On the duals of Lp spaces with 0<p<1

Given a measure space < Ω,m,μ >, a locally bounded, Hausdorff topological linear space < X, τ > and a real number 0<p<1, one can define the space L_p(Ω,m,μ,X), which is, under certain assumptions, a Fréchet space if endowed with a suitable topology. M.M. Day [1] has given a necessary and sufficient condition, in terms of the properties of the measure space < Ω,m,μ >, for the dual of L_p(Ω,m,μ,C) to be trivial. In this paper a different proof along with a slight generalization is given for this result, using standard and elementary measure theoretic arguments. This revised version was published online in June 2006 with corrections to the Cover Date.     

Iwasawa invariants of imaginary quadratic fields

LetK be an imaginary quadratic field andp an odd prime which splits inK. We study the Iwasawa invariants for ℤ _p -extensions ofK. This is motivated in part by a recent result of Sands. The main result is the following. Assumep does not divide the class number ofK. LetK _∞ be a ℤ _p -extension ofK. SupposeK _∞ is not totally ramified at the primes abovep. Then the μ-invariant forK _∞/K vanishes. We also show that if μ=0 for all ℤ _p -extensions ofK, then the λ-invariant is bounded asK _∞ runs through all such extensions.     

Vague convergence of sums of independent random variables

A sequence (μ _n) of probability measures on the real line is said to converge vaguely to a measureμ if∫ fdμ _n →∫ fdμ for every continuous functionf withcompact support. In this paper one investigates problems analogous to the classical central limit problem under vague convergence. Let ‖μ‖ denote the total mass ofμ andδ ₀ denote the probability measure concentrated in the origin. For the theory of infinitesimal triangular arrays it is true in the present context, as it is in the classical one, that all obtainable limit laws are limits of sequences of infinitely divisible probability laws. However, unlike the classical situation, the class of infinitely divisible laws is not closed under vague convergence. It is shown that for every probability measureμ there is a closed interval [0,λ], [0,e ⁻¹] ⊂ [0,λ] ⊂ [0, 1], such thatβμ is attainable as a limit of infinitely divisible probability laws iffβ ε [0,λ]. In the independent identically distributed case, it is shown that if (x ₁ + ... +x _n)/a _n, a_n → ∞, converges vaguely toμ with 0<‖μ‖<1, thenμ=‖μ‖δ ₀. If furthermore the ratiosa _n+1/a _n are bounded above and below by positive numbers, thenL(x)=P[|X ₁|>x] is a slowly varying function ofx. Conversely, ifL(x) is slowly varying, then for everyβ ε (0, 1) one can choosea _n → ∞ so that the limit measure=βδ ₀. To the memory of Shlomo Horowitz This research was partially supported by the National Science Foundation.     

On matrix transformations and sequence spaces

In this paper are given results on the spacesw _τ (μ) andc _τ (μ, μ′) the second one generalizing the well-known spacec _∞ (μ) of sequences that are strongly bounded. Then we deal with matrix transformations into these spaces. These results generalize those given in [7].     

A dimension gap for continued fractions with independent digits

Kinney and Pitcher (1966) determined the dimension of measures on [0, 1] which make the digits in the continued fraction expansion i.i.d. variables. From their formula it is not clear that these dimensions are less than 1, but this follows from the thermodynamic formalism for the Gauss map developed by Walters (1978). We prove that, in fact, these dimensions are bounded by 1−10⁻⁷. More generally, we considerf-expansions with a corresponding absolutely continuous measureμ under which the digits form a stationary process. Denote byE _δ the set of reals where the asymptotic frequency of some digit in thef-expansion differs by at leastδ from the frequency prescribed byμ. ThenE _δ has Hausdorff dimension less than 1 for anyδ>0.     

A generalized moment problem

Let {λ _n} (n≧0) satisfy (1.1) we are considering the following problems: What are the necessary and sufficient conditions on a sequence {μ_n} (n≧0) in order that it should possess the representation (1.2) wherea(t) is of bounded variation or the representation (1.3) wheref(t) ∈L _M[0, 1] orf(t) is essentially bounded.     

A Maximal Function Approach to Christoffel Functions and Nevai��s Operators

Let μ be a compactly supported positive measure on the real line, with associated Christoffel functions λ _n (d μ,⋅). Let g be a measurable function that is bounded above and below on supp[μ] by positive constants. We show that λ _n (g  d μ,⋅)/λ _n (d μ,⋅)→g in measure in {x:μ′(x)>0} and consequently in all L _p norms, p<∞. The novelty is that there are no local or global restrictions on μ. The main idea is a new maximal function estimate for the “tail” in Nevai’s operators.     

Finite quasihypermetric spaces

Let (X, d) be a compact metric space and let (X) denote the space of all finite signed Borel measures on X. Define I: (X) → ℝ by I(μ) = ∫ _X ∫ _X d(x, y)dμ(x)dμ(y), and set M(X) = sup I(μ), where μ ranges over the collection of measures in (X) of total mass 1. The space (X, d) is quasihypermetric if I(μ) ≦ 0 for all measures μ in (X) of total mass 0 and is strictly quasihypermetric if in addition the equality I(μ) = 0 holds amongst measures μ of mass 0 only for the zero measure. This paper explores the constant M(X) and other geometric aspects of X in the case when the space X is finite, focusing first on the significance of the maximal strictly quasihypermetric subspaces of a given finite quasihypermetric space and second on the class of finite metric spaces which are L ¹-embeddable. While most of the results are for finite spaces, several apply also in the general compact case. The analysis builds upon earlier more general work of the authors [11] [13].      

Convex Solid Subsets of <Emphasis Type="Italic">L</Emphasis><Subscript>0</Subscript>(<Emphasis Type="Italic">X</Emphasis>, μ)

If A⊂ L₀(X, μ) is a convex solid subset of L₀(X, μ), then there exist disjoint X₀ and X₁ with X = X₀∪ X₁ such that A_{| X_0} is dense in L₀(X₀, μ) and A_{|X_1} is bounded in measure in L₀(X₁, μ).     

SOME PROPERTIES OF Λω^p（X,μ）

In this paper, we shall give a necessary and sufficient condition for which the dual of Λ _ω ^p (X, M, μ) (0<p<∞) is zero, and a necessary and sufficient condition for which Λ _ω ^p (X, M, μ), (0<p<1) is normable. Supported by 973 project (G1999075105), RFDP (20030335019) and ZJNSF(RC97017).     

Ruzsa’s problem on sets of recurrence

The main purpose of this paper is to prove the existence of Poincaré sequences of integers which are not van der Corput sets. This problem was considered in I. Ruzsa’s expository article [R₁] (1982–83) on correlative and intersective sets. Thus the existence is shown of a positive non-continuous measureμ on the circle which Fourier transform vanishes on a set of recurrence, i.e.S={n ∈Z; (n)=0} is a set of recurrence but not a van der Corput set. The method is constructive and involves some combinatorial considerations. In fact, we prove that the generic density condition for both properties are the same.     

T 1 theorem for Besov spaces on nonhomogeneous spaces

Suppose μ is a Radon measure on ℝ ^d , which may be non doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀>0 such that for all x∈supp(μ) and r>0, μ(B(x, r))⪯C₀rⁿ, where 0<n⪯d. We prove T1 theorem for non doubling measures with weak kernel conditions. Our approach yields new results for kernels satisfying weakened regularity conditions, while recovering previously known Tolsa’s results. We also prove T1 theorem for Besov spaces on nonhomogeneous spaces with weak kernel conditions given in [7].     

On the Iwasawa invariants of totally real number fields

We shall show two sufficient conditions under which the Iwasawa invariants λ _k and μ _k of a totally real fieldk vanish for an odd primel, based on the results obtained in [1], [3] and [4]. LetK _n be the composite ofk and thel ⁿ-th cyclotomic extension of the fieldQ of rational numbers. LetC _n be the factor group of thel-class group ofK _n by a subgroup generated by ideals whose prime factors divide the principal ideal (l). Let ϕ₁ be an idempotent of the group ringZ _l[Gal(K ₁/k)] defined in the below. We shall prove λ _k = μ _k =0 if there is a natural numbern such that ε₁ C _n vanishes, under additional conditions concerning ramifications inK _n/k.     

<Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>(<Emphasis Type="Italic">μ</Emphasis>) for vector-valued measure <Emphasis Type="Italic">μ</Emphasis>

Let μ be a measure from a σ-algebra of subsets of a set T into a sequentially complete Hausdorff topological vector space X. Assume that μ is convexly bounded, i.e., the convex hull of its range is bounded in X, and denote by L ¹(μ) the space of scalar valued functions on T which are integrable with respect to the vector measure μ. We study the inheritance of some properties from X to L ¹(μ). We show that the bounded multiplier property passes from X to L ¹(μ). Answering a 1972 problem of Erik Thomas, we show that for a rather large class of F-spaces X the non-containment of c ₀ passes from X to L ¹(μ).     

Toeplitz Operators on Analytic Besov Spaces

We characterize complex measures μ on the unit disk for which the Toeplitz operator T _μ is bounded or compact on the analytic Besov spaces B _p with 1 ≤ p < ∞. Research supported in part by NSF grant, DMS 0200587 (first author); and by a NSERC grant (third author).     

Generalized Bounded Variation and Inserting Point Masses

Let d μ be a probability measure on the unit circle and d ν be the measure formed by adding a pure point to d μ. We give a formula for the Verblunsky coefficients of d ν following the method of Simon. Then we consider d μ ₀, a probability measure on the unit circle with ℓ ² Verblunsky coefficients (α _n (d μ ₀)) _n=0^∞ of bounded variation. We insert m pure points z _j into d μ ₀, rescale, and form the probability measure d μ _m . We use the formula above to prove that the Verblunsky coefficients of d μ _m are in the form , where the c _j ’s are constants of norm 1 independent of the weights of the pure points and independent of n; the error term E _n is in the order of o(1/n). Furthermore, we prove that d μ _m is of (m+1)-generalized bounded variation—a notion that we shall introduce in the paper. Then we use this fact to prove that lim  _n→∞ φ _n ^*(z,d μ _m ) is continuous and is equal to D(z,d μ _m )⁻¹ away from the pure points.      

A note on the quantization for probability measures with respect to the geometric mean error

We study the quantization with respect to the geometric mean error for probability measures μ on for which there exist some constants C, η > 0 such that for all ε > 0 and all . For such measures μ, we prove that the upper quantization dimension of μ is bounded from above by its upper packing dimension and the lower one is bounded from below by its lower Hausdorff dimension. This enables us to calculate the quantization dimension for a large class of probability measures which have nice local behavior, including the self-affine measures on general Sierpiński carpets and self-conformal measures. Moreover, based on our previous work, we prove that the upper and lower quantization coefficient for a self-conformal measure are both positive and finite.     

<Emphasis Type="Italic">BMO</Emphasis>-boundedness of the maximal operator for arbitrary measures

We show that in the one-dimensional case the weighted Hardy-Littlewood maximal operator M _μ is bounded on BMO(μ) for arbitrary Radon measure μ, and that this is not the case in higher dimensions.     

MAXIMAL CALDERON-ZYGMUND SINGULAR INTEGRAL ON RBMO

Given a positive Radon measure μ on R^d satisfying the linear growth condition μ（B（x,r））≤C0r^n,x∈R^d,r〉0,（1） where n is a fixed number and O〈n≤d. When d-1〈n,it is proved that if Tt,N1=0,then the corresponding maximal Calderon-Zygmund singular integral is bounded from RBMO to itself only except that it is infinite μ-a. e. on R^d.