Natasha Samko

Morrey-Campanato Spaces: an Overview

In this paper we overview known and recently obtained results on Morrey-Campanato spaces with respect to the properties of the spaces themselves, that is, we do not touch the study of operators in these spaces. In particular, we overview equivalent definitions of various versions of the spaces, the so-called ϕ- and θ-generalizations, structure of the spaces, embeddings, dual spaces, etc.

The maximal operator in weighted variable spaces Lp(

We study the boundedness of the maximal operator in the weighted spaces Lp(⋅)(ρ) over a bounded open set Ω in the Euclidean space ℝn or a Carleson curve Γ in a complex plane. The weight function may belong to a certain version of a general Muckenhoupt-type condition, which is narrower than the expected Muckenhoupt condition for variable exponent, but coincides with the usual Muckenhoupt class Ap in the case of constant p. In the case of Carleson curves there is also considered another class of weights of radial type of the form ρ(t)=∏k=1mwk(|t-tk|), tk∈Γ, where wk has the property that r1p(tk)wk(r)∈Φ10, where Φ10 is a certain Zygmund-Bari-Stechkin-type class. It is assumed that the exponent p(t) satisfies the Dini–Lipschitz condition. For such radial type weights the final statement on the boundedness is given in terms of the index numbers of the functions wk (similar in a sense to the Boyd indices for the Young functions defining Orlich spaces).

Commutators of Hardy operators in vanishing Morrey spaces

In this paper we study boundedness of commutators of the multi-dimensional Hardy type operators with BMO coefficients, in weighted global and/or local generalized Morrey spaces LΠp,φ(Rn,w) and vanishing local Morrey spaces VLlocp,φ(Rn,w) defined by an almost increasing function φ(r) and radial type weight w(|x|). This study is made in the perspective of posterior applications of the weighted results to some problems in the theory of PDE. We obtain sufficient conditions, in terms of some integral inequalities imposed on φ and w, and also in terms of the Matuszewska-Orlicz indices of φ and w, for such a boundedness.

What should have happened if Hardy had discovered this

First we present and discuss an important proof of Hardys inequality via Jensens inequality which Hardy and his collaborators did not discover during the 10 years of research until Hardy finally proved his famous inequality in 1925. If Hardy had discovered this proof, it obviously would have changed this prehistory, and in this article the authors argue that this discovery would probably also have changed the dramatic development of Hardy type inequalities in an essential way. In particular, in this article some results concerning power-weight cases in the finite interval case are proved and discussed in this historical perspective. Moreover, a new Hardy type inequality for piecewise constant p = p(x) is proved with this technique, limiting cases are pointed out and put into this frame.Mathematics Subject Classification: 26D15.

Fractional integrals and hypersingular integrals in variable order Holder spaces on homogeneous spaces

We consider non-standard Holder spaces Hλ(⋅)(X) of functions f on a metric measure space (X, d, μ), whose Holder exponent λ(x) is variable, depending on x ∈ X. We establish theorems on mapping properties of potential operators of variable order α(x), from such a variable exponent Holder space with the exponent λ(x) to another one with a “better” exponent λ(x)

Maximal, potential and singular operators in vanishing generalized Morrey spaces

We introduce vanishing generalized Morrey spaces

Parameter depending almost monotonic functions and their applications to dimensions in metric measure spaces

{V\mathcal{L}^{p,\varphi}_\Pi (\Omega), \Omega \subseteq \mathbb{R}^n}

Hardy Type Operators in Local Vanishing Morrey Spaces on Fractal Sets

with a general function