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The Campanato, Morrey and Hölder spaces on spaces of homogeneous type

Eiichi Nakai (2006)

Studia Mathematica

We investigate the relations between the Campanato, Morrey and Hölder spaces on spaces of homogeneous type and extend the results of Campanato, Mayers, and Macías and Segovia. The results are new even for the ℝⁿ case. Let (X,d,μ) be a space of homogeneous type and (X,δ,μ) its normalized space in the sense of Macías and Segovia. We also study the relations of these function spaces for (X,d,μ) and for (X,δ,μ). Using these relations, we can show that theorems for the Campanato, Morrey or Hölder spaces...

The class Bpfor weighted generalized Fourier transform inequalities

Chokri Abdelkefi, Mongi Rachdi (2015)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

In the present paper, we prove weighted inequalities for the Dunkl transform (which generalizes the Fourier transform) when the weights belong to the well-known class Bp. As application, we obtain the Pitt’s inequality for power weights.

The class of convolution operators on the Marcinkiewicz spaces

Ka-Sing Lau (1981)

Annales de l'institut Fourier

Let 𝒯 X denote the operator-norm closure of the class of convolution operators Φ μ : X → X where X is a suitable function space on R . Let ℳ r p be the closed subspace of regular functions in the Marinkiewicz space ℳ p , 1 ≤ p < ∞ . We show that the space 𝒯 ℳ r p is isometrically isomorphic to 𝒯 L p and that strong operator sequential convergence and norm convergence in 𝒯 ℳ r p coincide. We also obtain some results concerning convolution operators under the Wiener transformation. These are to improve a Tauberian theorem of Wiener on ℳ 2 .

The Complex Stone-Weierstrass Property

Kenneth Kunen (2004)

Fundamenta Mathematicae

The compact Hausdorff space X has the CSWP iff every subalgebra of C(X,ℂ) which separates points and contains the constant functions is dense in C(X,ℂ). Results of W. Rudin (1956) and Hoffman and Singer (1960) show that all scattered X have the CSWP and many non-scattered X fail the CSWP, but it was left open whether having the CSWP is just equivalent to being scattered. Here, we prove some general facts about the CSWP; in particular we show that if X is a compact ordered space,...

The concentration-compactness principle in the calculus of variations. The limit case, Part II.

Pierre-Louis Lions (1985)

Revista Matemática Iberoamericana

This paper is the second part of a work devoted to the study of variational problems (with constraints) in functional spaces defined on domains presenting some (local) form of invariance by a non-compact group of transformations like the dilations in RN. This contains for example the class of problems associated with the determination of extremal functions in inequalities like Sobolev inequalities, convolution or trace inequalities... We show how the concentration-compactness principle and method...

The concentration-compactness principle in the calculus of variations. The limit case, Part I.

Pierre-Louis Lions (1985)

Revista Matemática Iberoamericana

After the study made in the locally compact case for variational problems with some translation invariance, we investigate here the variational problems (with constraints) for example in RN where the invariance of RN by the group of dilatations creates some possible loss of compactness. This is for example the case for all the problems associated with the determination of extremal functions in functional inequalities (like for example the Sobolev inequalities). We show here how the concentration-compactness...

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