Absolutely L – Summing Norms of Diagonal Operators in lr and Limit Orders of Lexp – Summing Operators

We compute the absolutely L – summing norms of the diagonal operators acting on l_r (1 ≤ q, r < ∞) and determine the limit orders of the absolutely L_exp – summing operators.     

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.     

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.     

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.     

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     

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 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.     

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.     

Characterizations of the boundedness of generalized fractional maximal functions and related operators in Orlicz spaces

Given and a Young function η, we consider the generalized fractional maximal operator defined by where the supremum is taken over every ball B contained in . In this article, we give necessary and sufficient Dini type conditions on the functions , and η such that is bounded from the Orlicz space into the Orlicz space . We also present a version of this result for open subsets of with finite measure. Both results generalize those contained in 6 and 14 when , respectively. As a consequence, we obtain a characterization of the functions involved in the boundedness of the higher order commutators of the fractional integral operator with BMO symbols. Moreover, we give sufficient conditions that guarantee the continuity in Orlicz spaces of a large class of fractional integral operators of convolution type with less regular kernels and their commutators, which are controlled by .     

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.     

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.     

Tangent Conics at Quartic Surfaces and Conics in Quartic Double Solids

For a quartic double solid we study the parameter space of conics (i.e. of smooth rational curves C ⊂ Z such that C · φ*O(1) = 2). This parameter space is naturally fibred (with disconnected fibres) over Pˇ³. We study the monodromy of the fibres and determine this way the irreducible components of the parameter space.     

Une Famille Infinie d'Extensions Faiblement Ramifiées

We study weakly ramified extensions of ℚ with Galois group C ⋊ C_p, where p is an odd prime. In particular, we describe an infinite family of such extensions when p = 3.     

The Calderón approach to an elliptic boundary problem

We consider a boundary problem for an elliptic system in a bounded region Ω ? ?ⁿ and where the spectral parameter is multiplied by a discontinuous weight function ω (x) = diag(ω₁(x), …, ω_N (x)). The problem is considered under limited smoothness assumptions and under an ellipticity with parameter condition. Recently, this problem was studied under the assumption that the ω_j (x)^–1 are essentially bounded in Ω. In this paper we suppose that ω (x) vanishes identically in a proper subregion Ω of Ω and that the ω_j (x)^–1 are essentially bounded in . Then by using methods which are a variant of those used in constructing the Calderón projectors for the boundary Γ of Ω, we shall derive results here which will enable us in a subsequent work to apply the ideas of Calderón to develop the spectral theory associated with the problem under consideration here (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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

.     

Boundedness and compactness of Hardy operators on Lorentz‐type spaces

Given non‐negative measurable functions on we study the high dimensional Hardy operator between Orlicz–Lorentz spaces , where f is a measurable function of and is the ball of radius t in . We give sufficient conditions of boundedness of and . We investigate also boundedness and compactness of between weighted and classical Lorentz spaces. The function spaces considered here do not need to be Banach spaces. Specifying the weights and the Orlicz functions we recover the existing results as well as we obtain new results in the new and old settings.     

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.     

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.     

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.