An Algebra of Pseudodifferential Operators with Slowly Oscillating Symbols

Let VR denote the Banach algebra of absolutely continuous functionsof bounded total variation on R, and let B_p be the Banach algebraof bounded linear operators acting on the Lebesgue space L^pRfor 1 < p < . We study the Banach algebra A B_p generatedby the pseudodifferential operators of zero order with slowlyoscillating VR-valued symbols on R. Boundedness and compactnessconditions for pseudodifferential operators with symbols inL R, VR are obtained. A symbol calculus for the non-closed algebraof pseudodifferential operators with slowly oscillating VR-valuedsymbols is constructed on the basis of an appropriate approximationof symbols by infinitely differentiable ones and by use of thetechniques of oscillatory integrals. As a result, the quotientBanach algebra A = A K, where K is the ideal of compact operatorsin B_p, is commutative and involutive. An isomorphism betweenthe quotient Banach algebra A of pseudodifferential operatorsand the Banach algebra of their Fredholm symbols is established. A Fredholm criterionand an index formula for the pseudodifferential operators A A are obtained in terms of their Fredholm symbols. 2000 MathematicsSubject Classification 47G30, 47L15 (primary), 47A53, 47G10(secondary).     

A bifurcation problem governed by the boundary condition II

In this work we consider the problem u = a(x)u^p in on , where is a smooth bounded domain, isthe outward unit normal to , is regarded as a parameter and0 < p < 1. We consider both cases where a(x) > 0 in or a(x) is allowed to vanish in a whole subdomain ₀ of . Ourmain results include existence of non-negative non-trivial solutionsin the range 0 < < ₁, where ₁ is characterized by meansof an eigenvalue problem, uniqueness and bifurcation from infinityof such solutions for small , and the appearance of dead coresfor large enough .     

General Uniqueness Results and Variation Speed for Blow-Up Solutions of Elliptic Equations

Let be a smooth bounded domain in R^N. We prove general uniquenessresults for equations of the form – u = au – b(x)f(u) in , subject to u = on . Our uniqueness theorem is establishedin a setting involving Karamata's theory on regularly varyingfunctions, which is used to relate the blow-up behavior of u(x)with f(u) and b(x), where b 0 on and a certain ratio involvingb is bounded near . A key step in our proof of uniqueness usesa modification of an iteration technique due to Safonov. 2000Mathematics Subject Classification 35J25 (primary), 35B40, 35J60(secondary).     

Weak Covering Properties of Weak Topologies

We consider covering properties of weak topologies of Banachspaces, especially of weak or point-wise topologies of functionspaces C(K), for compact spaces K. We answer questions posedby A. V. Arkhangel'skii, S. P. Gul'ko, and R. W. Hansell. Ourmain results are the following. A Banach space of density atmost ₁ is hereditarily metaLindel of in its weak topology. Ifthe weight of a compact spaceK is at most ₁, then the spacesC_w(K) and C_p(K) are hereditarily metaLindel of. Let be the one-point compactificationof a treeT. Then the space is hereditarily -metacompact. If T is an infinitely branchingfull tree of uncountable height and of cardinality bigger thanc, then the weak topology of the unit sphere of is not -fragmented by any metric. The space C_p(rß₁)is neither metaLindel of nor -relatively metacompact. The spaceC_p(rß₂) is not -relatively metaLindel of. Under theset-theoretic axiom , there exists a scattered compact spaceK₁ such that the space C_p(K₁) is not -relatively metacompact,and under a related axiom , there exists a scattere compactspace K₂ such that the space C_p(K₂) is not -relatively metaLindelof. 1991 Mathematics Subject Classification: 54C35, 46B20, 54E20,54D30.     

Skeletons and Central Sets

Let be an open proper subset of Rⁿ. Its skeleton is the setof points with more than one nearest neighbour in the complementof its central set is the set of centres in maximal open ballsincluded in . Intuitively, if we think of as a land mass inwhich height is proportional to distance from the sea, its skeletonand central set can be thought of as corresponding to ridgesin the mountains of . In this note I discuss the metric andtopological properties of such sets. I show that any skeletonin Rⁿ is F, and has dimension at most n – 1, by any ofthe usual measures of dimension; that if is bounded and connected,its skeleton and central set are connected; and that separatesRⁿ iff its skeleton does iff its central set does. Any centralset in Rⁿ is a G set of topological dimension at most n –1. In the plane, I show that both skeletons and central setsare locally path-connected, and indeed include many paths offinite length. For any , its central set includes its skeleton;I give examples to show that the central set can be significantlylarger than the skeleton. 1991 Mathematics Subject Classification:54F99.     

A Joint Functional Calculus for Sectorial Operators with Commuting Resolvents

In this paper we study the notion of joint functional calculusassociated with a couple of resolvent commuting sectorial operatorson a Banach space X. We present some positive results when Xis, for example, a Banach lattice or a quotient of subspacesof a B-convex Banach lattice. Furthermore, we develop a notionof a generalized H-functional calculus associated with the extensionto (H) of a sectorial operator on a B-convex Banach lattice, where H is a Hilbert space. We apply our results to a newconstruction of operators with a bounded H-functional calculusand to the maximal regularity problem. 1991 Mathematics SubjectClassification: 47A60, 47D06, 46C15.     

Chaotic Expansion of Elements of the Universal Enveloping Algebra of a Lie Algebra Associated with a Quantum Stochastic Calculus

The functional Ito formula, in the form df() = f( + d ) –f(),is formulated and proved in the context of a Lie algebra L associatedwith a quantum (non-commutative) stochastic calculus. Here fis an element of the universal enveloping algebra U of L, andf() + d() – f() is given a meaning using the coproductstructure of U even though the individual terms of this expressionhave no meaning. The Ito formula is equivalent to a chaoticexpansion formula for f() which is found explicitly. 1991 MathematicsSubject Classification: primary 81S25; secondary 60H05; tertiary18B25.     

Global Branch of Solutions for Non-Linear Schrodinger Equations with Deepening Potential Well

We consider the stationary non-linear Schrödinger equation where > 0 and the functionsf and g are such that and for some bounded open set R^N. We use topological methods to establish the existenceof two connected sets D^± of positive/negative solutionsin R x W², ^p R^N where that cover the interval (, ()) in the sense that and furthermore, The number () is characterized as the unique value of in theinterval (, ) for which the asymptotic linearization has a positiveeigenfunction. Our work uses a degree for Fredholm maps of indexzero. 2000 Mathematics Subject Classification 35J60, 35B32,58J55.     

Ideal Spaces of Banach Algebras

The ideal space Id(A) of a Banach algebra A is studied as abitopological space Id(A), _u, _n, where _u is the weakest topologyfor which all the norm functions I || a + I|| (with a A andI Id(A)) are upper semi-continuous, and _n is the de Groot dualof _u. When A is separable, _nu is either a compact, metrizabletopology, or it is neither Hausdorff nor first countable. TAF-algebrasare shown to exhibit the first type of behaviour. Applicationsto Banach bundles (which motivate the study), and to PI-Banachalgebras, are given. 1991 Mathematics Subject Classification:46H10, 46J20.     

Multivariate R-O Varying Measures Part I: Uniform Bounds

A finite Borel measure µ on R^d is called R-O varying withindex F if there exist a GL(R^d)-valued function f varying regularlywith index (–F), an increasing function k: (0, ) (0,) with k(t) and k(t + 1)/k(t) c 1 as t , and a -finitemeasure on R^d\0 such that R-O varying measures generalize regularly varying measures introducedby Meerschaert (see M. M. Meerschaert, ‘Regular variationin R^k’, Proc. Amer. Math. Soc. 102 (1988) 341–348)and have numerous applications in limit theorems for probabilitymeasures. For an R-O varying measure µ and – < let denote the tail- andtruncated moment functions of µ in the direction || =1. The purpose of this paper is to show that R-O variation ofa measure implies sharp bounds on the growth rate of the tail-and truncated moment functions depending on the real parts ofthe eigenvalues of the index F along a compact set of directions.Furthermore, bounds on the ratio of these functions for certainvalues of a and b are obtained. 1991 Mathematics Subject Classification:60B10, 28C15.     

Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

We prove a nearly optimal bound on the number of stable homotopytypes occurring in a k-parameter semi-algebraic family of setsin R, each defined in terms of m quadratic inequalities. Ourbound is exponential in k and m, but polynomial in . More precisely,we prove the following. Let R be a real closed field and let = {P₁, ... , P_m} R[Y₁, ... ,Y,X₁, ... ,X_k], with deg_Y(P_i) 2, deg_X(P_i) d, 1 i m. Let S R^+k be a semi-algebraic set,defined by a Boolean formula without negations, with atoms ofthe form P 0, P 0, P . Let : R^+k R^k be the projection onthe last k coordinates. Then the number of stable homotopy typesamongst the fibers S_x = ^–1(x) S is bounded by (2^mkd)^O(mk).     

On the laplace equation with non-linear dynamical boundary conditions

The main part of the paper deals with local existence and globalexistence versus blow-up for solutions of the Laplace equationin bounded domains with a non-linear dynamical boundary condition.More precisely, we study the problem consisting in: (1) theLaplace equation in (0, ) x ; (2) a homogeneous Dirichlet condition(0, ) x ₀; (3) the dynamical boundary condition ; (4) the initial condition u(0, x) = u₀ (x) on . Here is a regular and bounded domain in Rⁿ, with n 1, and₀ and ₁ endow a measurable partition of . Moreover, m>1,2 p < r, where r = 2 (n – 1) / (n – 2) whenn 3, r = when n = 1,2, and u₀ H^1/2 , u₀ = 0 on ₀. The final part of the paper deals with a refinement of a globalnon-existence result by Levine, Park and Serrin, which is appliedto the previous problem. 2000 Mathematics Subject Classification35K55 (primary), 35K90, 35K77 (secondary).     

Hilbert Transforms Along Curves in the Heisenberg Group

We consider Hilbert transforms along curves on the Heisenberggroup. In particular, necessary and sufficient conditions forthe L²-boundedness are given for (t)=(t,(t),0) when is evenor odd and convex on R⁺. 1991 Mathematics Subject Classification:42B20, 42B25.     

Unknotting Tunnels and Seifert Surfaces

Let K be a knot with an unknotting tunnel and suppose thatK is not a 2-bridge knot. There is an invariant = p/q Q/2Z,with p odd, defined for the pair (K, ). The invariant has interesting geometric properties. It is oftenstraightforward to calculate; for example, for K a torus knotand an annulus-spanning arc, (K, ) = 1. Although is definedabstractly, it is naturally revealed when K is put in thinposition. If 1 then there is a minimal-genus Seifert surfaceF for K such that the tunnel can be slid and isotoped to lieon F. One consequence is that if (K, ) 1 then K > 1. Thisconfirms a conjecture of Goda and Teragaito for pairs (K, )with (K, ) 1. 2000 Mathematics Subject Classification 57M25,57M27.     

Ramsey and Freeness Properties of Polish Planes

Suppose that X is a Polish space which is not -compact. We provethat for every Borel colouring of X² by countably many colours,there exists a monochromatic rectangle with both sides closedand not -compact. Moreover, every Borel colouring of [X]² byfinitely many colours has a homogeneous set which is closedand not -compact. We also show that every Borel measurable functionf:X² X has a free set which is closed and not -compact. Ascorollaries of the proofs we obtain two results: firstly, theproduct forcing of two copies of superperfect tree forcing doesnot add a Cohen real, and, secondly, it is consistent with ZFCto have a closed subset of the Baire space which is not -compactand has the property that, for any three of its elements, noneof them is constructible from the other two. A similar proofshows that it is consistent to have a Laver tree such that noneof its branches is constructible from any other branch. Thelast four results answer questions of Goldstern and Brendle.2000 Mathematics Subject Classification: 03E15, 26B99, 54H05.     

Cohen-Macaulay Properties of Thom-Boardman Strata I: Morin's Ideal

Thom–Boardman strata ^I are fundamental tools in studyingsingularities of maps. The Zariski closures of the strata ^Iare components of the set of zeros of the ideals ^I defined by B. Morin using iterated jacobian extensions in his paper‘Calcul jacobien’ (Ann. Sci. École Norm.Sup.} 8 (1975) 1–98). In this paper, we consider the questionof when the Morin ideals ^I define Cohen–Macaulay spaces.We determine all I=(i₁...,i_k) such that ^I defines a Cohen–Macaulayspace alongthe stratum. 1991 Mathematics Subject Classification: 13D25, 14B05, 14M12, 58C25.     

Projection Methods for Discrete Schrodinger Operators

Let H be the discrete Schrödinger operator acting on l² Z⁺, where the potential v is real-valued and v(n) 0 as n . Let P be the orthogonal projection onto a closedlinear subspace l² Z⁺). In a recent paper E. B. Davies definesthe second order spectrum Spec₂(H, ) of H relative to as theset of z C such that the restriction to of the operator P(H- z)²P is not invertible within the space . The purpose of thisarticle is to investigate properties of Spec₂(H, ) when islarge but finite dimensional. We explore in particular the connectionbetween this set and the spectrum of H. Our main result providessharp bounds in terms of the potential v for the asymptoticbehaviour of Spec₂(H, ) as increases towards l² Z⁺). 2000 MathematicsSubject Classification 47B36 (primary), 47B39, 81-08 (secondary).     

On Lp boundedness of wave operators for 4-dimensional Schrodinger operators with threshold singularities

Let H=–+V(x) be a Schrödinger operator on L²(R⁴),H₀=–. Assume that |V(x)|+| V(x)|C x ^– for some>8. Let be the wave operators. It is known that W_± extend to bounded operators in L^p(R⁴)for all 1p, if 0 is neither an eigenvalue nor a resonance ofH. We show that if 0 is an eigenvalue, but not a resonance ofH, then the W_± are still bounded in L^p(R⁴) for all psuch that 4/3<p<4.