RESOLVENT ESTIMATES FOR ELLIPTIC OPERATORS ON COMPACT MANIFOLDS
|
|
- Ella Griffith
- 5 years ago
- Views:
Transcription
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)
Microlocal analysis and inverse problems Lecture 3 : Carleman estimates
Microlocal analysis and inverse problems ecture 3 : Carleman estimates David Dos Santos Ferreira AGA Université de Paris 13 Monday May 16 Instituto de Ciencias Matemáticas, Madrid David Dos Santos Ferreira
More informationMATH 205C: STATIONARY PHASE LEMMA
MATH 205C: STATIONARY PHASE LEMMA For ω, consider an integral of the form I(ω) = e iωf(x) u(x) dx, where u Cc (R n ) complex valued, with support in a compact set K, and f C (R n ) real valued. Thus, I(ω)
More informationA new class of pseudodifferential operators with mixed homogenities
A new class of pseudodifferential operators with mixed homogenities Po-Lam Yung University of Oxford Jan 20, 2014 Introduction Given a smooth distribution of hyperplanes on R N (or more generally on a
More informationFourier Transform & Sobolev Spaces
Fourier Transform & Sobolev Spaces Michael Reiter, Arthur Schuster Summer Term 2008 Abstract We introduce the concept of weak derivative that allows us to define new interesting Hilbert spaces the Sobolev
More informationChapter One. The Calderón-Zygmund Theory I: Ellipticity
Chapter One The Calderón-Zygmund Theory I: Ellipticity Our story begins with a classical situation: convolution with homogeneous, Calderón- Zygmund ( kernels on R n. Let S n 1 R n denote the unit sphere
More informationMicrolocal Methods in X-ray Tomography
Microlocal Methods in X-ray Tomography Plamen Stefanov Purdue University Lecture I: Euclidean X-ray tomography Mini Course, Fields Institute, 2012 Plamen Stefanov (Purdue University ) Microlocal Methods
More informationAnalysis in weighted spaces : preliminary version
Analysis in weighted spaces : preliminary version Frank Pacard To cite this version: Frank Pacard. Analysis in weighted spaces : preliminary version. 3rd cycle. Téhéran (Iran, 2006, pp.75.
More informationWave equation on manifolds and finite speed of propagation
Wave equation on manifolds and finite speed of propagation Ethan Y. Jaffe Let M be a Riemannian manifold (without boundary), and let be the (negative of) the Laplace-Beltrami operator. In this note, we
More informationMICROLOCAL ANALYSIS METHODS
MICROLOCAL ANALYSIS METHODS PLAMEN STEFANOV One of the fundamental ideas of classical analysis is a thorough study of functions near a point, i.e., locally. Microlocal analysis, loosely speaking, is analysis
More informationMicrolocal Analysis : a short introduction
Microlocal Analysis : a short introduction Plamen Stefanov Purdue University Mini Course, Fields Institute, 2012 Plamen Stefanov (Purdue University ) Microlocal Analysis : a short introduction 1 / 25 Introduction
More information1 Math 241A-B Homework Problem List for F2015 and W2016
1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let
More informationNew Proof of Hörmander multiplier Theorem on compact manifolds without boundary
New Proof of Hörmander multiplier Theorem on compact manifolds without boundary Xiangjin Xu Department of athematics Johns Hopkins University Baltimore, D, 21218, USA xxu@math.jhu.edu Abstract On compact
More informationA review: The Laplacian and the d Alembertian. j=1
Chapter One A review: The Laplacian and the d Alembertian 1.1 THE LAPLACIAN One of the main goals of this course is to understand well the solution of wave equation both in Euclidean space and on manifolds
More informationPOINTWISE BOUNDS ON QUASIMODES OF SEMICLASSICAL SCHRÖDINGER OPERATORS IN DIMENSION TWO
POINTWISE BOUNDS ON QUASIMODES OF SEMICLASSICAL SCHRÖDINGER OPERATORS IN DIMENSION TWO HART F. SMITH AND MACIEJ ZWORSKI Abstract. We prove optimal pointwise bounds on quasimodes of semiclassical Schrödinger
More informationZeta Functions and Regularized Determinants for Elliptic Operators. Elmar Schrohe Institut für Analysis
Zeta Functions and Regularized Determinants for Elliptic Operators Elmar Schrohe Institut für Analysis PDE: The Sound of Drums How Things Started If you heard, in a dark room, two drums playing, a large
More informationAverage theorem, Restriction theorem and Strichartz estimates
Average theorem, Restriction theorem and trichartz estimates 2 August 27 Abstract We provide the details of the proof of the average theorem and the restriction theorem. Emphasis has been placed on the
More informationFractal Weyl Laws and Wave Decay for General Trapping
Fractal Weyl Laws and Wave Decay for General Trapping Jeffrey Galkowski McGill University July 26, 2017 Joint w/ Semyon Dyatlov The Plan The setting and a brief review of scattering resonances Heuristic
More informationSingularities of affine fibrations in the regularity theory of Fourier integral operators
Russian Math. Surveys, 55 (2000), 93-161. Singularities of affine fibrations in the regularity theory of Fourier integral operators Michael Ruzhansky In the paper the regularity properties of Fourier integral
More informationWe denote the space of distributions on Ω by D ( Ω) 2.
Sep. 1 0, 008 Distributions Distributions are generalized functions. Some familiarity with the theory of distributions helps understanding of various function spaces which play important roles in the study
More informationAPPLICATION OF A FOURIER RESTRICTION THEOREM TO CERTAIN FAMILIES OF PROJECTIONS IN R 3
APPLICATION OF A FOURIER RESTRICTION THEOREM TO CERTAIN FAMILIES OF PROJECTIONS IN R 3 DANIEL OBERLIN AND RICHARD OBERLIN Abstract. We use a restriction theorem for Fourier transforms of fractal measures
More informationOn a class of pseudodifferential operators with mixed homogeneities
On a class of pseudodifferential operators with mixed homogeneities Po-Lam Yung University of Oxford July 25, 2014 Introduction Joint work with E. Stein (and an outgrowth of work of Nagel-Ricci-Stein-Wainger,
More informationThe oblique derivative problem for general elliptic systems in Lipschitz domains
M. MITREA The oblique derivative problem for general elliptic systems in Lipschitz domains Let M be a smooth, oriented, connected, compact, boundaryless manifold of real dimension m, and let T M and T
More informationCALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =
CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.
More informationOscillatory integrals
Chapter Oscillatory integrals. Fourier transform on S The Fourier transform is a fundamental tool in microlocal analysis and its application to the theory of PDEs and inverse problems. In this first section
More informationElliptic Regularity. Throughout we assume all vector bundles are smooth bundles with metrics over a Riemannian manifold X n.
Elliptic Regularity Throughout we assume all vector bundles are smooth bundles with metrics over a Riemannian manifold X n. 1 Review of Hodge Theory In this note I outline the proof of the following Fundamental
More informationTOOLS FROM HARMONIC ANALYSIS
TOOLS FROM HARMONIC ANALYSIS BRADLY STADIE Abstract. The Fourier transform can be thought of as a map that decomposes a function into oscillatory functions. In this paper, we will apply this decomposition
More informationEXPOSITORY NOTES ON DISTRIBUTION THEORY, FALL 2018
EXPOSITORY NOTES ON DISTRIBUTION THEORY, FALL 2018 While these notes are under construction, I expect there will be many typos. The main reference for this is volume 1 of Hörmander, The analysis of liner
More informationMATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5
MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5.. The Arzela-Ascoli Theorem.. The Riemann mapping theorem Let X be a metric space, and let F be a family of continuous complex-valued functions on X. We have
More information(1.2) Im(ap) does not change sign from to + along the oriented bicharacteristics of Re(ap)
THE RESOLUTION OF THE NIRENBERG-TREVES CONJECTURE NILS DENCKER 1. Introduction In this paper we shall study the question of local solvability of a classical pseudodifferential operator P Ψ m cl (M) on
More informationOn stable inversion of the attenuated Radon transform with half data Jan Boman. We shall consider weighted Radon transforms of the form
On stable inversion of the attenuated Radon transform with half data Jan Boman We shall consider weighted Radon transforms of the form R ρ f(l) = f(x)ρ(x, L)ds, L where ρ is a given smooth, positive weight
More informationStability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games
Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games Alberto Bressan ) and Khai T. Nguyen ) *) Department of Mathematics, Penn State University **) Department of Mathematics,
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationMath212a Lecture 5 Applications of the spectral theorem for compact self-adjoint operators, 2. Gårding s inequality and its conseque
Math212a Lecture 5 Applications of the spectral theorem for compact self-adjoint operators, 2. Gårding s inequality and its consequences. September 16, 2014 1 Review of Sobolev spaces. Distributions and
More informationGradient estimates for eigenfunctions on compact Riemannian manifolds with boundary
Gradient estimates for eigenfunctions on compact Riemannian manifolds with boundary Xiangjin Xu Department of athematics Johns Hopkins University Baltimore, D 21218 Abstract The purpose of this paper is
More informationFree Boundary Minimal Surfaces in the Unit 3-Ball
Free Boundary Minimal Surfaces in the Unit 3-Ball T. Zolotareva (joint work with A. Folha and F. Pacard) CMLS, Ecole polytechnique December 15 2015 Free boundary minimal surfaces in B 3 Denition : minimal
More informationGradient estimates for the eigenfunctions on compact manifolds with boundary and Hörmander multiplier Theorem
Gradient estimates for the eigenfunctions on compact manifolds with boundary and Hörmander multiplier Theorem Xiangjin Xu Department of athematics, Johns Hopkins University Baltimore, D, 21218, USA Fax:
More informationLaplace s Equation. Chapter Mean Value Formulas
Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic
More informationA Walking Tour of Microlocal Analysis
A Walking Tour of Microlocal Analysis Jeff Schonert August 10, 2006 Abstract We summarize some of the basic principles of microlocal analysis and their applications. After reviewing distributions, we then
More informationEigenvalues and eigenfunctions of the Laplacian. Andrew Hassell
Eigenvalues and eigenfunctions of the Laplacian Andrew Hassell 1 2 The setting In this talk I will consider the Laplace operator,, on various geometric spaces M. Here, M will be either a bounded Euclidean
More informationLECTURE 5: THE METHOD OF STATIONARY PHASE
LECTURE 5: THE METHOD OF STATIONARY PHASE Some notions.. A crash course on Fourier transform For j =,, n, j = x j. D j = i j. For any multi-index α = (α,, α n ) N n. α = α + + α n. α! = α! α n!. x α =
More informationTopics in Harmonic Analysis Lecture 6: Pseudodifferential calculus and almost orthogonality
Topics in Harmonic Analysis Lecture 6: Pseudodifferential calculus and almost orthogonality Po-Lam Yung The Chinese University of Hong Kong Introduction While multiplier operators are very useful in studying
More informationCHAPTER 3. Gauss map. In this chapter we will study the Gauss map of surfaces in R 3.
CHAPTER 3 Gauss map In this chapter we will study the Gauss map of surfaces in R 3. 3.1. Surfaces in R 3 Let S R 3 be a submanifold of dimension 2. Let {U i, ϕ i } be a DS on S. For any p U i we have a
More informationMicro-local analysis in Fourier Lebesgue and modulation spaces.
Micro-local analysis in Fourier Lebesgue and modulation spaces. Stevan Pilipović University of Novi Sad Nagoya, September 30, 2009 (Novi Sad) Nagoya, September 30, 2009 1 / 52 Introduction We introduce
More informationProblem: A class of dynamical systems characterized by a fast divergence of the orbits. A paradigmatic example: the Arnold cat.
À È Ê ÇÄÁ Ë ËÌ ÅË Problem: A class of dynamical systems characterized by a fast divergence of the orbits A paradigmatic example: the Arnold cat. The closure of a homoclinic orbit. The shadowing lemma.
More informationNotes for Elliptic operators
Notes for 18.117 Elliptic operators 1 Differential operators on R n Let U be an open subset of R n and let D k be the differential operator, 1 1 x k. For every multi-index, α = α 1,...,α n, we define A
More informationON STRICHARTZ ESTIMATES FOR SCHRÖDINGER OPERATORS IN COMPACT MANIFOLDS WITH BOUNDARY. 1. Introduction
ON STRICHARTZ ESTIMATES FOR SCHRÖDINGER OPERATORS IN COMPACT MANIFOLDS WITH BOUNDARY MATTHEW D. BLAIR, HART F. SMITH, AND CHRISTOPHER D. SOGGE 1. Introduction Let (M, g) be a Riemannian manifold of dimension
More informationJUHA KINNUNEN. Harmonic Analysis
JUHA KINNUNEN Harmonic Analysis Department of Mathematics and Systems Analysis, Aalto University 27 Contents Calderón-Zygmund decomposition. Dyadic subcubes of a cube.........................2 Dyadic cubes
More informationThe spectral zeta function
The spectral zeta function Bernd Ammann June 4, 215 Abstract In this talk we introduce spectral zeta functions. The spectral zeta function of the Laplace-Beltrami operator was already introduced by Minakshisundaram
More informationQualifying Exams I, 2014 Spring
Qualifying Exams I, 2014 Spring 1. (Algebra) Let k = F q be a finite field with q elements. Count the number of monic irreducible polynomials of degree 12 over k. 2. (Algebraic Geometry) (a) Show that
More informationWEYL S LEMMA, ONE OF MANY. Daniel W. Stroock
WEYL S LEMMA, ONE OF MANY Daniel W Stroock Abstract This note is a brief, and somewhat biased, account of the evolution of what people working in PDE s call Weyl s Lemma about the regularity of solutions
More informationDIEUDONNE AGBOR AND JAN BOMAN
ON THE MODULUS OF CONTINUITY OF MAPPINGS BETWEEN EUCLIDEAN SPACES DIEUDONNE AGBOR AND JAN BOMAN Abstract Let f be a function from R p to R q and let Λ be a finite set of pairs (θ, η) R p R q. Assume that
More informationLecture Notes Math 632, PDE Spring Semester Sigmund Selberg Visiting Assistant Professor Johns Hopkins University
Lecture Notes Math 63, PDE Spring Semester 1 Sigmund Selberg Visiting Assistant Professor Johns Hopkins University CHAPTER 1 The basics We consider the equation 1.1. The wave equation on R 1+n u =, where
More informationThe heat kernel meets Approximation theory. theory in Dirichlet spaces
The heat kernel meets Approximation theory in Dirichlet spaces University of South Carolina with Thierry Coulhon and Gerard Kerkyacharian Paris - June, 2012 Outline 1. Motivation and objectives 2. The
More informationPHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS
PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS OVIDIU SAVIN Abstract. We consider local minimizers of the Ginzburg-Landau energy functional 2 u 2 + 4 ( u2 ) 2 dx and prove that, if the level set is included
More informationRigidity and Non-rigidity Results on the Sphere
Rigidity and Non-rigidity Results on the Sphere Fengbo Hang Xiaodong Wang Department of Mathematics Michigan State University Oct., 00 1 Introduction It is a simple consequence of the maximum principle
More informationAsymptotics of generalized eigenfunctions on manifold with Euclidean and/or hyperbolic ends
Asymptotics of generalized eigenfunctions on manifold with Euclidean and/or hyperbolic ends Kenichi ITO (University of Tokyo) joint work with Erik SKIBSTED (Aarhus University) 3 July 2018 Example: Free
More informationSharp estimates for a class of hyperbolic pseudo-differential equations
Results in Math., 41 (2002), 361-368. Sharp estimates for a class of hyperbolic pseudo-differential equations Michael Ruzhansky Abstract In this paper we consider the Cauchy problem for a class of hyperbolic
More informationDeforming conformal metrics with negative Bakry-Émery Ricci Tensor on manifolds with boundary
Deforming conformal metrics with negative Bakry-Émery Ricci Tensor on manifolds with boundary Weimin Sheng (Joint with Li-Xia Yuan) Zhejiang University IMS, NUS, 8-12 Dec 2014 1 / 50 Outline 1 Prescribing
More informationThe heat equation in time dependent domains with Neumann boundary conditions
The heat equation in time dependent domains with Neumann boundary conditions Chris Burdzy Zhen-Qing Chen John Sylvester Abstract We study the heat equation in domains in R n with insulated fast moving
More informationSharp Gårding inequality on compact Lie groups.
15-19.10.2012, ESI, Wien, Phase space methods for pseudo-differential operators Ville Turunen, Aalto University, Finland (ville.turunen@aalto.fi) M. Ruzhansky, V. Turunen: Sharp Gårding inequality on compact
More informationRESTRICTION. Alex Iosevich. Section 0: Introduction.. A natural question to ask is, does the boundedness of R : L 2(r+1)
FOURIER TRANSFORM, L 2 THEOREM, AND SCALING RESTRICTION Alex Iosevich Abstract. We show, using a Knapp-type homogeneity argument, that the (L p, L 2 ) restriction theorem implies a growth condition on
More informationhere, this space is in fact infinite-dimensional, so t σ ess. Exercise Let T B(H) be a self-adjoint operator on an infinitedimensional
15. Perturbations by compact operators In this chapter, we study the stability (or lack thereof) of various spectral properties under small perturbations. Here s the type of situation we have in mind:
More informationLecture 2: Some basic principles of the b-calculus
Lecture 2: Some basic principles of the b-calculus Daniel Grieser (Carl von Ossietzky Universität Oldenburg) September 20, 2012 Summer School Singular Analysis Daniel Grieser (Oldenburg) Lecture 2: Some
More informationNOTES FOR CARDIFF LECTURES ON MICROLOCAL ANALYSIS
NOTES FOR CARDIFF LECTURES ON MICROLOCAL ANALYSIS JARED WUNSCH Note that these lectures overlap with Alex s to a degree, to ensure a smooth handoff between lecturers! Our notation is mostly, but not completely,
More informationA SHARP STABILITY ESTIMATE IN TENSOR TOMOGRAPHY
A SHARP STABILITY ESTIMATE IN TENSOR TOMOGRAPHY PLAMEN STEFANOV 1. Introduction Let (M, g) be a compact Riemannian manifold with boundary. The geodesic ray transform I of symmetric 2-tensor fields f is
More informationDEVELOPMENT OF MORSE THEORY
DEVELOPMENT OF MORSE THEORY MATTHEW STEED Abstract. In this paper, we develop Morse theory, which allows us to determine topological information about manifolds using certain real-valued functions defined
More informationMicrolocal analysis and inverse problems Lecture 4 : Uniqueness results in admissible geometries
Microlocal analysis and inverse problems Lecture 4 : Uniqueness results in admissible geometries David Dos Santos Ferreira LAGA Université de Paris 13 Wednesday May 18 Instituto de Ciencias Matemáticas,
More informationLocal smoothing and Strichartz estimates for manifolds with degenerate hyperbolic trapping
Local smoothing and Strichartz estimates for manifolds with degenerate hyperbolic trapping H. Christianson partly joint work with J. Wunsch (Northwestern) Department of Mathematics University of North
More informationINSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD
INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD () Instanton (definition) (2) ADHM construction (3) Compactification. Instantons.. Notation. Throughout this talk, we will use the following notation:
More informationCUTOFF RESOLVENT ESTIMATES AND THE SEMILINEAR SCHRÖDINGER EQUATION
CUTOFF RESOLVENT ESTIMATES AND THE SEMILINEAR SCHRÖDINGER EQUATION HANS CHRISTIANSON Abstract. This paper shows how abstract resolvent estimates imply local smoothing for solutions to the Schrödinger equation.
More informationExercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1
Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines
More informationComplex Analysis, Stein and Shakarchi The Fourier Transform
Complex Analysis, Stein and Shakarchi Chapter 4 The Fourier Transform Yung-Hsiang Huang 2017.11.05 1 Exercises 1. Suppose f L 1 (), and f 0. Show that f 0. emark 1. This proof is observed by Newmann (published
More informationfy (X(g)) Y (f)x(g) gy (X(f)) Y (g)x(f)) = fx(y (g)) + gx(y (f)) fy (X(g)) gy (X(f))
1. Basic algebra of vector fields Let V be a finite dimensional vector space over R. Recall that V = {L : V R} is defined to be the set of all linear maps to R. V is isomorphic to V, but there is no canonical
More informationON THE ASYMPTOTICS OF GREEN S FUNCTIONS OF ELLIPTIC OPERATORS WITH CONSTANT COEFFICIENTS. Shmuel Agmon
Séminaires & Congrès 9, 2004, p. 13 23 ON THE ASYMPTOTICS OF GREEN S FUNCTIONS OF ELLIPTIC OPERATORS WITH CONSTANT COEFFICIENTS by Shmuel Agmon Abstract. In this paper we discuss the following problem.
More informationRegularity of flat level sets in phase transitions
Annals of Mathematics, 69 (2009), 4 78 Regularity of flat level sets in phase transitions By Ovidiu Savin Abstract We consider local minimizers of the Ginzburg-Landau energy functional 2 u 2 + 4 ( u2 )
More informationPoisson Equation on Closed Manifolds
Poisson Equation on Closed anifolds Andrew acdougall December 15, 2011 1 Introduction The purpose of this project is to investigate the poisson equation φ = ρ on closed manifolds (compact manifolds without
More informationTaylor and Laurent Series
Chapter 4 Taylor and Laurent Series 4.. Taylor Series 4... Taylor Series for Holomorphic Functions. In Real Analysis, the Taylor series of a given function f : R R is given by: f (x + f (x (x x + f (x
More informationk=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula
20 VASILY PESTUN 3. Lecture: Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer Index theorems 3.. Index for a holomorphic vector bundle. For a holomorphic vector bundle E over a complex variety of dim
More informationSobolev resolvent estimates for the Laplace-Beltrami. operator on compact manifolds. Peng Shao
Sobolev resolvent estimates for the Laplace-Beltrami operator on compact manifolds by Peng Shao A dissertation submitted to The Johns Hopkins University in conformity with the requirements for the degree
More informationLagrangian submanifolds and generating functions
Chapter 4 Lagrangian submanifolds and generating functions Motivated by theorem 3.9 we will now study properties of the manifold Λ φ X (R n \{0}) for a clean phase function φ. As shown in section 3.3 Λ
More informationTraces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains
Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains Sergey E. Mikhailov Brunel University West London, Department of Mathematics, Uxbridge, UB8 3PH, UK J. Math. Analysis
More informationFriedrich symmetric systems
viii CHAPTER 8 Friedrich symmetric systems In this chapter, we describe a theory due to Friedrich [13] for positive symmetric systems, which gives the existence and uniqueness of weak solutions of boundary
More informationAlgebras of singular integral operators with kernels controlled by multiple norms
Algebras of singular integral operators with kernels controlled by multiple norms Alexander Nagel Conference in Harmonic Analysis in Honor of Michael Christ This is a report on joint work with Fulvio Ricci,
More informationTHE L 2 -HODGE THEORY AND REPRESENTATION ON R n
THE L 2 -HODGE THEORY AND REPRESENTATION ON R n BAISHENG YAN Abstract. We present an elementary L 2 -Hodge theory on whole R n based on the minimization principle of the calculus of variations and some
More informationMEAN CURVATURE FLOW OF ENTIRE GRAPHS EVOLVING AWAY FROM THE HEAT FLOW
MEAN CURVATURE FLOW OF ENTIRE GRAPHS EVOLVING AWAY FROM THE HEAT FLOW GREGORY DRUGAN AND XUAN HIEN NGUYEN Abstract. We present two initial graphs over the entire R n, n 2 for which the mean curvature flow
More informationRiemannian geometry of surfaces
Riemannian geometry of surfaces In this note, we will learn how to make sense of the concepts of differential geometry on a surface M, which is not necessarily situated in R 3. This intrinsic approach
More informationDECOUPLING INEQUALITIES IN HARMONIC ANALYSIS AND APPLICATIONS
DECOUPLING INEQUALITIES IN HARMONIC ANALYSIS AND APPLICATIONS 1 NOTATION AND STATEMENT S compact smooth hypersurface in R n with positive definite second fundamental form ( Examples: sphere S d 1, truncated
More informationMATH6081A Homework 8. In addition, when 1 < p 2 the above inequality can be refined using Lorentz spaces: f
MATH68A Homework 8. Prove the Hausdorff-Young inequality, namely f f L L p p for all f L p (R n and all p 2. In addition, when < p 2 the above inequality can be refined using Lorentz spaces: f L p,p f
More informationMathematical Tripos Part III Michaelmas 2017 Distribution Theory & Applications, Example sheet 1 (answers) Dr A.C.L. Ashton
Mathematical Tripos Part III Michaelmas 7 Distribution Theory & Applications, Eample sheet (answers) Dr A.C.L. Ashton Comments and corrections to acla@damtp.cam.ac.uk.. Construct a non-zero element of
More informationarxiv: v1 [math.ap] 20 Nov 2007
Long range scattering for the Maxwell-Schrödinger system with arbitrarily large asymptotic data arxiv:0711.3100v1 [math.ap] 20 Nov 2007 J. Ginibre Laboratoire de Physique Théorique Université de Paris
More informationPutzer s Algorithm. Norman Lebovitz. September 8, 2016
Putzer s Algorithm Norman Lebovitz September 8, 2016 1 Putzer s algorithm The differential equation dx = Ax, (1) dt where A is an n n matrix of constants, possesses the fundamental matrix solution exp(at),
More informationSCALE INVARIANT FOURIER RESTRICTION TO A HYPERBOLIC SURFACE
SCALE INVARIANT FOURIER RESTRICTION TO A HYPERBOLIC SURFACE BETSY STOVALL Abstract. This result sharpens the bilinear to linear deduction of Lee and Vargas for extension estimates on the hyperbolic paraboloid
More information8 Singular Integral Operators and L p -Regularity Theory
8 Singular Integral Operators and L p -Regularity Theory 8. Motivation See hand-written notes! 8.2 Mikhlin Multiplier Theorem Recall that the Fourier transformation F and the inverse Fourier transformation
More informationMath 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.
Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,
More informationClassical Fourier Analysis
Loukas Grafakos Classical Fourier Analysis Second Edition 4y Springer 1 IP Spaces and Interpolation 1 1.1 V and Weak IP 1 1.1.1 The Distribution Function 2 1.1.2 Convergence in Measure 5 1.1.3 A First
More informationRepresentations of moderate growth Paul Garrett 1. Constructing norms on groups
(December 31, 2004) Representations of moderate growth Paul Garrett Representations of reductive real Lie groups on Banach spaces, and on the smooth vectors in Banach space representations,
More informationHeat Kernel and Analysis on Manifolds Excerpt with Exercises. Alexander Grigor yan
Heat Kernel and Analysis on Manifolds Excerpt with Exercises Alexander Grigor yan Department of Mathematics, University of Bielefeld, 33501 Bielefeld, Germany 2000 Mathematics Subject Classification. Primary
More informationThe Borsuk Ulam Theorem
The Borsuk Ulam Theorem Anthony Carbery University of Edinburgh & Maxwell Institute for Mathematical Sciences May 2010 () 1 / 43 Outline Outline 1 Brouwer fixed point theorem 2 Borsuk Ulam theorem Introduction
More informationREPRESENTATION THEORY WEEK 7
REPRESENTATION THEORY WEEK 7 1. Characters of L k and S n A character of an irreducible representation of L k is a polynomial function constant on every conjugacy class. Since the set of diagonalizable
More information6. Applications to differential operators. The Sobolev theorem
6.1 6. Applications to differential operators. The Sobolev theorem 6.1. Differential and pseudodifferential operators on R n. As we saw in (5.39) (5.42), a differential operator P(D) (with constant coefficients)
More information