Some translation-invariant Banach function spaces which contain c₀
We produce several situations where some natural subspaces of classical Banach spaces of functions over a compact abelian group contain the space c₀.
Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures.
Spaces of generalized smoothness on h-sets and related Dirichlet forms
The paper is devoted to spaces of generalized smoothness on so-called h-sets. First we find quarkonial representations of isotropic spaces of generalized smoothness on ℝⁿ and on an h-set. Then we investigate representations of such spaces via differences, which are very helpful when we want to find an explicit representation of the domain of a Dirichlet form on h-sets. We prove that both representations are equivalent, and also find the domain of some time-changed Dirichlet form on an h-set.
Spherical means and measures with finite energy
We prove a restricted weak type inequality for the spherical means operator with respect to measures with finite α-energy, α ≤ 1. This complements recent results due to D. Oberlin.
Square functions associated to Schrödinger operators
We characterize geometric properties of Banach spaces in terms of boundedness of square functions associated to general Schrödinger operators of the form ℒ = -Δ + V, where the nonnegative potential V satisfies a reverse Hölder inequality. The main idea is to sharpen the well known localization method introduced by Z. Shen. Our results can be regarded as alternative proofs of the boundedness in H¹, and BMO of classical ℒ-square functions.
Strichartz's conjecture on Hardy-Sobolev spaces
We prove Strichartz's conjecture regarding a characterization of Hardy-Sobolev spaces.
Structure of the Hardy operator related to Laguerre polynomials and the Euler differential equation.
We present a direct proof of a known result that the Hardy operator Hf(x) = 1/x ∫0x f(t) dt in the space L2 = L2(0, ∞) can be written as H = I - U, where U is a shift operator (Uen = en+1, n ∈ Z) for some orthonormal basis {en}. The basis {en} is constructed by using classical Laguerre polynomials. We also explain connections with the Euler differential equation of the first order y' - 1/x y = g and point out some generalizations to the case with weighted Lw2(a, b) spaces.
Tempered Radon measures.
The Campanato, Morrey and Hölder spaces on spaces of homogeneous type
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 Characterization of the Triebel-Lizorkin Spaces for p = ...
The continuity of multilinear singular integral operators with variable Calderón-Zygmund kernel on Hardy and Herz spaces.
The Fourier transform in weighted Lorenz spaces.
The fractional maximal operator and fractional integrals on variable spaces.
The Hardy-Lorentz spaces
We deal with the Hardy-Lorentz spaces where 0 < p ≤ 1, 0 < q ≤ ∞. We discuss the atomic decomposition of the elements in these spaces, their interpolation properties, and the behavior of singular integrals and other operators acting on them.
The John-Nirenberg inequality for functions of bounded mean oscillation with bounded negative part
A version of the John-Nirenberg inequality suitable for the functions with is established. Then, equivalent definitions of this space via the norm of weighted Lebesgue space are given. As an application, some characterizations of this function space are given by the weighted boundedness of the commutator with the Hardy-Littlewood maximal operator.
The John-Nirenberg type inequality for non-doubling measures
X. Tolsa defined a space of BMO type for positive Radon measures satisfying some growth condition on . This new BMO space is very suitable for the Calderón-Zygmund theory with non-doubling measures. Especially, the John-Nirenberg type inequality can be recovered. In the present paper we introduce a localized and weighted version of this inequality and, as applications, we obtain some vector-valued inequalities and weighted inequalities for Morrey spaces.
The maximal function on variable spaces.
The minimal operator and the John--Nirenberg theorem for weighted grand Lebesgue spaces
We introduce the minimal operator on weighted grand Lebesgue spaces, discuss some weighted norm inequalities and characterize the conditions under which the inequalities hold. We also prove that the John-Nirenberg inequalities in the framework of weighted grand Lebesgue spaces are valid provided that the weight function belongs to the Muckenhoupt class.
The problem and singular integrals on Orlicz spaces
The Stein-Weiss Type Inequality for Fractional Integrals, Associated with the Laplace-Bessel Differential Operator
2000 Math. Subject Classification: Primary 42B20, 42B25, 42B35In this paper we study the Riesz potentials (B -Riesz potentials) generated by the Laplace-Bessel differential operator ∆B.* Akif Gadjiev’s research is partially supported by the grant of INTAS (project 06-1000017-8792) and Vagif Guliyev’s research is partially supported by the grant of the Azerbaijan–U.S. Bilateral Grants Program II (project ANSF Award / 16071) and by the grant of INTAS (project 05-1000008-8157).