On the Lp-Theory of C0-Semigroups Associated with Second-Order Elliptic Operators, II

We study positive C₀-semigroups on L_p associated with second-order uniformly elliptic divergence-type operators with singular lower-order terms, subject to a wide class of boundary conditions. We obtain an interval (p_min, p_max) in the L_p-scale where these semigroups can be defined, including the case 2∉(p_min, p_max). We present an example showing that the result is optimal. We also show that the semigroups are analytic with angles of analyticity and spectra of the generators independent of p, for the whole range of p where the semigroups are defined.     

Functional calculi of second-order elliptic partial differential operators with bounded measurable coefficients

Consider a second-order elliptic partial differential operatorL in divergence form with real, symmetric, bounded measurable coefficients, under Dirichlet or Neumann conditions on the boundary of a strongly Lipschitz domain Ω. Suppose that 1 <p < ∞ and μ > 0. ThenL has a bounded H_∞ functional calculus in L^p(Ω), in the sense that ¦¦f (L +cI)u¦¦_p ≤C sup_¦arλ¦<μ ^¦f¦ ¦‖u¦‖_p for some constantsc andC, and all bounded holomorphic functionsf on the sector ¦ argλ¦ < μ that contains the spectrum ofL +cI. We prove this by showing that the operatorsf(L + cI) are Calderón-Zygmund singular integral operators.     

Stark-Heegner points on modular Jacobians

We present a construction which lifts Darmon's Stark-Heegner points from elliptic curves to certain modular Jacobians. Let N be a positive integer and let p be a prime not dividing N. Our essential idea is to replace the modular symbol attached to an elliptic curve E of conductor Np with the universal modular symbol for Γ₀(Np). We then construct a certain torus T over Qp and lattice L⊂T, and prove that the quotient T/L is isogenous to the maximal toric quotient J₀(Np)^p-new of the Jacobian of X₀(Np). This theorem generalizes a conjecture of Mazur, Tate, and Teitelbaum on the p-adic periods of elliptic curves, which was proven by Greenberg and Stevens. As a by-product of our theorem, we obtain an efficient method of calculating the p-adic periods of J₀(Np)^p-new.     

Generation of the integrated semigroups by superelliptic differential operators

Let A be a superelliptic differential operator of order 2m introduced by E.B. Davies [E.B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995) 141-169]. In the case of 2m>N, he obtained the upper Gaussian bound of the integral kernel representing (e−zA)_z∈C+ and the estimates of the Lp-operator norm of the semigroup for all p∈[1,∞). The purpose of the present paper is to show that −i(A+k) (for some constant k>0) generates an integrated semigroup on Lα,p (weighted Lp space) and lp(Lα,q). To prove this we need norm estimates of (e−zA)_z∈C+ on each of these spaces. Also we get another norm estimate of (e−zA)_z∈C+ on Lp when 2m>N without using the integral kernel. This norm estimate is better than that in [E.B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995) 141-169] and gives a better “times of the integration” of the integrated semigroup.     

Uniqueness of norm on L(G) and C(G) when G is a compact group

Let G be a compact group. If the trivial representation of G is not weakly contained in the left regular representation of G on L₀²(G) and X is either L^p(G) for 1<p?∞ or C(G), then we show that every complete norm |·| on X that makes translations from (X,|·|) into itself continuous is equivalent to ||·||_p or ||·||_∞ respectively. If 1<p?∞ and every left invariant linear functional on L^p(G) is a constant multiple of the Haar integral, then we show that every complete norm |·| on L^p(G) that makes translations from (L^p(G),|·|) into itself continuous and that makes the map t?L_t from G into bounded is equivalent to ||·||_p.     

Isomorphisms between Lp-function spaces when p < 1

We prove a number of results concerning isomorphisms between spaces of the type L_p(X), where X is a separable p-Banach space and 0 < p < 1. Our results imply that the quotient of L_p([0, 1] × [0, 1]) by the subspace of functions depending only on the first variable is not isomorphic to L_p, answering a question of N. T. Peck. More generally if $\text{B}$₀ is a sub-σ-algebra of the Borel sets of [0, 1], then $\text{L}{}_{\mathrm{p}}\text{([0, 1])}\text{L}{}_{\mathrm{p}}\text{([0, 1]}$, $\text{B}$₀) is isomorphic to L_p if and only if L_p([0, 1], B₀) is complemented. We also show that L_p has, up to isomorphism, at most one complemented subspace non-isomorphic to L_p and classify completely those spaces X for which $\text{L}{}_{\mathrm{p}}\text{(X) ? L}{}_{\mathrm{p}}$. In particular if $\text{L(L}{}_{\mathrm{p}}\text{, X) = {0}}\text{and}\text{L}{}_{\mathrm{p}}\text{(X) ? L}{}_{\mathrm{p}}$ then $\text{X ? l}{}_{\mathrm{p}}$ or is finite-dimensional. If X has trivial dual and $\text{L}{}_{\mathrm{p}}\text{(X) ? L}{}_{\mathrm{p}}\text{then}\text{X ? L}{}_{\mathrm{p}}$.     

Interpolation orbits in couples of Lp spacesOrbites d'interpolation pour les couples d'espaces Lp

We consider linear operators T mapping a couple of weighted L_p spaces {L_p0(U₀), L_p1(U₁)} into {L_q0(V₀),L_q1(V₁)} for any 1?p₀, p₁, q₀, q₁?∞, and describe the interpolation orbit of any a∈L_p0(U₀)+L_p1(U₁) that is we describe a space of all {Ta}, where T runs over all linear bounded mappings from {L_p0(U₀),L_p1(U₁)} into {L_q0(V₀),L_q1(V₁)}. We show that interpolation orbit is obtained by the Lions–Peetre method of means with functional parameter as well as by the K-method with a weighted Orlicz space as a parameter. To cite this article: V.I. Ovchinnikov, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 881–884.     

Determining L-functions by twists

Suppose that L₁(s) and L₂(s) are two L-functions whose twists by a set of Dirichlet characters simultaneously vanish (vanish mod p) at a critical point. We examine the extent to which this property determines the L-functions in the cases of L-functions of elliptic curves, of number fields, and of curves over finite fields.     

Translation-invariant linear forms on C0(G), C(G), Lp(G) for noncompact groups

Suppose G is a locally compact noncompact group. For abelian such G's, it is shown in this paper that L¹(G), C(G), and L^∞(G) always have discontinuous translation-invariant linear forms(TILF's) while C₀(G) and L^p(G) for 1 < p < ∞ have such forms if and only if $\text{G}\text{H}$ is a torsion group for some open σ-compact subgroup H of $\text{G}$. For σ-compact amenable G's, all the above spaces have discontinuous left TILF's.     

Convergence in L space for the homogenization problems of elliptic and parabolic equations in the plane

We study the convergence rate of an asymptotic expansion for the elliptic and parabolic operators with rapidly oscillating coefficients. First we propose homogenized expansions which are convolution forms of Green function and given force term of elliptic equation. Then, using local L^p-theory, the growth rate of the perturbation of Green function is found. From the representation of elliptic solution by Green function, we estimate the convergence rate in L^p space of the homogenized expansions to the exact solution. Finally, we consider L2(0,T:H1(Ω)) or L∞(Ω×(0,T)) convergence rate of the first order approximation for parabolic homogenization problems.     

Kato's square root problem in Banach spaces

Let L be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces Lp(Rn;X) of X -valued functions on Rn. We characterize Kato's square root estimates and the H^∞-functional calculus of L in terms of R-boundedness properties of the resolvent of L, when X is a Banach function lattice with the UMD property, or a noncommutative Lp space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. In the special case X=C, we get a new approach to the Lp theory of square roots of elliptic operators, as well as an Lp version of Carleson's inequality.     

A new proof of local C1,α regularity for solutions of certain degenerate elliptic p.d.e.

We prove C_loc^1,α estimates for solutions u?W^{1,p + 2} of the degenerate elliptic p.d.e. $\text{div}\text{(¦Du¦}{}^{\mathrm{p}}\text{Du) = 0}$ (p > 0).     

Dual semigroups and second order linear elliptic boundary value problems

It is shown that general second order elliptic boundary value problems on bounded domains generate analytic semigroups onL ₁. The proof is based on Phillips’ theory of dual semigroups. Several sharp estimates for the corresponding semigroups inL _p, 1≦p<∞, are given.     

Linear-topological classification of separableL p-spaces associated with von Neumann algebras of type I

We classify, up to a linear-topological isomorphism, all separableL _p-spaces, 1≤p<∞, associated with von Neumann algebras of type I. In particular, anyL _p-space associated with an infinite-dimensional atomic von Neumann algebra is isomorphic tol _p, or toC _p, or to . Further, anyL _p-space,p∈[1,∞),p∈2 associated with an infinite-dimensional von Neumann algebraM of type I is isomorphic to one of the following nine Banach spaces: l_p, L_p, S_P, C_p, S_p ⊕ L_p, L_p(S_p), C_p ⊕ L_p, L_p(C_p), C_p ⊕ L_p(S_p). In the casep=1 all the spaces in this list are pairwise non-isomorphic. Research supported by the Australian Research Council.     

A sharp nonasymptotic bound and phase diagram of L 1/2 regularization

We derive a sharp nonasymptotic bound of parameter estimation of the L1/2 regularization.The bound shows that the solutions of the L1/2 regularization can achieve a loss within logarithmic factor of an ideal mean squared error and therefore underlies the feasibility and effectiveness of the L1/2regularization.Interestingly,when applied to compressive sensing,the L1/2 regularization scheme has exhibited a very promising capability of completed recovery from a much less sampling information.As compared with the Lp(0 p 1) penalty,it is appeared that the L1/2 penalty can always yield the most sparse solution among all the Lp penalty when 1/2 ≤ p 1,and when 0 p 1/2,the Lp penalty exhibits the similar properties as the L1/2 penalty.This suggests that the L1/2 regularization scheme can be accepted as the best and therefore the representative of all the Lp(0 p 1) regularization schemes.     

Subordination in the sense of Bochner of L p -semigroups and associated Markov processes

We show that the subordination induced by a convolution semigroup（subordination in the sense of Bochner）of a C0-semigroup of sub-Markovian operators on an Lpspace is actually associated to the subordination of a right（Markov）process.As a consequence,we solve the martingale problem associate with the Lp-infinitesimal generator of the subordinate semigroup.We also prove quasi continuity properties for the elements of the domain of the Lp-generator of the subordinate semigroup.It turns out that an enlargement of the base space is necessary.A main step in the proof is the preservation under such a subordination of the property of a Markov process to be a Borel right process.We use several analytic and probabilistic potential theoretical tools.     

An integral transform in L2-cohomology for the ladder representations of U(p, q)

Starting from the realization of the Fock space as L²-cohomology of $\text{C}$^{p + q}, $\text{H}$^0,p($\text{C}$^{p + q}) = ⊕_m?$\text{Z}$$\text{H}$_m^0,p($\text{C}$^{p + q}), an integral transform is constructed which is a direct-image mapping from $\text{H}$_m^{0,p($\text{C}$p + q) into the space of holomorphic sections of some vector bundle E_m over M ≈ U(p, q)/(U(q) × U(p)), m ? 0}. The transform intertwines the natural actions of U(p, q) and is injective if m ? 0, so it provides a geometric realization of the ladder representations of U(p, q). The sections in the image of the transform satisfy certain linear differential equations, which are explicitly described. For example, Maxwell's equations are of this form if p = q = 2 and m = 2. Thus, this transform is analogous to the Penrose correspondence.     

An analogue of Hartshorne and Serre problems for 1-convex surfaces

Let C be an elliptic curve and let L∈Pic(C). If c₁(L)<0, a well known result of Grauert tells us that L is rigid. On the other hand, Arnold provided a criterion for the rigidity of L when c₁(L)=0. However, a concrete example of such a bundle is hard to come by. In this paper, we construct explicitly such an L which turns out to be the line bundle associated to some toroidal group , viewed as topologically trivial -bundle over C. This example turns out to be the counterexample to the following analogue of a problem of Serre for 1-convex surfaces:Let X be a compactifiable surface such that - for all p?0 and q?1. Is X always 1-convex?Also a cohomological characterization of toroidal groups of finite type is established, as well as an analogue of a problem of Hartshorne for 1-convex surfaces will be discussed.     

On a Lp-estimate for the linearized compressible Navier-Stokes equations with the Dirichlet boundary conditions

In this paper a special L_p-estimate for the linearized compressible Navier-Stokes in the Lagrangian coordinates for the Dirichlet boundary conditions is obtained. The constant in the estimate does not depend on the length of time interval [0,T]. The result is essential to obtain an existence for regular solutions for the nonlinear problem with the lowest class of regularity in L_p-spaces.     

Tranversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the restricted three-body problem

The restricted three-body problem is considered for values of the Jacobi constant C near the value C₂ associated to the Euler critical point L₂. A Lyapunov family of periodic orbits near L₂, the so-called family (c), is born for C = C₂ and exists for values of C less than C₂. These periodic orbits are hyperbolic. The corresponding invariant manifolds meet transversally along homoclinic orbits. In this paper the variation of the transversality is analyzed as a function of the Jacobi constant C and of the mass parameter μ. Asymptotical expressions of the invariant manifolds for C ? C₂ and μ ? 0 are found. Several numerical experiments provide accurate information for the manifolds and a good agreement is found with the asymptotical expressions. Symbolic dynamic techniques are used to show the existence of a large class of motions. In particular the existence of orbits passing in a random way (in a given sense) from the region near one primary to the region near the other is proved.