Multipliers spaces, Muckenhoupt weights and pseudo-differential operators

Let σ(x,ξ) be a sufficiently regular function defined on Rd×Rd. The pseudo-differential operator with symbol σ is defined on the Schwartz class by the formula

     

Pseudodifferential operators on Bochner spaces and an application

We consider pseudodifferential operators with operator valued symbols a(x,ξ) acting on a UMD Banach space X. Assuming some regularity of Hölder type in x and Mihlin type in ξ we prove L _p (? ⁿ ;X) boundedness of such operators. This result is then applied to the study of L _p -maximal regularity for nonautonomous parabolic evolution equations.     

Milnor open books and Milnor fillable contact 3-manifolds

We say that an oriented contact manifold (M,ξ) is Milnor fillable if it is contactomorphic to the contact boundary of an isolated complex-analytic singularity (X,x). In this article we prove that any three-dimensional oriented manifold admits at most one Milnor fillable contact structure up to contactomorphism. The proof is based on Milnor open books: we associate an open book decomposition of M with any holomorphic function f:(X,x)→(C,0), with isolated singularity at x and we verify that all these open books carry the contact structure ξ of (M,ξ)—generalizing results of Milnor and Giroux.     

ω-hypoelliptic differential operators of constant strength

We study ω-hypoelliptic differential operators of constant strength. We show that any operator with constant strength and coefficients in which is homogeneous ω-hypoelliptic is also σ-hypoelliptic for any weight function σ=O(ω). We also present a sufficient condition in order to ensure that a differential operator admits a parametrix and, as a consequence, we obtain some conditions on the weights (ω,σ) to conclude that, for any operator P(x,D) with constant strength, the σ-hypoellipticity of the frozen operator P(x₀,D) implies the ω-hypoellipticity of P(x,D). This requires the use of pseudodifferential operators.     

О функциях, удовлетворяюших ослабленному условию асимптотической моногенности

We consider functionsf(z),z∈D, of one complex variable that satisfy the following weakened asymptotic monogeny condition: for some positiveσ<1/2,f(z) is monogenic at each pointξ∈D with respect to some setG(ξ) such that the lower density ofG(ξ) atξ is greater than 1/2+σ. We show that if for somep _σ ≥1 the function (log⁺|?(z)|) ^p _σ is locally integrable inD with respect to the plane Lebesgue measure, thenf(z) is holomorphic inD.     

Pseudo-Differential Calculus on Homogeneous Trees

To study concentration and oscillation properties of eigenfunctions of the discrete Laplacian on regular graphs, we construct in this paper a pseudo-differential calculus on homogeneous trees, their universal covers. We define symbol classes and associated operators. We prove that these operators are bounded on L ² and give adjoint and product formulas. Finally, we compute the symbol of the commutator of a pseudo-differential operator with the Laplacian.     

Spectral properties of zeroth-order pseudodifferential operators

Let Q be a self-adjoint, classical, zeroth order pseudodifferential operator on a compact manifold X with a fixed smooth measure dx. We use microlocal techniques to study the spectrum and spectral family, {E_S}_{S∈$\text{R}$} as a bounded operator on L²(X, dx).Using theorems of Weyl (Rend. Circ. Mat. Palermo, 27 (1909), 373–392) and Kato (“Perturbation Theory for Linear Operators,” Springer-Verlag, 1976) on spectra of perturbed operators we observe that the essential spectrum and the absolutely continuous spectrum of Q are determined by a finite number of terms in the symbol expansion. In particular Spec_ESSQ = range(q(x, ξ)) where q is the principal symbol of Q. Turning the attention to the spectral family {E_S}_{S∈$\text{R}$}, it is shown that if $\text{dE}\text{ds}$ is considered as a distribution on $\text{R}$×X×X it is in fact a Lagrangian distribution near the set $\text{{σ=0}?}\text{T}{}^1\text{(}\text{R}\text{×X×X)}\text{0}$ where (s, x, y, σ, ξ,η) are coordinates on T¹($\text{R}$×X×X) induced by the coordinates (s, x, y) on $\text{R}$×X×X. This leads to an easy proof that$\text{?(Q)}$ is a pseudodifferential operator if ?∈C^∞($\text{R}$) and to some results on the microlocal character of E_s. Finally, a look at the wavefront set of $\text{dE}\text{ds}$ leads to a conjecture about the existence of absolutely continuous spectrum in terms of a condition on q(x, ξ).     

An implicit-function theorem for inclusions

We consider the question of the solvability of an inclusion F(x, σ) ∈ A, i.e., of determining a mapping (implicit function) σ ? x(σ) defined on a set such that F(x(σ), σ) ∈ A for any σ from this set. Results of this kind play a key role in the different branches of analysis and, especially, in the theory of extremal problems, where they are the main tool for deriving conditions for an extremum.     

Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point

We study the microlocal kernel of h-pseudodifferential operators Oph(p)−z, where z belongs to some neighborhood of size O(h) of a critical value of its principal symbol p₀(x,ξ). We suppose that this critical value corresponds to a hyperbolic fixed point of the Hamiltonian flow Hp₀. First we describe propagation of singularities at such a hyperbolic fixed point, both in the analytic and in the C^∞ category. In both cases, we show that the null solution is the only element of this microlocal kernel which vanishes on the stable incoming manifold, but for energies z in some discrete set. For energies z out of this set, we build the element of the microlocal kernel with given data on the incoming manifold. We describe completely the operator which associate the value of this null solution on the outgoing manifold to the initial data on the incoming one. In particular it appears to be a semiclassical Fourier integral operator associated to some natural canonical relation.     

Functional ergodic limits for occupation time processes of site-dependent branching Brownian motions in ℝ

We consider a kind of site-dependent branching Brownian motions whose branching laws depend on the site-branching factor σ(·). We focus on the functional ergodic limits for the occupation time processes of the models in ?. It is proved that the limiting process has the form of λξ(·), where λ is the Lebesgue measure on ? and ξ(·) is a real-valued process which is non-degenerate if and only if σ is integrable. When ξ(·) is non-degenerate, it is strictly positive for t > 0. Moreover, ξ converges to 0 in finite-dimensional distributions if the integral of σ tends to infinity.     

A system of two inhomogeneous linear functional equations

Given real nonzero coefficients a, A, b, B and additive functions $\phi,\psi : {\mathbb {R}}\to {\mathbb {R}}$ , necessary and sufficient conditions for existence and uniqueness of additive solutions of the system ξ(ax)?Aξ(x)=?(x), ξ(bx)?Bξ(x)=ψ(x) are presented. According to the various possibilities concerning the arithmetic nature of the four coefficients and the algebraic relationships between them, the additive solution(s) of the system are given explicitly for each case.     

Rings and semigroups with permutable zero products

We consider rings R, not necessarily with 1, for which there is a nontrivial permutation σ on n letters such that x₁?x_n=0 implies x_σ(1)?x_σ(n)=0 for all x₁,…,x_n∈R. We prove that this condition alone implies very strong permutability conditions for zero products with sufficiently many factors. To this end we study the infinite sequences of permutation groups P_n(R) consisting of those permutations σ on n letters for which the condition above is satisfied in R. We give the full characterization of such sequences both for rings and for semigroups with 0. This enables us to generalize some recent results by Cohn on reversible rings and by Lambek, Anderson and Camillo on rings and semigroups whose zero products commute. In particular, we prove that rings with permutable zero products satisfy the Köthe conjecture.     

A boundedness result for twisted convolution

We consider twisted convolution operators with kernels having singularities on a sphere and having as Fourier transform the oscillatory symbol m_α(|ξ|) = |ξ|^–αe^i|ξ|, 0 ≤ ??α < 2n. We give integral representations for such operators and, as a principal result, we study L_p–L_q estimates for them.     

Lp-spectra of strongly Carleman pseudo-differential operators

The spectrum and essential spectrum in L^p($\text{R}$ⁿ) of a strongly Carleman pseudo-differential operator with symbol of class S_?,0^m, 0 ? p ? 1, are shown to coincide with the range of the symbol for some (but need not be all) p different from 2. The absolutely continuous spectra of perturbations of operators with strongly Carleman symbols (but without the assumption of being in S_?,0^m) are also investigated.     

Sharp semiclassical estimates for the number of eigenvalues below a totally degenerate critical level

We consider the semiclassical asymptotic behaviour of the number of eigenvalues smaller than E for elliptic operators in L²(Rd). We describe a method of obtaining remainder estimates related to the volume of the region of the phase space in which the principal symbol takes values belonging to the intervals [E^′;E^′+h], where E^′ is close to E. If the volume of this region is O(h), then we obtain remainder estimates O(h1−d) with no assumptions on the Hessian of the principal symbol at the critical level. Moreover we do not assume that the coefficients are smooth—all results hold if second order derivatives of coefficients are Hölder continuous.     

A necessary condition for the local solvability of the operator Pm2(x, D) + P2m − 1(x, D)

It is shown that a necessary condition for the local solvability of the operator P(x, D) = P_m²(x, D) + P_{2m ? 1}(x, D), where P_m(x, D) is an mth-order homogeneous differential operator of principal type with real coefficients, is that along any null-bicharacteristic strip of P_m(x, ξ) the imaginary part of the sub-principal symbol cannot have an odd-order zero where its real part does not vanish.     

Renormalized solutions of degenerate elliptic problems

We consider a general class of degenerate elliptic problems of the form Au+g(x,u,Du)=f, where A is a Leray-Lions operator from a weighted Sobolev space into its dual. We assume that g(x,s,ξ) is a Caratheodory function verifying a sign condition and a growth condition on ξ. Existence of renormalized solutions is established in the L¹-setting.     

Finite metrics in switching classes

Let g:D×D→R be a symmetric function on a finite set D satisfying g(x,x)=0 for all x∈D. A switch g^σ of g w.r.t. a local valuation σ:D→R is defined by g^σ(x,y)=σ(x)+g(x,y)+σ(y) for x≠y and g^σ(x,x)=0 for all x. We show that every symmetric function g has a unique minimal semimetric switch, and, moreover, there is a switch of g that is isometric to a finite Manhattan metric. Also, for each metric on D, we associate an extension metric on the set of all nonempty subsets of D, and we show that this extended metric inherits the switching classes on D.     

Quantum mechanics and partial differential equations

This paper develops the basic theory of pseudo-differential operators on Rⁿ, through the Calderón-Vaillancourt (0, 0) L²-estimate, as a natural part of the harmonic analysis on the Heisenberg group, the group-theoretic embodiment of Heisenberg's Canonical Commutation Relations. The symbol mapping is given a group-theoretic interpretation consistent with the Kirillov method of orbits. By comparing different well-known realizations of the unique irreducible representation of the Heisenberg group, the Toeplitz operators on the complex n-ball are shown essentially to be pseudo-differential operators. The proof of the Calderón-Vaillancourt estimate is almost purely group-theoretic. Criteria for positivity, and for compactness are also given.     

Integro-local theorems for sums of independent random vectors in the series scheme

Let S(n) = ξ(1)+?+ξ(n) be a sum of independent random vectors ξ(i) = ξ _(n)(i) with general distribution depending on a parameter n. We find sufficient conditions for the uniform version of the integro-local Stone theorem to hold for the asymptotics of the probability P(S(n) ∈ Δ[x), where Δ[x) is a cube with edge Δ and vertex at a point x.