ e itτ t=+ t=0 = i ) e ( 2πi/m = 0. iτmz m Here we have used the fact that ρ and ρ are compactly supported, and ρ(0) = ρ(0) =. Furthermore, since all the derivatives of ρ and ρ vanish at the origin, when integrating by parts m times in (2.6), the only possible contribution to the boundary terms may be written in the form m l= B l, where B l = i ( e 2πi/m ( ) l ρ( z t)ρ(t)( iτ (iτ) l mz m ) l e itτ e itτ t=0 t= ) + ρ( z t)ρ(t)(iτ ) l e itτ e itτ t=+ t=0 = i e 2πi/m ( ) l (( iτ (iτ) l mz m ) l (iτ ) l ). When l is odd, it is clear that B l = 0. Recall now that m is even. When l is even and l m, we also have B l = 0 due to the fact that e 2πi/m (τ ) l = z l (e 2πli/m ) = z l e πli = 0. e2πli/m Here we have used that τ = ze 2πi/m and the fact that e 2πli/m when 2 l m 2. Hence, when integrating by parts m times in (2.6), the only possible

14 4 KRUPCHYK AND UHMANN contribution to the boundary terms is of the form, B m = 2 τ m mz m e 2πi/m (τ ) m = 2 τ m m e 2πi = τ m. (2.2) et us explain how to estimate the integrals arising after having integrated by parts m times in (2.6). The worst case scenario occurs when no derivatives fall on ρ(t), and the corresponding contribution can be estimated by a constant times 0 z l (d l τ m t ρ)( z t)ρ(t)( iτ ) l 2 e itτ e itτ dt C z m τ. (2.22) m / z Here l + l 2 = m. Then it follows from (2.6), (2.22) and (2.2) that S z (τ) C τ m, which shows (2.20) for j = 0 in the case τ >. To establish (2.20) for j =, 2,... in the case τ >, we write d j τ S z (τ) = i ( 0 e 2πi/m mz m + 0 ρ( z t)ρ(t)e itτ (it) j e itτ dt ) ρ( z t)ρ(t)e itτ (it) j e itτ dt, (2.23) and integrate by parts (m+j) times in (2.23). Due to the appearance of the terms t j in the integrands in (2.23), no boundary terms arise when integrating by parts the first j times. Integrating by parts further, the contributions to the boundary terms that one has to consider would be similar to those in the case j = 0, and therefore, we need only to discuss the integrals obtained after an integration by parts m + j times in (2.23). The worst case scenario here occurs when no derivatives fall on ρ(t), and the corresponding contribution to the integrals can be bounded by a constant times 0 z l (d l τ m+j t ρ)( z t)ρ(t)( iτ ) l 2 e itτ t j l 3 e itτ dt C z m τ. m+j / z Here l + l 2 + l 3 = m + j, 0 l 3 j. Together with (2.23) this implies (2.20). The proof is complete. Combing emma 2.5 with the fact that Q Ψ cl (M) is elliptic and self-adjoint, we conclude from [4, Theorem 4.3.] that S z (Q) is a pseudodifferential operator of order m, with the symbol seminorms uniformly bounded in z C, z. et S z (Q)(x, y) D (M M) be the Schwartz ernel of the operator S z (Q). Then S z (Q)(x, y) is C away from the diagonal {(x, x) : x M}. By [6, Proposition

15 p RESOVENT ESTIMATES 5, p. 24], since n m > 0, we have near the diagonal, in local coordinates, S z (Q)(x, y) C x y m n, uniformly in z C, z. An application of the Hardy-ittlewood-Sobolev inequality gives the estimate (2.9). et us now prove the estimate (2.8). By the Riesz Thorin interpolation theorem, (2.8) follows, if we show that that there is a constant C = C(δ) so that for all z Ξ δ {z C : z }, we have and S z,j (Q)f 2 (M) C z m 2 j f 2 (M), j = 0,,..., (2.24) S z,j (Q)f (M) C z n m 2 (n ) 2 j f (M), j = 0,,.... (2.25) Here the interpolation parameter θ = n m, and n ( z m 2 j ) θ ( z n m 2 (n ) 2 j ) θ = 2 j m nm. When proving the estimate (2.24), we use the identity e itq f 2 (M) = f 2 (M), t R, the fact that β(2 j z t) = 0 when t / [2 j / z, 2 j+ / z ], and Minowsi s inequality, to get S z,j (Q)f 2 (M) uniformly in z, which shows (2.24). C e itq f z m 2 (M)dt t [2 j / z,2 j+ / z ] C z m 2j f 2 (M), Now we are left with proving (2.25). et us denote by K z,j (x, y) the Schwartz ernel of the operator S z,j (Q). The estimate (2.25) is implied by the estimate K z,j (x, y) C z n m 2 (n ) 2 j, x, y M, (2.26) for all z Ξ δ {z C : z }, uniformly in z. By (2.5), we have K z,j (x, y) = i e 2πi/m mz m β(2 j z t)ρ(t)e i t τ e itq (x, y)dt, (2.27) where e itq (x, y) is the Schwartz ernel of the half-wave operator e itq. To proceed, we shall mae use of the Hörmander ax parametrix for the the half-wave operator e itq, see [6], [4, Theorem 4..2]. emma 2.6. et Q Ψ cl (M) be elliptic and self-adjoint with respect to a positive C density dµ, and q(x, ξ) be the principal symbol of Q. Then there is ε > 0 small, depending on M and Q, so that if t < ε, e itq = G(t) + R(t), where the remainder R(t) has the ernel R(t, x, y) C ([ ε, ε] M M), and the ernel G(t, x, y) is supported in a small neighborhood of the diagonal in M M, for t < ε. Furthermore, suppose that local coordinates are chosen in a patch

16 6 KRUPCHYK AND UHMANN Ω M so that dµ agrees with the ebesque measure in the corresponding open subset of R n. If ω Ω is relatively compact, G(t, x, y) has the form, G(t, x, y) = (2π) n R n e i[ϕ(x,y,ξ)+tq(y,ξ)] g(t, x, y, ξ)dξ when (t, x, y) [ ε, ε] M ω. Here g S 0,0, i.e. α ξ β t β 2 x β 3 y g(t, x, y, ξ) C α,β,β 2,β 3 ( + ξ ) α, for all multi-indices α, β, β 2, β 3, and g is supported in a small neighborhood of the diagonal in ω ω, and ϕ is a real function which is homogeneous of degree one in ξ, C for ξ 0, and satisfies i.e. for all multi-indices α. ϕ(x, y, ξ) = x y, ξ + O S ( x y 2 ξ ), (2.28) α ξ (ϕ(x, y, ξ) x y, ξ ) C α x y 2 ξ α, In what follows, we shall mae the choice of ε in the definition (2.2) of the function ρ(t) so that emma 2.6 is applicable. We assume that 2 j z >, as otherwise S z,j = 0, cf. (2.7). et us write where K () i z,j (x, y) = K (2) i z,j (x, y) = K z,j (x, y) = K () z,j mz m mz m (x, y) + K(2) z,j (x, y), e 2πi/m β(2 j z t)ρ(t)e i t τ G(t, x, y)dt, e 2πi/m β(2 j z t)ρ(t)e i t τ R(t, x, y)dt. Since R(t, x, y) C ([ ε, ε] M M), we have K (2) z,j (x, y) C z m t [2 j / z,2 j+ / z ] dt 2j C z. (2.29) m As 2 j z >, the estimate (2.29) is better than the desired bound (2.26) for K z,j. et us now estimate K () z,j. Setting r = 2j z, z r <,

17 p RESOVENT ESTIMATES 7 and assuming that the local coordinates are chosen as in emma 2.6, we write K () z,j (x, y) = i mz m R n e 2πi/m (2π) n β(t/r)ρ(t)e i t τ e i[ϕ(x,y,ξ)+tq(y,ξ)] g(t, x, y, ξ)dtdξ, (2.30) for (x, y) M ω. We would lie to replace ϕ by the Euclidean phase function ϕ 0 = x y, ξ. In doing so, we shall follow [] and notice that both ϕ and ϕ 0 parametrize the trivial agrangian manifold {(x, ξ, x, ξ)}. This is due to the fact that when (x, y) is in a neighborhood of the diagonal, we have ϕ ξ = 0 precisely when x = y, and then ϕ x = ϕ y = ξ. Following [], we can use the following result of [7, Theorem 3..6]. emma 2.7. Suppose that ϕ is as in emma 2.6, i.e. ϕ satisfies (2.28). Then, for (x, y) close to the diagonal, there is a C for ξ 0 homogeneous of degree one change of coordinates η = κ x,y (ξ) so that ϕ(x, y, κ x,y(η)) = x y, η. The transformation κ x,y depends smoothly on the parameters x, y, and in addition, κ x,y = Identity, when x = y. (2.3) emma 2.7 implies that (2.30) can be rewritten as K () z,j where (x, y) = i with D(κ x,y)(η) Dη mz m R n e 2πi/m (2π) n β(t/r)ρ(t)e i t τ e i[ x y,η +t q(x,y,η)] g(t, x, y, η)dtdη, D(κ g(t, x, y, η) = g(t, x, y, κ x,y(η)) x,y)(η) Dη, (2.32) being the Jacobian of the transformation κ x,y, has the same properties as g, in particular g S 0,0. Also, q(x, y, η) = q(y, κ x,y(η)) depends smoothly on x, y. Furthermore, since strict convexity is preserved under diffeomorphisms that are sufficiently close to the identity in the C sense, the surface Σ x,y = {η R n : q(x, y, η) = }

18 8 KRUPCHYK AND UHMANN is strictly convex. Maing the change of variables t t/r in (2.32), we get K () z,j (x, y) = ir mz m R n e 2πi/m (2π) n β(t)ρ(rt)e ir t τ e i x y,η e itr q(x,y,η) g(rt, x, y, η)dtdη. As q and κ x,y are homogeneous of degree one, we have r q(x, y, η) = q(x, y, rκ x,y(η)) = q(x, y, rη). Maing further change of variables η rη in (2.33), we obtain that K () ir n z,j (x, y) = mz m R n e 2πi/m (2π) n β(t)ρ(rt)e ir t τ e x y i r,η e it q(x,y,η) g(rt, x, y, η/r)dtdη. (2.33) (2.34) As q(x, y, η) is not smooth at η = 0, it will be convenient to write J (x, y, t, r) = e i[ x y r,η +t q(x,y,η)] χ(η) g(rt, x, y, η/r)dη, R n x y J 2 (x, y, t, r) = r,η +t q(x,y,η)] ( χ(η)) g(rt, x, y, η/r)dη, R n e i[ where χ C 0 (R n ) and χ = when η. Here t [/2, 2] and 0 < r. As g S 0,0, we see that for all x, y ω, x y small enough, uniformly in r. J (x, y, t, r) C, (2.35) et us now estimate the absolute value of the oscillatory integral J 2 (x, y, t, r) when t [/2, 2]. To that end, consider η [ x y, η + t q(x, y, η)], t [/2, 2]. r As q(x, y, η) is homogeneous of degree one in η, by the Euler homogeneity relation, we have η η q(x, y, η) = q(x, y, η). This and the ellipticity of q imply that η q(x, y, η) 0 for all η R n \ {0}. Thus, there is a constant A > /2 such that η q(x, y, η) A for all η S n, and therefore, by the fact that η q is homogeneous of degree zero, we conclude that η q(x, y, η) A for all η R n \ {0}.

19 p RESOVENT ESTIMATES 9 On the other hand, since η q S 0,0, for η, we have η q(x, y, η) A. Hence, for t [/2, 2], if x, y are such that then x y r / [A /4, 4A], (2.36) η [ x y, η + t q(x, y, η)] A /2. (2.37) r Assume first that (2.36) holds. Then we shall integrate by parts in the oscillatory integral J 2, see [7, emma.2.]. To that end, setting ψ(t, x, y, η) = x y, η + t q(x, y, η), r we consider the operator = n j= a j ηj, a j = η j ψ i η ψ 2. We have N (e iψ(η) ) = e iψ(η) for any N N, and the transpose of is given by n = a j ηj div a, a = (a,..., a n ). (2.38) j= Hence, we get J 2 (x, y, t, r) = e iψ(η) ( ) N (( χ(η)) g(rt, x, y, η/r))dη. R n We observe that ( χ(η)) g(rt, x, y, η/r) S 0,0 (2.39) uniformly in 0 < r. This follows from the facts that when η, α η β t β 2 x β 3 y g(rt, x, y, η/r) rβ r α C α,β,β 2,β 3 (+ η /r) α C α,β,β 2,β 3 (+ η ) α, for all β N 0 := N {0} and all α, β 2, β 3 N n 0, and for all α N n 0 and all N > 0. et us now show that α η χ(η) C α,n ( + η ) N, a j (η) S 0,0, η, (2.40) uniformly in r, x, y and t satisfying (2.36). Indeed, first using (2.37), we have a j (η) = η j ψ 2A. (2.4) η ψ 2

20 20 KRUPCHYK AND UHMANN et α N n be such that α. Then by eibniz formula, we get η α a j (η) = ( ) c β,γ η β ( ηj ψ) η γ, (2.42) η ψ 2 with constants c β,γ. Here and hence, for β, we have β+γ=α ηj ψ = x j y j r + t ηj q(x, y, η), β η ( ηj ψ) C β ( + η ) β, (2.43) uniformly in r. To estimate the absolute value of η γ (/ η ψ 2 ) for γ, we shall use the Faà di Bruno formula, see [8, p. 94], ( ) η γ = η γj b C b b γ,...,γ b. (2.44) γ γ = γ + + γ γ j j= For γ j, using eibniz formula and (2.43), we have γj η ( η ψ 2 ) C γ j η ψ ( + η ) γj. Therefore, (2.44) implies that for γ N n 0, ( ) γ η C η ψ 2 γ η ψ ( + 2 η ) γ (2.45) uniformly in r. We conclude from (2.42) with the help of (2.43) and (2.45) that for all a N n, α, α η a j (η) C α ( + η ) α, (2.46) uniformly in r. Hence, (2.40) follows from (2.4) and (2.46). Using (2.46), we obtain that div a S,0, η, (2.47) uniformly in r, x, y and t satisfying (2.36). Thus, it follows from (2.38) with the help of (2.40), (2.47) and (2.39) that ( ) N (( χ(η)) g(rt, x, y, η/r)) S N,0 uniformly in r, x, y and t satisfying (2.36). Hence, choosing N sufficiently large, we conclude that J 2 (x, y, t, r) C. (2.48)

21 p RESOVENT ESTIMATES 2 Therefore, it follows from (2.34), (2.35) and (2.48) that when x, y are such that x y r the desired estimate (2.26). K () r n z,j (x, y) C z = m 2j( n) z n m, (2.49) / [A /4, 4A]. The estimate (2.49) is better than Assume now that x y [A /4, 4A] and let us estimate the absolute value of r (x, y) in this case. As above, we only need to estimate the absolute value of K () z,j K (,2) z,j (x, y) = ir n e 2πi/m mz m (2π) n R n e x y i r β(t)ρ(rt)e ir t τ,η e it q(x,y,η) ( χ(η)) g(rt, x, y, η/r)dtdη, where χ C0 (R n ) is such that χ = when η. Using (2.), we get K (,2) z,j (x, y) = r n e it( rτ+ q(x,y,η)) (2π) n+ τ m z dτ m β(t)ρ(rt)e x y i r R n,η ( χ(η)) g(rt, x, y, η/r)dηdt. (2.50) Maing the change of variables τ rτ + q(x, y, η), we obtain that where K (,2) z,j (x, y) = r n (2π) n h r (τ, x, y, η) = 2π Rn h r (τ, x, y, η)e x y i r,η dηdτ, (2.5) ( q(x,y,η) τ ) r m zm e itτ β(t)ρ(rt)( χ(η)) g(rt, x, y, η/r)dt (2.52) is the inverse Fourier transform of the compactly supported smooth function t β(t)ρ(rt)( χ(η)) g(rt, x, y, η/r). We have γ η h r (τ, x, y, η) C N,γ ( + τ ) N ( + η ) γ, (2.53) uniformly in r, for all N > 0 and γ N n 0. This can be seen by using (2.39) in the case τ, and by integrating by parts N times in (2.52) and using (2.39) in the case τ. We write ( q(x, y, η) τ r ) m m z m = ( q(x, y, η) τ ze ), 2πi/m r

22 22 KRUPCHYK AND UHMANN and using a partial fraction decomposition, we get where ( q(x,y,η) τ ) r m z = r m A = ( m Thus, it follows from (2.5) that K (,2) z,j (x, y) = r n m (2π) n z m z m m A q(x, y, η) τ rze 2πi/m, (e 2πi/m e )) 2πli/m. l=0 l Rn x y i h r (τ, x, y, η)e r,η A q(x, y, η) (τ + rze 2πi/m ) dηdτ. (2.54) Recalling that arg(z) (0, 2π/m), we see that τ + rze 2πi/m avoids the real axis, for = 0,..., m. To proceed further, we shall need the following result, similar to [, Proposition 2.4]. emma 2.8. et n 2 and let h C (R n \ {0}) satisfy the Mihlin-type condition, α ξ h(ξ) H α ξ α, ξ 0, α N n 0. (2.55) et a C (R n \ {0}) be homogeneous of degree one. Assume that a(ξ) > 0 for all ξ R n \ {0} and that the cosphere Σ = {ξ R n : a(ξ) = } is strictly convex. Then there is a constant C > 0 such that for all x R n, x 0, and all w C \ [0, ), we have Rn h(ξ)e i x,ξ a(ξ) w dξ C( x n + ( w / x ) n 2 ). (2.56) Proof. First notice that since a C (R n \ {0}) is homogeneous of degree one, we have α ξ a(ξ) A α ξ α, ξ 0, α N n 0. et b C (R n \ {0}) be such that α ξ b(ξ) B α ξ α, ξ 0, α N n 0. Then it follows from [6, p. 245] that the Fourier transform of b(ξ) satisfies b(ξ)e i x,ξ dξ C x n, x 0. (2.57) R n Assume first that w is outside of a small but fixed conic neighborhood of the positive real axis [0, ), i.e. arg w [θ, 2π θ] for some θ > 0 small but fixed,

23 and w =. et us establish that satisfies uniformly in w. b w (ξ) = To that end, let us show that p RESOVENT ESTIMATES 23 h(ξ) a(ξ) w C (R n \ {0}), α ξ b w (ξ) B α ξ α, ξ 0, α N n 0, (2.58) a(ξ) w C θ ( ξ + ), (2.59) with a constant C θ > 0 uniformly in w. When doing so, we notice there is a constant δ > 0 such that a(ξ) δ ξ, and then (2.59) follows for all ξ large enough. It remains to consider the case when ξ is bounded. Then if arg w [θ, π θ] [π + θ, 2π θ], we get a(ξ) w Im(w) C θ. If arg w (π θ, π + θ), we write arg w = π + O(θ). Then w = O(θ), and therefore, a(ξ) w = a(ξ) + + O(θ) 2, for θ small enough. The bound (2.59) follows. By the eibniz formula we write α ξ (b w (ξ)) = β+γ=α C β,γ β ξ (h(ξ)) γ ξ ( ), (2.60) a(ξ) w with constants C β,γ. It follows from the Faà di Bruno formula (2.44) and (2.59) that for γ 0, ( ) γ ξ C γ,θ ξ γ, ξ 0, (2.6) a(ξ) w uniformly in w. Hence, we conclude from (2.60), with the help of (2.55) and (2.6), that (2.58) holds. Thus, applying (2.57) for b w, we obtain that Rn h(ξ)e i x,ξ a(ξ) w dξ C x n, x 0, (2.62) uniformly in w C, arg w [θ, 2π θ] with θ > 0 small but fixed, and w =.

24 24 KRUPCHYK AND UHMANN Assume now that w C, arg w [θ, 2π θ] with θ > 0 small but fixed, and w. etting w = w/ w, we have Rn h(ξ)e i x,ξ a(ξ) w dξ = Rn h(ξ)e i x,ξ Rn h( w ξ)e i w x,ξ dξ = w n w a(ξ/ w ) w a(ξ) w dξ. Since the dilate h( w ξ) of h(ξ) satisfies exactly the same bounds as in (2.55), as above, we obtain the uniform estimate (2.62), for all w C, arg w [θ, 2π θ] with θ > 0 small but fixed. Assume now that w C \ [0, ), arg w ( θ, θ) with θ > 0 small but fixed, and w =. Then w = + O(θ), and therefore, a(ξ) w = a(ξ) O(θ) C, for ξ / a ([/2, 2]), uniformly in w. Hence, letting 0 χ C0 ((0, )) be such that χ(t) = when t [/2, 2] and supp (χ) [/4, 4], by the above argument, we conclude that h(ξ)( χ(a(ξ))) b w (ξ) := a(ξ) w satisfies the bound (2.58) uniformly in w. Therefore, Rn h(ξ)( χ(a(ξ)))e i x,ξ dξ a(ξ) w C x n, uniformly in w C \ [0, ), arg w ( θ, θ) with θ > 0 small but fixed, and w =. et us now write, Rn h(ξ)χ(a(ξ))e i x,ξ I(x) = dξ = I (x) + I 2 (x), (2.63) a(ξ) w where Rn h(ξ)χ(a(ξ))(a(ξ) w )e i x,ξ Rn ih(ξ)χ(a(ξ))w 2 e i x,ξ I (x) := dξ, I (a(ξ) w ) 2 + w2 2 2 (x) = dξ. (a(ξ) w ) 2 + w2 2 Here w = Re w = + O(µ 2 ), w 2 = Im w = µ + O(µ 2 ), and µ := arg w, µ small. Using the coarea formula in the integral I 2 (x), we get dξ I 2 (x) C w 2 a ([/4,4]) (a(ξ) w ) 2 + w2 2 4 ds E de = C w 2, /4 a(ξ)=e ξ a(ξ) (E w ) 2 + w2 2 where ds E is the ebesque measure on the surface a(ξ) = E. et us notice that by Euler homogeneity relations for a(ξ) = E, we have ξ a(ξ) /C, (2.64)

25 uniformly in E [/4, 4]. Therefore, I 2 (x) C w 2 uniformly in µ. 4 /4 p RESOVENT ESTIMATES 25 de (E w ) 2 + w 2 2 C w 2 de E 2 + w 2 2 Appealing to the coarea formula in the integral I (x), we get h(ξ)χ(a(ξ))(a(ξ) w )e i x,ξ I (x) = dξ a ([/4,4]) (a(ξ) w ) 2 + w2 2 4 (E w ) = J(E, x)de, /4 (E w ) 2 + w2 2 where J(E, x) = χ(e) a(ξ)=e h(ξ)e i x,ξ ξ a(ξ) ds E = E n χ(e) a(ξ)= C, (2.65) (2.66) h(eξ)e i x,eξ ds. ξ a(ξ) We see that J(E, x) is C in E, x. Maing the change of variables E E w in (2.66), we get ( 0 w /4 4 w ) E I (x) = + + J(E + w /4 w 0 w /4 E 2 + w2 2, x)de = + w /4 0 4 w w /4 E(J(E + w, x) J( E + w, x)) de E 2 + w2 2 E J(E + w E 2 + w2 2, x)de. As f(e) = J(E + w, x) J( E + w, x) is C in E, w, and x, and f(0) = 0, it follows that f(e) = Eg(E) with a function g which is C in E, w, and x. Hence, recalling that w = + O(µ 2 ), for x, we get I (x) C 2 0 E 2 de + C E 2 + w2 2 4 uniformly in µ with 0 < µ θ, where θ is sufficiently small. We conclude from (2.63), (2.65) and (2.67) that I(x) C, /4 de C, (2.67) E for x, uniformly in µ with 0 < µ θ, where θ is sufficiently small. et us now show that when x, we get I(x) C x (n ) 2, (2.68)

26 26 KRUPCHYK AND UHMANN uniformly in µ. First using the coarea formula in (2.63), we get 4 h(ξ)χ(e)e i x,ξ ds E I(x) = /4 a(ξ)=e (E w) ξ a(ξ) de 4 E n χ(e) h(eξ) = E w ξ a(ξ) ei Ex,ξ ds de. /4 a(ξ)= To proceed recall that a(ξ) is homogeneous of degree one, C for ξ 0, and a(ξ) > 0 on R n \ {0}. Then ξ a 0 along the cosphere Σ = {ξ R n : a(ξ) = }, which is therefore is a C compact hypersurface. Furthermore, Σ is homeomorphic to the sphere S n via the homeomorphism S n Σ, ω ω/a(ω). Hence, Σ is connected. The assumption that the Gaussian curvature of Σ never vanishes implies that the Gauss map is a diffeomorphism from Σ to S n. Thus, given x R n \ {0}, there are exactly two points ξ (x), ξ 2 (x) Σ with normal x. Since ξ (x), ξ 2 (x), are homogeneous of degree zero and smooth in R n \ {0}, it follows that the functions x, ξ (x), x, ξ 2 (x) are also smooth for x 0 and homogeneous of degree one. We shall need the following result concerning the inverse Fourier transform of a smooth measure carried by the cosphere Σ, which is an application of the stationary phase theorem, see [4, Theorem.2., p. 50] and [4, p. 68]. emma 2.9. et dσ(ξ) = β(ξ)ds(ξ) with β C (Σ) and ds is the surface measure on Σ. Then under the above assumptions, the inverse Fourier transform of the measure dσ satisfies (2π) n e i x,ξ dσ(ξ) = b (x)e i x,ξ (x) + b 2(x)e i x,ξ2(x), x, x (n )/2 x (n )/2 Σ where the functions b j are such that α x b j (x) C α x α, x, α N n 0. As ξ j (x) is homogeneous of degree zero, by emma 2.9, for x, we get I(x) = (2π) n x (n ) 2 2 j= 4 /4 E (n )/2 χ(e)b j (x, E) e ie x,ξj(x) de, E w with some functions b j C for x and E [/4, 4], and E l x α b j (x, E) C l,α x α, x, E [/4, 4], l N 0, α N n 0. (2.69) The estimate (2.68) would follow if we could show that 4 /4 E (n )/2 χ(e)b j (x, E) E w e ie x,ξj(x) de C, (2.70)

27 p RESOVENT ESTIMATES 27 uniformly in µ, 0 < µ θ. To show (2.70), we let f(e, x) = E (n )/2 χ(e)b j (x, E), ϕ(x) = x, ξ j (x). For x, the function f(, x) is C with compact support in E [/4, 4], and (2.69) yields that l Ef(E, x) C l. (2.7) We write J(x) = 4 /4 = 2πi f(e, x)e ieϕ(x) E w de = 2π f(t, x)e iw (t+ϕ(x)) f(t, x) e iτ(t+ϕ(x)) w 2 iτ dτdt, e ie(t+ϕ(x)) E w iw 2 dedt where f(t, x) is the Fourier transform of f(e, x). We shall use the following fact: for all α R, α 0, + e iτt 2π α iτ dτ = sgnαh(αt)e αt, where H(t) is the Heaviside function which equals one for t 0 and zero for t < 0, see [, emma 2.]. As w 2 0, we get J(x) = f(t, x)ie iw (t+ϕ(x)) sgn(w 2 )H(w 2 (t + ϕ(x)))e w 2(t+ϕ(x)) dt, and therefore, using that f has compact support in E and (2.7), we obtain that J(x) C f(t, x) dt C ( + t 2 ) f(t, x) t C( f(e, x) E + 2 Ef(E, x) E ) C, uniformly in w. This establishes (2.70), and hence, (2.68). Thus, for w C \ [0, ), argw ( θ, θ), θ > 0 small but fixed, and w =, we get Rn h(ξ)e i x,ξ a(ξ) w dξ C( x n + x (n ) 2 ), x 0, (2.72) uniformly in w. In the case when w C \ [0, ), argw ( θ, θ), θ > 0 small but fixed, and w, the estimate (2.56) follows from (2.72) by a change of scale. The proof of emma 2.8 is complete. Now using emma 2.8, the estimate (2.53), and the fact that x y [A /4, 4A], r we obtain that Rn x y i h r (τ, x, y, η)e r,η q(x, y, η) (τ + rze 2πi/m ) dη C N( + τ ) N ( + τ + r z ) n 2, (2.73)

28 28 KRUPCHYK AND UHMANN for = 0,,..., m and N > 0. It follows from (2.54) and (2.73) that for N > 0 sufficiently large, K (,2) z,j (x, y) C r n z m ( + τ ) Cr (n ) 2 z n+ 2m 2. N+ n n 2 ( + r z ) 2 dτ Here we have used that r z. Recalling that r = 2 j / z, the above estimate completes the proof of the estimate (2.26), and therefore, the estimates (2.25) and (2.8). As m nm j=0 2j = /( 2 m nm ), we have obtained the (2.4) for the local operator Uniform estimate for the non-local operator in the case of unbounded z. et τ R and consider the multipliers r z (τ) = m z (τ) m loc z (τ) = for all z Ξ δ {z C : z }. i mz m e 2πi/m ( ρ(t))e i t τ e itτ dt, (2.74) In order to prove (.5), we are left with establishing that r z (Q)f n m (M) C f n+m (M), (2.75) for all z Ξ δ {z C : z }, uniformly in z. et us first show that r z (τ) is bounded for all z Ξ δ {z C : z }, uniformly in z. Indeed, we have r z (τ) C ( ε/2 ) e timτ dt + e timτ dt C. z m ε/2 Imτ (2.76) Recall that τ = ze 2πi/m, and therefore, 0 < arg(τ ) < π, = 0,..., m/2. If now 0 < arg(τ ) π/2, then Imτ z = sin(arg(τ )) sin(arg(z)), and thus, using the fact that z Ξ δ, we get If π/2 < arg(τ ) < π, then Imτ z and therefore, Imτ Imz δ. (2.77) = sin(π arg(τ )) sin(π arg(τ )) = sin(arg(z) 2π/m), Imτ Im(ze 2πi/m ) δ. (2.78)

29 Hence, it follows from (2.76), (2.77) and (2.78) that for all z Ξ δ {z C : z }, uniformly in z. p RESOVENT ESTIMATES 29 r z (τ) Cδ, (2.79) To obtain the decay of r z (τ), let us integrate by parts N times, N =, 2,..., in (2.74). We have i ( r z (τ) = e 2πi/m ( ) N mz m i N ( τ + τ) N + ( )N i N (τ + τ) N 0 0 ( N t ρ(t))e it( τ +τ) dt ) ( t N ρ(t))e it(τ+τ) dt. Notice that all the boundary terms disappear when integrating by parts due to the presence of the term ( ρ(t)) in (2.74) and the fact that Imτ > 0. As ± τ + τ = ± Re τ + τ 2 + Im τ 2 ± Re τ + τ 2 + δ 2 δ 2 ( + ± Re τ + τ ), where δ <, we obtain that r z (τ) C (( + Re τ z m + τ ) N + ( + Re τ + τ ) N ), uniformly in z. Thus, for τ 0, we get r z (τ) C ( z m We have,..., Reτ 0 r z (Q)f = (+ Re τ +τ ) N + r z (µ j )E j f = j=,..., Reτ <0 (+ Re τ +τ ) N ) (2.80) rz(q)f, l f C (M), (2.8) l= where rz(q)f l = r z (µ j )E j f, l =, 2,.... µ j [l,l)

30 30 KRUPCHYK AND UHMANN Using emma 2.2 and (2.80) with N = m +, we obtain that rz(q)f l n m (M) Clm ( sup r z (τ) ) f Clm τ [l,l) n+m (M) z m ( Reτ 0 ( + Re τ + l ) + m+ Reτ <0 ( + Re τ + l ) m+ Here we have used the fact that for l τ l, we have ± Re τ + l ± Re τ + τ + l τ ± Re τ + τ +. ) f n+m (M). (2.82) Hence, (2.75) would follow from (2.8) and (2.82), if we could show that Σ := l m z m ( + a + l ) C, a = Re τ, (2.83) m+ l= with some constant C > 0 uniform in z C, z. et us now show (2.83). Assume first that a. Then Σ = l m z m ( a + l) m+ z m l C, 2 l= with a constant C > 0 uniform in z C, z. Consider now the case a >. Then denoting [a] the integer part of a, we write where Σ := z m Σ 2 := z m Σ 3 := z m l [a] ( l [a]+2 Σ = Σ + Σ 2 + Σ 3, l m ( + a l) m+, [a] m l= ( + a + [a] ) + ([a] + ) m m+ ( + a + [a] + ) m+ l m ( a + l). m+ Using the fact that a z, we see that Σ 2 C, uniformly in z C, z. We shall next estimate Σ 3. As the function t m /( a + t) m+ is decreasing for t > 0, we get Σ 3 z m C m z m t m dt = [a]+ ( a + t) m+ ( dt + + (a )m t2 z m dt t m+ 2+[a] a ) C, ), (t + a ) m dt t m+

31 uniformly in z C, z. p RESOVENT ESTIMATES 3 et us now estimate Σ. Since the function t m /( + a t) m+ is increasing for t > 0, we obtain that Σ [a] t m a + a t m dt dt z m ( + a t) m+ z m +a [a] t m+ C m z m (( + a) m dt t m+ + dt t 2 ) C, uniformly in z C, z. This completes the proof of (2.83) and hence, of Theorem.. Finally let us remar that the a priori estimate (.5) implies the following simple result concerning the 2 resolvent of P, (P ζ). Proposition 2.0. et ζ C\[0, ). Then the resolvent (P ζ) is a bounded operator: n+m (M) n m (M). Proof. et ζ / {λ, λ 2,... } so that (P ζ) : 2 (M) 2 (M) is bounded. By elliptic regularity, we have (P ζ) C (M) C (M), and therefore, the linear continuous operator P ζ : C (M) C (M) is bijective. By the open mapping theorem, (P ζ) : C (M) C (M) is continuous. We have next the linear continuous map P ζ : D (M) D (M) given by (P ζ)u, ϕ = u, (P ζ)ϕ, ϕ C (M), which is bijective, with continuous inverse (P ζ) : D (M) D (M). By Remar 2.4, when ζ C \ [0, ), we have the following a priori estimate u n m (M) C ζ (P ζ)u n+m (M), for all u C (M). Thus, for any f C (M), we get (P ζ) f n m (M) C ζ f n+m (M). (2.84) Now let f n+m (M). Then there is a sequence fj C (M), converging to f in n+m (M) as j. It follows from (2.84) that (P ζ) f j is a Cauchy sequence in n m (M), and therefore, it converges in n m (M). As (P ζ) : D (M) D (M) continuous, we have (P ζ) f n m (M) and (P ζ) f j converges to (P ζ) f in n m (M) as j. Hence, (2.84) is valid for any f n+m (M), which shows the claim of Proposition 2.0.

32 32 KRUPCHYK AND UHMANN 3. Saturation of the resolvent estimates. Proof of Theorem.2 We shall need the following Bernstein type inequality, established in [, emma 3.]. emma 3.. et β C 0 (R) be such that 0 / supp (β). Then if q r, there is a constant C = C(r, q) so that β(q/α)f r (M) Cα n( q r ) f q (M), α. In Theorem. we obtained the uniform estimate (.5) for all z in the sector Ξ of the complex plane such that dist( Ξ, z) δ for some δ > 0. The next result shows that removing the eigenvalues of the operator Q = P /m in some interval [α, α + ] allows us to obtain the uniform estimate (.5) for all z Ξ with Re z = α or Re (ze 2πi/m ) = α. emma 3.2. et χ [α,α+) f = Then we have the uniform estimate: µ j [α,α+) E j f. (I χ [α,α+) ) (P z m ) f n m (M) C f n+m (M), (3.) with z Ξ, Re z = α, and 0 < Im z, and the uniform estimate: (I χ [α,α+) ) (P z m ) f n m (M) C f n+m (M), (3.2) with z Ξ, Re (ze 2πi/m ) = α, and 0 < Im (ze 2πi/m ). Proof. et us start by proving (3.). et z Ξ, Re z = α, and assume first that δ Im z = β for some δ > 0. We write χ [α,α+) (P z m ) f = (µ m j z m ) E j f. By (2.5), we get χ [α,α+) (P z m ) f n m (M) Cαm ( Writing we have µ j [α,α+) sup τ [α,α+) z m = (α + iβ) m = α m ( + miβ/α + O(β 2 /α 2 )), (τ m z m ) ) f n+m (M), (3.3) Im z m = mβα m + O(β 2 α m 2 ) m 2 βαm m 2 δαm, (3.4) for α sufficiently large. Therefore, it follows from (3.3), (3.4) and (.5) that (I χ [α,α+) ) (P z m ) f n m (M) C f n+m (M), (3.5)

33 p RESOVENT ESTIMATES 33 for all z Ξ, Re z = α, and δ Im z, uniformly in z. et z Ξ, Re z = α, and 0 < Im z = β /2. Then using the fact that α + i Ξ for α sufficiently large and (3.5), we see that (3.) follows once we establish that (I χ [α,α+) ) ((P z m ) (P (α + i) m ) )f n m (M) C f n+m (M), uniformly in z. We have (I χ [α,α+) ) ((P z m ) (P (α + i) m ) )f ( )( = + µ j [0,α ) ( = µ j [0,α ) + µ j [α+,+ ) =2 µ j [α+,α+) µ m j z m µ m j (α + i) m )( ) E j f ) µ m j z E m µ m j (α + i) m j f. (3.6) (3.7) By (2.5), for = 2, 3..., we get ( ) µ m µ j [α+,α+) j z E m µ m j (α + i) m j f C(α + )m n m (M) sup z m (α + i) m τ [α+,α+) (τ m z m )(τ m (α + i) m ) f n+m. (M) (3.8) We have, for α sufficiently large, that and therefore, z m (α + i) m = α m mi(β ) + O(α m 2 ), As Re z m = α m + O(α m 2 ), we obtain that τ m z m τ m α m O(α m 2 ) z m (α + i) m Cα m. (3.9) = (τ α)(τ m + τ m 2 α + + τα m 2 + α m ) O(α m 2 ) ( )(τ m + α m ) O(α m 2 ) ( )τ m ( )(α + ) m /C, (3.0) for τ [α +, α + ), = 2, 3,..., and α sufficiently large. Thus, it follows from (3.8), (3.9), and (3.0) that µ j [α+,α+) ( µ m j z m µ m j (α + i) m ) E j f n m (M) C ( ) 2 f n+m (M), (3.)

34 34 KRUPCHYK AND UHMANN for = 2, 3,.... Using (2.5) and rescaling, we get ( ) µ m j z E m µ m j (α + i) m j f C f n m (M) n+m. (3.2) (M) µ j [0,α ) Hence, (3.6) follows from (3.7), (3.), and (3.2). The proof of (3.) is complete. et us now show (3.2). To that end, letting w = ze 2πi/m, we have w m = z m, and therefore, (3.2) is a consequence of the uniform estimate, (I χ [α,α+) ) ((P w m ) (P (α + i) m ) )f n m (M) C f n+m (M), with z Ξ, w = ze 2πi/m, Re w = α, and 0 < Im w. This is obtained similarly to the derivation of (3.6). The proof of emma 3.2 is complete. et N(α) = #{j : µ j < α} be the counting function for the eigenvalues of the operator Q. We have N(α) = S α (x, x)dµ(x), (3.3) where is the spectral function. M S α (x, y) = µ j <α e j (x)e j (y) Similarly to [, Theorem.2] we obtain the following result which gives a sufficient condition for the optimality of the region Ξ δ in the uniform resolvent estimate (.5) for operators of order m, in terms of the density of eigenvalues in shrining intervals of the form [α β, α + β ), α, 0 < β 0 as. emma 3.3. Assume that there exist sequences α and 0 < β 0 as such that (β α n ) [N(α + β ) N(α β )],. (3.4) et z () = α + iβ and z (2) = e 2πi/m (α iβ ). Then we have (P (z (j) )m ) n+m (M) n m,, j =, 2. (3.5) (M) Proof. In what follows we shall only establish (3.5) for j =, the proof in the other case being similar. We shall then write z = z (). et us notice that z Ξ for large enough. By (3.), we now that for large, (I χ [α,α +)) (P z m ) n+m (M) n m (M) C,

35 uniformly in. Thus, we only need to show that p RESOVENT ESTIMATES 35 χ [α,α +) (P z m ) n+m (M) n m,. (3.6) (M) et g C 0 (R) be such that 0 / supp (g) and g(τ) = for τ [/2, 2]. Then for large, we have χ [α,α +) = g(q/α ) χ [α,α +) g(q/α ). (3.7) Using (3.7) and emma 3., we obtain χ [α,α +) (P z m ) f (M) = g(q/α ) χ [α,α +) (P z m ) g(q/α )f (M) Cα n m 2 χ [α,α +) (P z m ) n+m (M) n m g(q/α )f (M) n+m (M) Cα n m χ [α,α +) (P z m ) n+m (M) n m f (M) (M). Thus, in order to show (3.6) it suffices to chec that α (n m) χ [α,α +) (P z m ) (M) (M),. (3.8) The ernel of the operator χ [α,α +) (P z m ) is given by We have α (n m) K(x, y) = µ j [α,α +) µ m j z m e j (x)e j (y). χ [α,α +) (P z m ) (M) (M) = α (n m) sup x,y M K(x, y) α (n m) sup e x M µ m µ j [α,α +) j z m j (x) 2 α (n m) sup Im µ m m j z µ m j z e j(x) 2 m 2 x M µ j [α,α +) α (n m) Im ( z m ) sup x M µ j [α β,α +β ) for sufficiently large. Writing z m = (α iβ ) m, we get Im ( z m ) = mβ α m µ m j z m 2 e j(x) 2 :=, + O(βα 2 m 2 ) mβ α m /2, (3.9) for sufficiently large. Using the fact that µ j [α β, α + β ) in the last sum, we obtain that µ m j z m = µ j z µ m j + µ m 2 j z + + µ j z m 2 + z m Cβ α m, (3.20)