Function Spaces with Exponential Weights II

This paper is a continuation of [8]. We study weighted function spaces of type B and F on the Euclidean space Rⁿ, where u is a weight function of at most exponential growth. In particular, u(χ (±|χ|) is an admissible weight. We deal with atomic decompositions of these spaces. Furthermore, we prove that the spaces B and F are isomorphic to the corresponding unweighted spaces B and F.     

Entropy Numbers in Weighted Function Spaces and Eigenvalue Distributions of Some Degenerate Pseudodifferential Operators II

This paper is the continuation of [17]. We investigate mapping and spectral properties of pseudodifferential operators of type Ψ with χ χ ? ? and 0 ≤ γ ≤ 1 in the weighted function spaces B (?ⁿ, w(x)) and F (?ⁿ, w(x)) treated in [17]. Furthermore, we study the distribution of eigenvalues and the behaviour of corresponding root spaces for degenerate pseudodifferential operators preferably of type b₂(x) b(x, D) b₁(x), where b₁(x) and b₂(x) are appropriate functions and b(x, D) ? Ψ. Finally, on the basis of the Birman-Schwinger principle, we deal with the “negative spectrum” (bound states) of related symmetric operators in L₂.     

On the Lorentz-Marcinkiewicz Operator Ideal

This article deals with the LORENTZ-MARCINKIEWICZ operator ideal ?? generated by an additive s-function and the LORENTZ-MARCINKIEWICZ sequence space λ^q(φ). We give eigenvalue distributions for operators belonging to ?? (E, E) and we show the interpolation properties of ??-ideals. Furthermore, we study certain SCHAUDER bases in ?? (H, K), H and K Hilbert spaces.     

On invariant measures of extensions of Markov transition functions

By using the LITTLEWOOD matrices A_2n we generalize CLARKSON' S inequalities, or equivalently, we determine the norms ‖A_2n: l(L_P) → l(L_P)‖ completely. The result is compared with the norms ‖A_2n: l → l‖, which are calculated implicitly in PIETSCH [6].     

Entropy Numbers in Weighted Function Spaces and Eigenvalue Distributions of Some Degenerate Pseudodifferential Operators I

In this paper we study weighted function spaces of type B(?ⁿ, Q(x)) and F(?ⁿ, Q(x)), where Q(x) is a weight function of at most polynomial growth. Of special interest are the weight functions Q(x) = (1 + |x|²)^α/2 with α ? ?. The main result deals with estimates for the entropy numbers of compact embeddings between spaces of this type.     

Subelliptic Estimates for a Class of Degenerate Elliptic Integro-Differential Operators

In this paper we prove subelliptic estimates for operators of the form Δ_x + λ² (x)S in ?^N = ? × ?, where the operator S is an elliptic integro - differential operator in ?^N and λ is a nonnegative Lipschitz continuous function.     

Sharp Sobolev Embeddings and Related Hardy Inequalities: The Sub -- Critical Case

The paper deals with sharp embeddings of the spaces B and F into rearrangement-variant spaces and related Hardy inequalities. Here (1/_p, s) belongs to the interior of the shaded invariant spaces region in the Figure     

Countable Direct Sums of Banach Spaces

Let (X_n) be a sequence of infinite-dimensional BANACH spaces. We prove that has a non-locally complete quotient if X₁ is not quasi-reflexive.     

Several Dimensional θ-Summability and Hardy Spaces

The d-dimensional Hardy spaces H_p ( T × … × T ) (d = d₁ + … + d_kand a general summability method of Fourier series and Fourier transforms are introduced with the help of integrable functions θ_j having integrable Fourier transforms. Under some conditions on θ_j we show that the maximal operator of the θ-means of a distribution is bounded from H_p ( T × … × T ) to L_p ( T ^d) where p₀ < p < ∞ and p₀ < 1 is depending only on the functions θ_j. By an interpolation theorem we get that the maximal operator is also of weak type ( L₁) (i = 1, …, k) where the Hardy space is defined by a hybrid maximal function and if k = 1. As a consequence we obtain that the θ-means of a function (log L)^k–1 converge a.e. to the function in question. If k = 1 then we get this convergence result for all f ∈ L₁. Moreover, we prove that the θ-means are uniformly bounded on the spaces H_p ( T × … × T ) whenever p₀ <p < ∞, thus the θ-means converge to f in ( T × … × T ) norm. The same results are proved for the conjugate θ-means and for d-dimensional Fourier transforms, too. Some special cases of the θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, Riemann, de La Vallée-Poussin, Rogosinski and Riesz summations.     

Cylinder Homomorphisms and Chow Groups

Let X be a projective algebraic manifold of dimension n (over C), CH¹(X) the Chow group of algebraic cycles of codimension l on X, modulo rational equivalence, and A¹(X) ? CH¹(X) the subgroup of cycles algebraically equivalent to zero. We say that A¹(X) is finite dimensional if there exists a (possibly reducible) smooth curve T and a cycle z∈CH¹(Γ × X) such that z_*:A¹(Γ)-A¹(X) is surjective. There is the well known Abel-Jacobi map λ₁:A¹(X)-J(X), where J(X) is the lth Lieberman Jacobian. It is easy to show that A¹(X)→J(X) A¹(X) finite dimensional. Now set with corresponding map A^*(X)→J(X). Also define Level . In a recent book by the author, there was stated the following conjecture: where it was also shown that (?) in (**) is a consequence of the General Hodge Conjecture (GHC). In this present paper, we prove A^*(X) finite dimensional ?? Level (H^*(X)) ≤ 1 for a special (albeit significant) class of smooth hypersurfaces. We make use of the family of k-planes on X, where ([…] = greatest integer function) and d = deg X; moreover the essential technical ingredients are the Lefschetz theorems for cohomology and an analogue for Chow groups of hypersurfaces. These ingredients in turn imply very special cases of the GHC for our choice of hypersurfaces X. Some applications to the Griffiths group, vanishing results, and (universal) algebraic representatives for certain Chow groups are given.     

Hardy-Littlewood Maximal Operator,Singular Integral Operators and the Riesz Potentials on Generalized Morrey Spaces

Any continuous linear operator T: L^p → L^q has a natural vector-valued extension T: L^p(l) → L^q(l) which is automatically continuous. Relations between the norms of these operators in the cases of p = q and r = 2 were considered by Marcinkiewicz -Zygmund [28], Herz [14] and Krivine [19] - [21]. In this paper we study systematically these relations and given some applications. It turns out that some known results can be proved in a simple way as a consequence of these developments.     

Selbstadjungierte Relations-Erweiterungen für eine Klasse von symmetrischen Differential-Funktional-Operatoren

Symmetry- and selfadjointness-conditions are derived for ordinary differential-integral-interface operators under integral-interface conditions. Criteria for the existence of selfadjoint extensions in L×L are given. These extensions are characterized in a constructive way. The main tools are some extension-theorems for linear relations (subspaces), wich are developed in section 2.     

On a Relation between Intrinsic and Extrinsic Dirichlet Forms for Interacting Particle Systems

In this paper we extend the result obtained in [AKR98] (see also [AKR96a]) on the representation of the intrinsic pre–Dirichlet form ℰ^Γ of the Poisson measure π_σ in terms of the extrinsic one ℰ^P. More precisely, replacing π_σ by a Gibbs measure μ on the configuration space Γ_X we derive a relation between the intrinsic prend–Dirichlet form ℰ^Γ_μ of the measure μ and the extrinsic one ℰ^P. As a consequence we prove the closability of ℰ^Γ_μ on L²(Γ_X, μ) under very general assumptions on the interaction potential of the Gibbs measures μ.     

On Decomposition in Exponential Orlicz Spaces

Various characterizations are given of the exponential Orlicz space L and the Orlicz‐Lorentz space L. By way of application we give a simple proof of the celebrated theorem of Brézis and Wainger concerning a limiting case of a Sobolev imbedding theorem.     

A Reuniformization for Contractive Mappings in Uniform Spaces

Let X be a complete uniform HAUSDORFF space with a uniformity generated by a saturated family of pseudometrics ?? = {?_α(x, y): α ? A} and let T: X → X be a continuous mapping. The paper contains necessary and sufficient conditions for the existence of a new family of pseudometrics ??*={?*(x, y): α*?A*} generated the same topology such that T is contractive with respect to ??*.     

Hardy Classes of Banach-Space-Valued Distributions

It is shown that for 0<p ≥ 1, the trigonometric polynomials are dense in H, the space of B-valued harmonic functions with non-tangential maximal function in L_p, if and only if the Banach space B has the Radon-Nikodym property (R.N.P.). This extends known results for 1 <p < ∞. We also show that H coincides with the corresponding atomic space if and only if B has the R.N.P.     

Non-linear parabolic problem associated with the penetration of a magnetic field into a substance

We study the following initial and boundary value problem: In section 1, with u₀ in L²(Ω), f continuous such that f(u) + ? non-decreasing for ? positive, we prove the existence of a unique solution on (0,T), for each T > 0. In section 2 it is proved that the unique soluition u belongs to L²(0, T; H ∩ H²) ∩ L^∞(0, T; H) if we assume u₀ in H and f in C¹(?,?). Numerical results are given for these two cases.     

Real Interpolation for Families of Banach Spaces and Convexity

We show how the geometrical properties of uniform convexity and uniformly non-?? are inherited by real interpolation spaces for infinite families.     

On Hardy - Littlewood Maximal Functions in Orlicz Spaces

Let Φ(t) and Ψ(t) be the functions having the following representations Φ(t) = ∫a(s)ds and Ψ(t) = ∫b(s) ds, where a(s) is a positive continuous function such that ∫a(s)/s ds = + ∞ and b(s) is an increasing function such that lim_s→ ∞ b(s) = + ∞. Then the following statements for the Hardy - Littlewood maximal function M f (x) are equivalent:

• 1 (i) there exist positive constants c₁ and s₀ such that
• 1 (ii) there exist positive constant c₂ and c₃ such that

.     

The Triebel - Lizorkin Scale of Function Spaces for the Fourier - Helgason Transform

We define and investigate the Triebel - Lizorkin scale of function spaces F, with 1< p < ∞, 1< q ≤ ∞ for the Fourier-Helgason transform on symmetric Riemannian manifolds of the noncompact type.