Stefan Samko

On a progress in the theory of lebesgue spaces with variable exponent: maximal and singular operators

This paper represents a broadened version of the plenary lecture presented by the author at the conference Analytic Methods of Analysis and Differential Equations (AMADE-2003), September 4–9, 2003, Minsk, Belarus. We give a survey of investigations on ‘the variable exponent business’, concentrating mainly on recent advances in the operator theory and harmonic analysis in the generalized Lebesgue and Sobolev spaces L p(·) and W m, p(·).

Completely monotonic functions

In this expository article we survey some properties of completely monotonic functions and give various examples, including some famous special functions. Such function are useful, for example, in probability theory. It is known, [1, p.450], for example, that a function w is the Laplace transform of an infinitely divisible probability distribution on (0,∞), if and only if w = e-h , where the derivative of h is completely monotonic and h(0+) = 0.

Hypersingular integrals and their applications

Part I: Hypersingular Integrals and Their Applications. Some Basics from the Theory of Special Functions and Operator Theory. The Reisz Potential Operator and the Lizorkin Type Invarient Spaces. Hypersingular Integrals with Constant Characteristics Potentials and Hypersingular Integrals with Non-Homogenous Characteristics. Hypersingular Integrals on the Unit Sphere. Part II: Applications of Hypersingular Integrals. Hypersingular Integrals in the Theory of Functional Spaces. Solution of Multidimensional Integral Equations of the First Kind with a Potential Type Kernel. Hypersingular Operators as Positive Fractional Powers of Some Operators of Mathematical Physics. Regularization of Integral Equations of the First Kind with a Potential Type Kernel Applications to Function Spaces on a Sphere. Some Modifications of Hypersingular Integrals and Their Applications.

Integration and differentiation to a variable fractional order

Interpretation and differentiation of functions to a variable order (d/dx)nf(x) is studied in two ways: 1) using the Riemann-Liouville definition, 2) using Fourier transforms. Some properties and the inversion formula are obtained.

Maximal and Fractional Operators in Weighted

We study the boundedness of the maximal operator, potential type operators and operators with fixed singularity (of Hardy and Hankel type) in the spaces Lp(·)(ρ, Ω) over a bounded open set in Rn with a power weight ρ(x) = |x−x0| , x0 ∈ Ω, and an exponent p(x) satisfying the Dini–Lipschitz condition.

L^{p(x)}

AbstractИзучается дробное ин тегрированиеIα(x) и дифференциро ваниеDα(x) переменного порядка. Важность изу чения пространств фу нкций переменной гладкост и, напримерLpα(x), связана с т ем, что оператор (Iα(x))−1, совпадающий с (Iα(x))−1, предс тавим в виде свертки с обрат имым интегральным оп ератором Вольтерра с квазидиф ференцируемым ядром, что является кл ючевым моментом для характеризацииIα(x)(Lp).

Spaces

Introduction * Notation * I. On Fredholmness of Singular and Convolution Operators * II. On Fredholmness of Other Singular-Type Operators * III. Functional and Singular Integral Equations with Carleman Shifts in the Case of Continuous Coefficients * IV. Two Term Equations

Fractional integration and differentiation of variable order

(A+Qb)\Varphi = f

Equations with Involutive Operators

with an Involutive Operator

Convolution and potential type operators in Lp(x) (Rn )

Q