On the Structure of L 1 of a Vector Measure via its Integration Operator

Geometric and summability properties of the integration operator associated to a vector measure m can be translated in terms of structure properties of the space L¹(m). In this paper we study the cases of the integration operator being: (i) p-concave on L^p(m), or (ii) positive p-summing on L¹(m) (where ). We prove that (i) is equivalent to saying that L¹(m) contains continuously the L^p space of a (non-negative scalar) control measure for m. On the other hand, we show that (ii) holds if and only if L¹(m) is order isomorphic to the L¹ space of a non-negative scalar measure. J.M. Calabuig was supported by MEC and FEDER (MTM2005-08350-C03-03) and Generalitat Valenciana (GV/2007/191). J. Rodríguez was supported by MEC and FEDER (MTM2005-08379) and Generalitat Valenciana (GVPRE/2008/312). E.A. Sánchez-Pérez was supported by MEC and FEDER (MTM2006-11690-C02-01).     

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Let (E,H,μ) be an abstract Wiener space and let D_V:=VD, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space . Given a bounded operator B on , coercive on the range , we consider the operators A:=V^*BV in H and in , as well as the realisations of the operators and in L^p(E,μ) and respectively, where 1<p<∞. Our main result asserts that the following four assertions are equivalent:

(1) with for ;

(2) admits a bounded H^∞-functional calculus on ;

(3) with for ;

(4) admits a bounded H^∞-functional calculus on .

Moreover, if these conditions are satisfied, then . The equivalence (1)–(4) is a non-symmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where , V=I, ). A one-sided version of (1)–(4), giving L^p-boundedness of the Riesz transform in terms of a square function estimate, is also obtained. As an application let −A generate an analytic C₀-contraction semigroup on a Hilbert space H and let −L be the L^p-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A. The boundedness of the Riesz transform is used to obtain an L^p-domain characterisation for the operator L.

Multicyclicity of unbounded normal operators and polynomial approximation in C

A remarkable and much cited result of Bram [J. Bram, Subnormal operators, Duke Math. J. 22 (1955) 75-94] shows that a star-cyclic bounded normal operator in a separable Hilbert space has a cyclic vector. If, in addition, the operator is multiplication by the variable in a space L²(m) (not only unitarily equivalent to it), then it has a cyclic vector in L^∞(m). We extend Bram's result to the case of a general unbounded normal operator, implying by this that the (classical) multiplicity and the multicyclicity of the operator (cf. [N.K. Nikolski, Operators, Functions and Systems: An Easy Reading, vol. 2, Math. Surveys Monogr., vol. 93, Amer. Math. Soc., Providence, 2002]) coincide. It follows that if m is a sigma-finite Borel measure on C (possibly with noncompact support), then there is a nonnegative finite Borel measure τ equivalent to m and such that L²(C,τ) is the norm-closure of the polynomials in z.     

The Estimates for Sharp Maximal Functions of Multilinear Strongly Singular Integral Operators

In this paper, the author obtains that the multilinear operators of strongly singular integral operators and their dual operators are bounded from some L^p（R^n） to L^p（R^n） when the m-th order derivatives of A belong to L^p（R^n） for r large enough. By this result, the author gets the estimates for the Sharp maximal functions of the multilinear operators with the m-th order derivatives of A being Lipschitz functions. It follows that the multilinear operators are （L^p, L^p）-type operators for 1 〈 p 〈 ∞.     

Large time behavior for nonlinear higher order convection–diffusion equations

We study the large time asymptotic behavior, in L^p (1p∞), of higher derivatives D^γu(t) of solutions of the nonlinear equation

(1)

where the integers n and θ are bigger than or equal to 1, a is a constant vector in with . The function ψ is a nonlinearity such that and ψ(0)=0, and is a higher order elliptic operator with nonsmooth bounded measurable coefficients on . We also establish faster decay when .     

On Representations Associated with Completely n-Positive Linear Maps on Pro-C^＊-Algebras

It is shown that an n × n matrix of continuous linear maps from a pro-C^＊-algebra A to L（H）, which verifies the condition of complete positivity, is of the form [V^＊TijФ（·）V]^n i,where Ф is a representation of A on a Hilbert space K, V is a bounded linear operator from H to K, and j=1,[Tij]^n i,j=1 is a positive element in the C^＊-algebra of all n×n matrices over the commutant of Ф（A） in L（K）. This generalizes a result of C. Y.Suen in Proc. Amer. Math. Soc., 112（3）, 1991, 709-712. Also, a covariant version of this construction is given.     

Some Results about Dissipativity of Kolmogorov Operators

Given a Hilbert space H with a Borel probability measure , we prove the m-dissipativity in L ¹(H, ) of a Kolmogorov operator K that is a perturbation, not necessarily of gradient type, of an Ornstein-Uhlenbeck operator.     

Stable extendibility of vector bundles over lens spaces mod 3 and the stable splitting problem

Let Lⁿ(3) denote the (2n+1)-dimensional standard lens space mod 3. In this paper, we study the conditions for a given real vector bundle over Lⁿ(3) to be stably extendible to L^m(3) for every mn, and establish the formula on the power ζ^k=ζζ (k-fold) of a real vector bundle ζ over Lⁿ(3). Moreover, we answer the stable splitting problem for real vector bundles over Lⁿ(3) by means of arithmetic conditions.     

<Emphasis Type="Italic">m</Emphasis>-Dissipativity of Some Gradient Systems with Measurable Potential

We consider the operator in L ²(B, ν) and in L ¹(B, ν) with Neumann boundary condition, where U is an unbounded function belonging to for some q ∈(1, ∞), B is the possibly unbounded convex open set in where U is finite and ν(dx) = C exp (−2U (x))dx is a probability measure, infinitesimally invariant for N ₀. We prove that the closure of N ₀ is a m-dissipative operator both in L ²(B, ν) and in L ¹(B, ν). Moreover we study the properties of ergodicity and strong mixing of the measure ν in the L ² case.      

Difference estimates for continuous and discrete operator semigroups

For suitable bounded operator semigroups (e ^tA ) _t≥0 in a Banach space, we characterize the estimate ‖Ae ^tA ‖≤c/F(t) for large t, where F is a function satisfying a sublinear growth condition. The characterizations are by holomorphy estimates on the semigroup, and by estimates on powers of the resolvent. We give similar characterizations of the difference estimate ‖T ⁿ −T ⁿ⁺¹‖≤c/F(n) for a power-bounded linear operator T, when F(n) grows faster than n ^1/2 for large n.     

Extended Best Polynomial Approximation Operator in Orlicz Spaces

In this article we consider the best polynomial approximation operator, defined in an Orlicz space L ^Φ(B), and its extension to L ^?(B) where ? is the derivative function of Φ. A characterization of these operators and several properties are obtained.     

Banach function subspaces of L of a vector measure and related Orlicz spaces

Given a vector measure ν with values in a Banach space X, we consider the space L¹(ν) of real functions which are integrable with respect to ν. We prove that every order continuous Banach function space Y continuously contained in L¹(ν) is generated via a certain positive map related to ν and defined on X* x M, where X* is the dual space of X and M the space of measurable functions. This procedure provides a way of defining Orlicz spaces with respect to the vector measure ν.     

Dense Subsets of -Solutions to Linear Elliptic Partial Differential Equations

Let Ω ^N (N2) be an unbounded domain, and L_m be a homogeneous linear elliptic partial differential operator with constant coefficients. In this paper we show, among other things, that rapidly decreasing ¹-solutions to L_m (in Ω) approximate all ¹-solutions to L_m (in Ω), provided there exist real numbers R_j→∞, 0, and a sequence {y^j} such that B(y^j, )∩Ω= and where |·| means the volume and

Full-size image

for z ^N, R>0 and D ^N. For m=2, we can replace the volume density by the capacity-density. It appears that the problem is related to the characterization of largest sets on which a nonzero polynomial solution to L_m may vanish, along with its (m−1)-derivatives. We also study a similar approximation problem for polyanalytic functions in .     

On -topological vector spaces generated by a co-tower of L-topological vector spaces

In this paper, a characterization for an I(L)-topological space to be generated by a given co-tower of L-topological spaces is obtained. Moreover, the relationship between some properties of an I(L)-topological vector space generated by a co-tower of L-topological vector spaces and the corresponding properties of the given co-tower of L-topological vector spaces is investigated. Our results show that if an I(L)-topological vector space generated by a co-tower of L-topological vector spaces has some properties, such as local convexity and local boundedness, then all L-topological vector spaces in the co-tower also have the same properties. But the converse is incorrect even in the case of I-topological vector space generated by a co-tower of classical topological vector spaces. Finally, we supply a necessary and sufficient condition for an I(L)-topological vector space generated by a co-tower of L-topological vector spaces with some properties, such as local convexity and local boundedness, to have such properties too.     

Sub-Laplacians of Holomorphic L-type on Rank One AN-Groups and Related Solvable Groups

Consider a right-invariant sub-Laplacian L on an exponential solvable Lie group G, endowed with a left-invariant Haar measure. Depending on the structure of G and possibly also that of L, L may admit differentiable L^p-functional calculi, or may be of holomorphic L^p-type for a given p≠2, as recent studies of specific classes of groups G and sub-Laplacians L have revealed. By “holomorphic L^p-type” we mean that every L^p-spectral multiplier for L is necessarily holomorphic in a complex neighborhood of some point in the L²-spectrum of L. This can only arise if the group algebra L¹(G) is non-symmetric. In this article we prove that, for large classes of exponential groups, including all rank one AN-groups, a certain Lie algebraic condition, which characterizes the non-symmetry of L¹(G) [37], also suffices for L to be of holomorphic L¹-type. Moreover, if this condition, which was first introduced by J. Boidol [6] in a different context, holds for generic points in the dual * of the Lie algebra of G, then L is of holomorphic L^p-type for every p≠2. Besides the non-symmetry of L¹(G), also the closedness of coadjoint orbits plays a crucial role. We also discuss an example of a higher rank AN-group. This example and our results in the rank one case suggest that sub-Laplacians on exponential Lie groups may be of holomorphic L¹-type if and only if there exists a closed coadjoint orbit Ω * such that the points of Ω satisfy Boidol's condition. In the course of the proof of our main results, whose principal strategy is similar as in [8], we develop various tools which may be of independent interest and largely apply to more general Lie groups. Some of them are certainly known as “folklore” results. For instance, we study subelliptic estimates on representation spaces, the relation between spectral multipliers and unitary representations, and develop some “holomorphic” and “continuous” perturbation theory for images of sub-Laplacians under “smoothly varying” families of irreducible unitary representations.     

β级星像函数类和凸像函数类的扩展问题

本文利用文[2,3]的引理和算子L(a,c)f(z)的一些性质.结合Hadamard乘积,研究了算子L(a,c)f(z),获得了L(a,c)f(z)∈S*(β)和L(a,c)f(z)∈K(β)的充分条件,推广了文[2,3]的相关结论.     

Ordered representations of spaces of integrable functions

Let X be a Banach space, (Ω,Σ) a measurable space and let m : Σ → X be a (countably additive) vector measure. Consider the corresponding space of integrable functions L¹(m). In this paper we analyze the set of (countably additive) vector measures n satisfying that L¹(n) = L¹(m). In order to do this we define a (quasi) order relation on this set to obtain under adequate requirements the simplest representation of the space L¹(m) associated to downward directed subsets of the set of all the representations. This research has been partially supported by La Junta de Andalucía. The support of D.G.I. under project MTM2006–11690–C02 (M.E.C. Spain) and FEDER is gratefully acknowledged.     

-Operator frames for a Banach space

In this paper, (p,Y)-Bessel operator sequences, operator frames and (p,Y)-Riesz bases for a Banach space X are introduced and discussed as generalizations of the usual concepts for a Hilbert space and of the g-frames. It is proved that the set of all (p,Y)-Bessel operator sequences for a Banach space X is a Banach space and isometrically isomorphic to the operator space B(X,ℓ^p(Y)). Some necessary and sufficient conditions for a sequence of operators to be a (p,Y)-Bessel operator sequence are given. Also, a characterization of an independent (p,Y)-operator frame for X is obtained. Lastly, it is shown that an independent (p,Y)-operator frame for X is just a (p,Y)-Riesz basis for X and has a unique dual (q,Y^*)-operator frame for X^*.     

Construction of markov process with markov resolvent onL P (E, m)

LetE be a Hausdorff topological space and let β(E) be its Borel σ-field. Letm be a σ-finite measure on (E, β(E)). A necessary and sufficient condition for a Markov resolvent onL ^P (E,m) to be associated with anm-tightm-special standard process (with state spaceE) is given. Furthermore some new examples which do not belong to the framework of Dirichlet space are also given. Project supported by the Youth Foundation of Chinese Academy of Sciences.     

The isometric extension of the into mapping from the unit sphere S 1( E ) to S 1( l ∞(Γ))

This paper considers the isometric extension problem concerning the mapping from the unit sphere S ₁(E) of the normed space E into the unit sphere S ₁(l ^∞(Γ)). We find a condition under which an isometry from S ₁(E) into S ₁(l ^∞(Γ)) can be linearly and isometrically extended to the whole space. Since l ^∞(Γ) is universal with respect to isometry for normed spaces, isometric extension problems on a class of normed spaces are solved. More precisely, if E and F are two normed spaces, and if V ₀: S ₁(E) → S ₁(F) is a surjective isometry, where c ₀₀(Γ) ⊆ F ⊆ l ^∞(Γ), then V ₀ can be extended to be an isometric operator defined on the whole space. This work is supported by Natural Science Foundation of Guangdong Province, China (Grant No. 7300614)