Bounded imaginary powers of differential operators on manifolds with conical singularities

 We study the minimal and maximal closed extension of a differential operator A on a manifold B with conical singularities, when A acts as an unbounded operator on weighted L _p -spaces over B, 1<p<∞. Under suitable ellipticity assumptions we can define a family of complex powers A ^z , zℂ. We also obtain sufficient information on the resolvent of A to show the boundedness of the purely imaginary powers. Examples concern unique solvability and maximal regularity for the solution of the Cauchy problem for the Laplacian on conical manifolds as well as certain quasilinear diffusion equations. Received: 12 June 2001; in final form: 3 June 2002 / Published online: 1 April 2003 Mathematics Subject Classification (2000): 35J70, 47A10, 35K57     

Calabi-Yau manifolds with isolated conical singularities

Let \(X\) be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let \(L\) be an ample line bundle on \(X\). Assume that the pair \((X,L)\) is the flat limit of a family of smooth polarized Calabi-Yau manifolds. Assume that for each singular point \(x \in X\) there exist a Kähler-Einstein Fano manifold \(Z\) and a positive integer \(q\) dividing \(K_{Z}\) such that \(-\frac{1}{q}K_{Z}\) is very ample and such that the germ \((X,x)\) is locally analytically isomorphic to a neighborhood of the vertex of the blow-down of the zero section of \(\frac{1}{q}K_{Z}\). We prove that up to biholomorphism, the unique weak Ricci-flat Kähler metric representing \(2\pi c_{1}(L)\) on \(X\) is asymptotic at a polynomial rate near \(x\) to the natural Ricci-flat Kähler cone metric on \(\frac{1}{q}K_{Z}\) constructed using the Calabi ansatz. In particular, our result applies if \((X, \mathcal{O}(1))\) is a nodal quintic threefold in \(\mathbf {P}^{4}\). This provides the first known examples of compact Ricci-flat manifolds with non-orbifold isolated conical singularities.     

On the completeness of the generalized eigenfunctions of elliptic operators on manifolds with conical singularities

We show the completeness of the system of generalized eigenfunctions of closed extensions of elliptic cone operators under suitable conditions on the symbols (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Complex powers of degenerate differential operators connected with the Klein-Gordon-Fock operator

We develop a theory of complex powers of the generalized Klein-Gordon-Fock operator

$m^2 - \square - i\lambda \frac{{\partial ^2 }}{{\partial x_1^2 }},\lambda > 0.$

. The negative powers of this operator are realized as potential-type integrals with nonstandard metrics, while positive powers inverse to negative ones are realized as approximative inverse operators.

     

The index of elliptic operators on manifolds with conical points

For general elliptic pseudodifferential operators on manifolds with singular points, we prove an algebraic index formula. In this formula the symbolic contributions from the interior and from the singular points are explicitly singled out. For two-dimensional manifolds, the interior contribution is reduced to the Atiyah-Singer integral over the cosphere bundle while two additional terms arise. The first of the two is one half of the "eta" invariant associated to the conormal symbol of the operator at singular points. The second term is also completely determined by the conormal symbol. The example of the Cauchy-Riemann operator on the complex plane shows that all the three terms may be nonzero. Moreover, we introduce a natural symmetry condition for a pseudodifferential operator on a manifold with cylindrical ends ensuring that the operator admits a doubling across the boundary. For such operators we prove an explicit index formula containing, apart from the Atiyah-Singer integral, a finite number of residues of the logarithmic derivative of the conormal symbol.     

The fractional porous medium equation on manifolds with conical singularities II

This is the second of a series of two papers that studies the fractional porous medium equation, ${\partial }_{t}u+{(-\mathrm{\Delta })}^{\sigma }{(|u|}^{m-1}u)=0$ with $m>0$ and $\sigma \in (0,1]$, posed on a Riemannian manifold with isolated conical singularities. The first aim of the article is to derive some useful properties for the Mellin–Sobolev spaces including the Rellich–Kondrachov theorem and Sobolev–Poincaré, Nash and Super Poincaré type inequalities. The second part of the article is devoted to the study the Markovian extensions of the conical Laplacian operator and its fractional powers. Then based on the obtained results, we establish existence and uniqueness of a global strong solution for $L_{\infty }-$initial data and all $m>0$. We further investigate a number of properties of the solutions, including comparison principle, $L_p-$contraction and conservation of mass. Our approach is quite general and thus is applicable to a variety of similar problems on manifolds with more general singularities.     

Complex powers of some degenerate differential operators related to the Helmholtz operator

In the L _p -spaces, we study the complex powers of the operator

$G_\lambda = m^2 I + \Delta - i\lambda \frac{{\partial ^2 }}{{\partial x_1^2 }},0 < \lambda < 1,m > 0,$

where δ is the Laplace operator. The complex powers G _λ ^?α/2 , Reα > 0, are realized as potential type operators B _λ ^α with a nonstandard metric. We obtain L _p → L _p + L _s -estimates for the operator B _λ ^α . By using the method of approximate inverse operators, we construct the inversion of the potentials B _λ ^α φ with L _p -densities and describe the range B _λ ^α (L _p ) in terms of the inversion constructions.

     

Complex powers of classical SG-pseudodifferential operators

Abstract Under a suitable ellipticity condition, we show that classical SG-pseudodifferential operators of nonnegative order possess complex powers. We show that the powers are again classical and derive an explicit formula for all homogeneous components. Keywords: Complex power, Weighted symbols, Noncompact manifolds     

Complex powers of elliptic pseudodifferential operators

The aim of this paper is the construction of complex powers of elliptic pseudodifferential operators and the study of the analytic properties of the corresponding kernels k_S (x,y). For x=y, the case of principal interest, the domain of holomorphy and the singularities of k_S (x,x) are shown to depend on the asymptotic expansion of the symbol. For classical symbols, k_S (x,x) is known to be meromorphic on with simple poles in a set of equidistant points on the real axis. In the more general cases considered here, the singularities may be distributed over a half plane and k_S (x,x) can not always be extended to337-2. An example is given where k_S (x,x) has a vertical line as natural boundary.Dedicated to Professor H.G. Tillmann on the occasion of his 60th birthday     

Complex powers of hypoelliptic pseudodifferential operators

Complex powers of a class of hypoelliptic pseudodifferential operators in Rn, as well as their heat kernels are studied. An application to the Schatten-von Neumann property of pseudodifferential operators is given.     

Invariant differential operators on symplectic grassmann manifolds

LetM _2n,r denote the vector space of real or complex2n×r matrices with the natural action of the symplectic group Sp _2n , and letG=G _n,r =Sp _2n ×M _2n,r denote the corresponding semi-direct product. For any integerp with 0≤p≤n−1, letH denote the subgroupG _p,r ×Sp _2n−2p ofG. We explicitly compute the algebra of left invariant differential operators onG/H, and we show that it is a free algebra if and only ifr≤2n−2p+1. We also give orthogonal analogues of these results, generalizing those of Gonzalez and Helgason [3]. Partially supported by NSF grant DMS-9101358 This article was processed by the author using the Springer-Verlag T_EX mamath macro package 1990.     

C*-algebras of singular integral operators in domains with oscillating conical singularities

 For a domain with singular points on the boundary, we consider a C ^* -algebra of operators acting in the weighted space . It is generated by the operators of multiplication by continuous functions on and the operators where σ is a homogeneous function. We show that the techniques of limit operators apply to define a symbol algebra for . When combined with the local principle, this leads to describing the Fredholm operators in . Received: 21 December 2000     

Existence and maximal Lp-regularity of solutions for the porous medium equation on manifolds with conical singularities

We consider the porous medium equation on manifolds with conical singularities and show existence, uniqueness, and maximal L^p-regularity of a short-time solution. In particular, we obtain information on the short time asymptotics of the solution near the conical point. Our method is based on bounded imaginary powers results for cone differential operators on Mellin–Sobolev spaces and R-sectoriality perturbation techniques.     

Reconstruction of Sturm-Liouville differential operators with singularities inside the interval

The inverse spectral problem for Sturm-Liouville differential operators on a finite interval is studied for an arbitrary and finite number of regular singular points inside the interval. A uniqueness theorem is proved; necessary and sufficient conditions and a procedure for the solution of the inverse problem are obtained.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 143–156, July, 1998.This research was supported by the Ministry of Education (KTsFE) under grant No. 96-1.7-4 and by the Russian Foundation for Basic Research under grant No. 97-01-00566.