Traces of weighted function spaces: Dyadic norms and Whitney extensions

The trace spaces of Sobolev spaces and related fractional smoothness spaces have been an active area of research since the work of Nikolskii, Aronszajn, Slobodetskii, Babich and Gagliardo among others in the 1950’s. In this paper, we review the literature concerning such results for a variety of weighted smoothness spaces. For this purpose, we present a characterization of the trace spaces (of fractional order of smoothness), based on integral averages on dyadic cubes, which is well-adapted to extending functions using the Whitney extension operator.     

Prehomogeneous vector spaces and field extensions

Summary Letk be an infinite field of characteristic not equal to 2, 3, 5. In this paper, we construct a natural map from the set of orbits of certain prehomogeneous vector spaces to the set of isomorphism classes of Galois extensions ofk which are splitting fields of equations of certain degrees, and prove that the inverse image of this map corresponds bijectively with conjugacy classes of Galois homomorphisms.Oblatum 24-I-1992 & 23-IV-1992Both authors are supported by NSF Grant DMS-8803085, DMS-9101091; The first author was partially supported by NSA grant MDA904-91-H-0041     

Shape preserving approximation in vector ordered spaces

The aim of this note is to extend some classical results on the shape preserving approximation of real functions (of real variables) to functions with values in ordered vector spaces.     

Disjointness preserving operators on normed pre-Riesz spaces: extensions and inverses

We explore inverses of disjointness preserving bijections in infinite dimensional normed pre-Riesz spaces by several methods. As in the case of Banach lattices, our aim is to show that such inverses are disjointness preserving. One method is extension of the operator to the Riesz completion, which works under suitable denseness and continuity conditions. Another method involves a condition on principle bands. Examples illustrate the differences to the Riesz space theory.

     

Interpolation of weighted Orlicz-valued function spaces

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This paper is to be part of the author's doctoral dissertation written at the University of Campinas under the supervision of Prof. D. L. Fernandez.     

A reverse weighted inequality for the Hardy-Littlewood maximal function in Orlicz spaces

Let Φ(t)= ∫_0^t a(s) ds and Ψ(t)= ∫_0^t b(s) ds, where a(s) is a positive continuous function such that ∫_0^1 \frac{a(s)}{s} ds < ∞and ∫_1^{\∞}\frac{a(s)}{s} ds= +\∞, and b(s) is an increasing function such that \lim_{s\to\∞}b(s)= +\∞. Letw be a weight function and suppose that w∈A₁\∩ A_∞'. Then the following statements for the Hardy-Littlewood maximal function Mf(x) are equivalent:(I) there exist positive constants C ₁ and C ₂ such that $$\int_0^s {\frac{{a\left( t \right)}}{t}dt \geqq } C_1 b\left( {C_2 s} \right)foralls > 0;$$ (II) there exist positive constants C ₃ and C ₄ such that $$\int {_{R^n } } \Psi \left( {C_3 \left| {f\left( x \right)} \right|} \right)w\left( x \right)dx \leqq C_4 \int {_{R^n } } \Phi \left( {Mf\left( x \right)} \right)w\left( x \right)dxforallf \in {\mathcal{R}}_0 \left( w \right)$$      

Radon-Nikodým derivatives for vector measures belonging to Köthe function spaces

Let m, n be a couple of vector measures with values on a Banach space. We develop a separation argument which provides a characterization of when the Radon-Nikodým derivative of n with respect to m—in the sense of the Bartle-Dunford-Schwartz integral—exists and belongs to a particular sublattice Z(μ) of the space of integrable functions L¹(m). We show that this theorem is in fact a particular feature of our separation argument, which can be applied to prove other results in both the vector measure and the function space settings.     

Correction to: Disjointness preserving operators on normed pre-Riesz spaces: extensions and inverses

Positivity - In the original publication, due to an error in Theorem 6.10 (page 21 of article).     

Entropy numbers in sequence spaces with an application to weighted function spaces

We determine the exact asymptotic behaviour of entropy numbers of diagonal operators from ℓ_p to ℓ_q, 0<q<p∞, under mild regularity conditions on the generating diagonal sequence. On one hand, this is a quantitative version of Pitt's theorem for diagonal operators, and on the other hand it is a limiting case of results by Carl. An application to embeddings of weighted Besov and Triebel–Lizorkin spaces is also given.     

A priori bounds in weighted spaces

In unbounded domains we state some a priori bounds for solutions of the Dirichlet problem for linear second order elliptic differential equations in nondivergence form with discontinuous coefficients in weighted spaces. The weight function is related to the distance function from a fixed subset S of ∂Ω.     

A convolution operator in weighted spaces

Convolution operators in R ^m with a homogeneous function of an arbitrary complex order are considered in spaces with weighted norms. The role of the weight is played by some power of the distance to a k-dimensional plane. The fundamental content of the paper consists in the investigation of the transversal operators (TO), which arise from convolutions as a result of a Fourier transform along the mentioned plane and a change of variables. Formulas for the composition of TO are derived and the (finite-dimensional) kernels and cokernels of elliptic TO are described. Problems for TO are indicated, containing additional boundary conditions or potentials. The properties of the convolution operators are obtained from the corresponding properties of the TO.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 208–237, 1990.     

On weak uniform normal structure in weighted Orlicz sequence spaces

It is proved that a weighted Orlicz sequence space ?M(w), equipped with Luxemburg or Amemiya norm has weak uniform normal structure iff ?M(w)≅hM(w) for wide class of weight sequences . An example is constructed, where M has not Δ₂-condition but by choosing a suitable weight sequence limn→∞wn=∞ we get that ?M(w) has weak uniform normal structure.     

A remark on extensions of hausdorff spaces

It is shown that any Hausdorff space X which has extensions E ⊇ X in a productive and closed-hereditary class A of Hausdorff spaces whose outgrowth E−X belongs to a productive and hereditary class C of Hausdorff spaces also has a maximal extension of this kind.     

A function with prescribed initial derivatives in different Banach spaces

Let X₀ ? X₁ ? ··· ? X_p be Banach spaces with continuous injection of X_k into X_{k + 1} for 0 ? k ? p ? 1, and with X₀ dense in X_p. We seek a function u: [0, 1] → X₀ such that its kth derivative u^(k), k = 0, 1,…, p, is continuous from [0, 1] into x_k, and satisfies the initial condition u^(k)(0) = a_k?X_k. It is shown that such a function exists if and only if the initial values a₀, a₁, …, a_p satisfy a certain condition reminiscent of interpolation theory. This condition always holds when p = 1; when p ? 2, the spaces X_k (k = 0, 1, …, p) may or may not be such that the desired function exists for any given initial values a_k?X_k.     

A description of normal extensions

Translated from Matematicheskie Zametki, Vol. 53, No. 5, pp. 21–28, May, 1993.