Tables of integrals

In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.

General Properties of Interpolation Spaces

I am presenting a survey of regularity results for both minima of variational integrals, and solutions to non-linear elliptic, and sometimes parabolic, systems of partial differential equations. I will try to take the reader to the Dark Side...

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Regularity of minima: an invitation to the Dark Side of the Calculus of Variations

The author (Appl. Math. Comput. 218:860-865, 2011) introduced a new fractional integral operator given by, � I � a+f � (x) = � 1 � which generalizes the well-known Riemann-Liouville and the Hadamard fractional integrals. In this paper we present a new fractional derivative which generalizes the familiar Riemann-Liouville and the Hadamard fractional derivatives to a single form. We also obtain two representations of the generalized derivative in question. An example is given to illustrate the results.

A new approach to generalized fractional derivatives

Multiple integrals in the calculus of variations and nonlinear elliptic systems

We study the boundedness of the maximal operator, potential type operators and operators with fixed singularity (of Hardy and Hankel type) in the spaces $L^{p(\cdot)}(\rho,\Omega)$ over a bounded open set in $\mathbb{R}^n$ with a power weight $\rho(x)=|x-x_0|^\gamma$, $x_0\in \overline{\Omega}$, and an exponent $p(x)$ satisfying the Dini-Lipschitz condition.

https://ems.press/content/serial-article-files/4995

Maximal and Fractional Operators in Weighted $L^{p(x)}$ Spaces

In this survey, results on the representation of a function as an absolutely convergent Fourier integral are collected, classified and discussed. Certain applications are also given.

The Wiener algebra of absolutely convergent Fourier integrals: an overview

We introduce grand Lebesgue spaces on open sets Ω of infinite measure in R n , controlling the integrability of |f(x)| p - e at infinity by means of a weight (depending also on e); in general, such spaces are different for different ways to introduce dependence of the weight on e We prove some properties of these spaces We introduce also a version of weighted grand Lebesgue spaces, different from the usual ones, for bounded sets, in which together with the passage from p to p−e we introduce a weight also depending on e In both versions we show that every linear operator bounded in a Lebesgue space with Muckenhoupt weights is also bounded in the corresponding grand Lebesgue space a Muckenhoupt weight (for bounded or unbounded Ω)

On Iwaniec-Sbordone Spaces on Sets Which May Have Infinite Measure (pp. 67-84)

We consider generalized Orlicz-Morrey spaces MΦ,φ(R) including their weak versions. In these generalized spaces we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on φ(r) without assuming any monotonicity property of φ(r), or in terms of supremal operators, related to φ(r). Mathematics Subject Classification (2010). Primary 42B20, 42B25, 42B35.

Boundedness of Maximal and Singular Oper- ators on Generalized Orlicz-Morrey Spaces

We consider generalized Orlicz–Morrey spaces $M_{\Phi},\varphi\,(\mathbb{R}_{n})$ including their weak versions. In these generalized spaces we prove the boundedness of the Hardy–Littlewood maximal operator and Calderon–Zygmund singular operators with standard kernel. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on $\varphi(r)$ without assuming any monotonicity property of $\varphi(r)$ , or in terms of supremal operators, related to $\varphi(r)$.

Stefan Samko

Papers

Maximal and Fractional Operators in Weighted $L^{p(x)}$ Spaces

The Wiener algebra of absolutely convergent Fourier integrals: an overview

On Iwaniec-Sbordone Spaces on Sets Which May Have Infinite Measure (pp. 67-84)

Boundedness of Maximal and Singular Oper- ators on Generalized Orlicz-Morrey Spaces

Boundedness of the Maximal and Singular Operators on Generalized Orlicz–Morrey Spaces