Advances in Harmonic Analysis and Operator Theory

Frontmatter

Stefan G. Samko – Mathematician, Teacher and Man

Abstract

It is a great honour for me, to contribute this article on the occasion of a remarkable anniversary. The goal of this paper is to present, in a concise manner, the scientific achievements of a leading authority in mathematical analysis, Professor Stefan Samko.

V. Kokilashvili

The Role of S.G. Samko in the Establishing and Development of the Theory of Fractional Differential Equations and Related Integral Operators

Abstract

The aim of this work is to describe main aspects of the modern theory of fractional differential equations, to present elements of classification of fractional differential equations, to formulate basic components of investigations related to fractional differential equations, to pose some open problems in the study of fractional differential equations. A survey of results by S.G. Samko on different problems of modern mathematical analysis is given. Main results of S.G. Samko having an essential influence on the establishing and development of the theory of fractional differential equations are singled out.

Sergei V. Rogosin

Energy Flow Above the Threshold of Tunnel Effect

Abstract

We consider the Klein-Gordon equation on two half-axes connected at their origins. We add a potential that is constant but different on each branch. In a previous paper, we studied the L ^∞-time decay via Hörmander’s version of the stationary phase method. Here we apply these results to show that for initial conditions in an energy band above the threshold of the tunnel effect a fixed portion of the energy propagates between group lines. Further we consider the situation that the potential difference tends to infinity while the energy band of the initial condition is shifted upwards such that the particle stays above the threshold of the tunnel effect. We show that the total transmitted energy as well as the portion between the group lines tend to zero like \( {{a_{2}}^{-1/2}} \)in the branch with the higher potential a2 as a2 tends to infinity. At the same time the cone formed by the group lines inclines to the t-axis while its aperture tends to zero.

F. Ali Mehmeti, R. Haller-Dintelmann, V. Régnier

Some New Hardy-type Integral Inequalities on Cones of Monotone Functions

Abstract

Some new Hardy-type inequalities with Hardy-Volterra integral operators on the cones of monotone functions are obtained. The case 1 < p ≤ q < ∞ is considered and the involved kernels satisfy conditions which are less restrictive than the classical Oinarov condition.

L. S. Arendarenko, R. Oinarov, L.-E. Persson

On a Boundary Value Problem for a Class of Generalized Analytic Functions

Abstract

For solutions of the Bers-Vekua equation \( {\rm D} v : {v\bar{z}}-c(z,\bar{z})\bar{v}=0\) defined in a domain \( \mathbb{D} \subset \mathbb{C} \) we consider Riemann-Hilbert type boundary conditions.

In the case of the existence of certain differential operators with which all the solutions of Dv = 0 defined in \( \mathbb{D} \) can be generated from a function f holomorphic in \( \mathbb{D} \) the boundary value problem is reduced to a Goursat problem for f in essence. For certain classes of coefficients c and domains \( \mathbb{D} \) we show how this problem can be solved explicitly.

For the poly-pseudoanalytic functions obeying the differential equation \( D^{n}v = 0, n \in \mathbb{N}, \) we investigate an appropriate boundary value problem and show the equivalence of this problem to n boundary value problems for generalized analytic functions.

P. Berglez, T. T. Luong

The Factorization Problem: Some Known Results and Open Questions

Abstract

This is a concise survey of some results and open problems concerning Wiener-Hopf factorization and almost periodic factorization of matrix functions. Several classes of discontinuous matrix functions are considered. Also sketched is the abstract framework which unifies the two types of factorization.

Albrecht Böttcher, Ilya M. Spitkovsky

A Class of Sub-elliptic Equations on the Heisenberg Group and Related Interpolation Inequalities

Abstract

We firstly prove the existence of least energy solutions to a class of sub–elliptic equations on the Heisenberg group. Then we use this least energy solution to give a sharp estimate to the smallest positive constant in the Gagliardo–Nirenberg inequality on the Heisenberg group. Finally we point out some extensions to the quasilinear sub–elliptic case.

Jianqing Chen, Eugénio M. Rocha

New Types of Solutions of Non-linear Fractional Differential Equations

Abstract

Using the Riemann-Liouville and Caputo Fractional Standard Maps (FSM) and the Fractional Dissipative Standard Map (FDSM) as examples, we investigate types of solutions of non-linear fractional differential equations. They include periodic sinks, attracting slow diverging trajectories (ASDT), attracting accelerator mode trajectories (AMT), chaotic attractors, and cascade of bifurcations type trajectories (CBTT). New features discovered include attractors which overlap, trajectories which intersect, and CBTTs.

Mark Edelman, Laura Anna Taieb

Stability, Structural Stability and Numerical Methods for Fractional Boundary Value Problems

Abstract

In this work, we investigate the stability and the structural stability of a class of fractional boundary value problems. We approximate the solution by using a wide range of numerical methods illustrating our theoretical results.

Neville J. Ford, M. Luísa Morgado

On the Boundedness of the Fractional Maximal Operator, Riesz Potential and Their Commutators in Generalized Morrey Spaces

Abstract

In the paper the authors find conditions on the pair \( (\varphi_{1},\varphi_{2}) \) which ensure the Spanne type boundedness of the fractional maximal operator \( M_{\alpha} \) and the Riesz potential operator \( I_{\alpha} \) from one generalized Morrey spaces \( M_{p,{\varphi_{1}}} \) to another \( M_{q,{\varphi_{2}}}, 1 < p < q < \infty, 1/p-1/q = \alpha/n, \) and from \( M_{1,{\varphi_{1}}} \) to the weak space W \( M_{q,{\varphi_{2}}}, 1 < p < q < \infty, 1- 1/q = \alpha/n, \) We also find conditions on \( \varphi \) which ensure the Adams type boundedness of the \( M_{\alpha}\; {\rm and}\; I_{\alpha}\; {\rm from} \; M_{p,{\varphi}^{\frac {1}{p}}}\; \rm{to}\; M_{q,{\varphi}^{\frac {1}{q}}}\;\rm {for 1 < p < q < \infty \; and\; from\; M_{1,{\varphi}}\; to \;W\;M_{q,{\varphi}^{\frac{1}{p}}} \; for \; 1 < q < \infty.}\) As applications of those results, the boundeness of the commutators of operators \( I_{\alpha} and I_{\alpha} \) on generalized Morrey spaces is also obtained. In the case \( b \in BMO{\mathbb{(R)}^{n}}\; \rm and \;1 < p < q < \infty,\) we find the sufficient conditions on the pair \( (\varphi_{1},\varphi_{2}) \) which ensures the boundedness of the operators \( {M_{b,\alpha}}\; \rm {and \;[b,I_{\alpha}] \; from \; M_{p,\varphi_{1}}\; to \; M_{q,\varphi_{2}}\; with\; 1/p - 1/q = \alpha/n.} \) We also find the sufficient conditions on \( \varphi \) which ensures the boundedness of the operators \( {M_{b,\alpha}}\; \rm {and \;[b,I_{\alpha}] \; from \; M_{p,{\varphi^{\frac{1}{p}}}}\; to \; M_{q,\varphi^{\frac{1}{p}}}\; for\; 1 < p < q < \infty.} \) In all cases conditions for the boundedness are given in terms of Zygmund-type integral inequalities on \( \rm {(\varphi_{1},\varphi_{2}) \;and \;\varphi} ,\)which do not assume any assumption on monotonicity of \( \rm {\varphi_{1},\varphi_{2} \;and \;\varphi} \;\rm{in\; r} ,\) As applications, we get some estimates for Marcinkiewicz operator and fractional powers of the some analytic semigroups on generalized Morrey spaces.

Vagif S. Guliyev, Parviz S. Shukurov

Existence of Solutions of a Class of Nonlinear Singular Equations in Lorentz Spaces

Abstract

We consider the following nonlinear elliptic Dirichlet problem involving a Leray-Lions type differential operator \( \rm {-div}(\psi(x,u(x),\bigtriangledown u(x))) + a(x)u(x)=f(x), \quad \rm {in} \; \Omega, \;\; u \in W^{1,p}_{0}(\Omega)\) where \( \Omega \subset \mathbb{R}^{N} \) is a bounded domain with smooth boundary, \( 2 \leq p < N, a \in L^{\infty}_{\rm loc}(\Omega;)\mathbb{R}^{+}_{0}\; \rm{and}\; f \in L^{q,q_{1}(\Omega)} \) is a function in a Lorentz space. We show the existence of a solution \( u \in W^{1,p}_{0}(\Omega) \cap L ^{r,s}(\Omega) \) and an a priori estimate for the solution with respect to the Lorentz space norm of \( f \in L^{q,q_{1}}(\omega) \) Ω), for suitable values \( p,q,q_{1},r \; \rm{and}\; s \)

L. Huang, K. Murillo, E. M. Rocha

Growth of Schrödingerian Subharmonic Functions Admitting Certain Lower Bounds

Abstract

Matsaev’s theorem on the growth of entire functions admitting some lower bounds is extended to subsolutions of the stationary Schrödinger equation

Alexander I. Kheyfits

The Riemann and Dirichlet Problems with Data from the Grand Lebesgue Spaces

Abstract

In Section 1, we present a solution of the following boundary value problem: find an analytic function Φon the plane cut along a closed piecewisesmooth curve Γ which is represented by a Cauchy type integral with a density from the Grand Lebesgue Space \( L^{p)},\theta(\Gamma)(1 < p < \infty, 0 < \theta < \infty) \) and whose boundary values satisfy the conjugacy condition

$$ \Phi^{+}(t)=G(t)\Phi^{-}(t)+g(t),\quad t \in \Gamma $$

Here G and g are functions defined on Γ such that G is a piecewise continuous function, \( G(t)\neq 0 \) and \( g \in L^{{p}),\theta}(\Gamma) \) The conditions for the problem to be solvable are established and the solutions are constructed in explicit form.

In Section 2, the Dirichlet problem for harmonic functions, real parts of Cauchy type integrals with densities from weighted generalized Grand Lebesgue Spaces is studied when boundary data belong to the same space.

V. Kokilashvili, V. Paatashvili

Overview of Fractional h-difference Operators

Abstract

Fractional difference operators and their properties are discussed. We give a characterization of three operators that we call Grünwald-Letnikov, Riemann-Liouville and Caputo like difference operators. We show relations among them. In the paper, linear fractional h-difference equations are described. We give formulas of solutions to initial value problems. Crucial formulas are gathered in the tables presented in the last section of the paper.

Dorota Mozyrska, Ewa Girejko

A Singularly Perturbed Dirichlet Problem for the Poisson Equation in a Periodically Perforated Domain. A Functional Analytic Approach

Abstract

Let Ω be a sufficiently regular bounded open connected subset of \( \mathbb{R}^{n} \) such that 0 ϵ Ω and that \( \mathbb{R}^{n}\setminus \rm {cl}\Omega \) is connected. Then we take \( (q_{11},...,q_{nn})\in]0,+\infty{[^{n}}\; \rm {and} \;p \in Q \equiv \prod\nolimits^{n}_{j=1}]0,q_{jj}[.\) If є is a small positive number, then we define the periodically perforated domain \( \mathbb{S}[\Omega_{p,\epsilon}]^{-} \equiv \mathbb{R}^{n} \setminus \cup_{z\in\mathbb{Z}^{n}}\rm {cl}(p+\epsilon\Omega\;+\;\sum\nolimits^{n}_{j=1}(q_{jj}z_{j})e_{j}) \), where \(\left\{e_{1},...,e_{n}\right\}\) is the canonical basis of \( \mathbb{R}^{n}\). For є small and positive, we introduce a particular Dirichlet problem for the Poisson equation in the set \( \mathbb{S}[\Omega_{p,\epsilon}]^{-}\). . Namely, we consider a Dirichlet condition on the boundary of the set \( p \; + \; \epsilon\Omega\) , together with a periodicity condition. Then we show real analytic continuation properties of the solution as a function of є , of the Dirichlet datum on \( p \; + \; \epsilon\partial\Omega\) , and of the Poisson datum, around a degenerate triple with є = 0.

Paolo Musolino

Fractional Variational Calculus of Variable Order

Abstract

We study the fundamental problem of the calculus of variations with variable order fractional operators. Fractional integrals are considered in the sense of Riemann–Liouville while derivatives are of Caputo type.

T. Odzijewicz, A. B. Malinowska, D. F. M. Torres

Improving Bounds for Singular Operators via Sharp Reverse Höolder Inequality for $$ A_\infty $$

Abstract

In this expository article we collect and discuss some recent results on different consequences of a Sharp Reverse Hölder Inequality for \( A\infty \) weights. For two given operators T and S, we study \( L^{p}(w) \) bounds of Coifman– Fefferman type:

$$ \parallel T\;f \parallel_{L^p}(w)\;\leq \; c_{n,w,p}\parallel S\;f \parallel _{L^p}(w),$$

that can be understood as a way to control T by S.

We will focus on a quantitative analysis of the constants involved and show that we can improve classical results regarding the dependence on the weight w in terms of Wilson’s \( A\infty \) constant

$$ [w]A_{\infty}\; := \; {\rm sup_Q}\frac{1}{w(Q)}\int_{Q}{M({w_\mathcal{X}}_Q)} .$$

We will also exhibit recent improvements on the problem of finding sharp constants for weighted norm inequalities involving several singular operators. In the same spirit as in [10], we obtain mixed \( A_{1}-A_{\infty}\) estimates for the commutator [b,T] and for its higher–order analogue \( T^{k}_{b}\) . A common ingredient in the proofs presented here is a recent improvement of the Reverse Hölder Inequality for \( A_{\infty}\) weights involving Wilson’s constant from [10].

Carmen Ortiz-Caraballo, Carlos Pérez, Ezequiel Rela

Potential Type Operators on Weighted Variable Exponent Lebesgue Spaces

Abstract

We consider double-layer potential type operators acting in weighted variable exponent Lebesgue space \( L^{p(.)}(\Gamma,w)\) on some composed curves with oscillating singularities. We obtain a Fredholm criterion for operators \( A=aI+bD_{g.\Gamma}:L^{p(.)}(\Gamma,w)\rightarrow L^{p(.)}(\Gamma,w) \; {\rm where}{D_{g,\Gamma}} \) is the operator of the form

$${D_{g,\Gamma}{u(t)}}=\frac{1}{\pi}\int_{\Gamma}\frac{g(t,\tau)(\nu(\tau),\tau-t)u(\tau)dl_{t}}{|t-\tau|^{2}},t \in \Gamma $$

\(\nu(\tau)\) is the inward unit normal vector to Γ at the point \( \tau \in \Gamma \setminus \mathcal{F},dl_{\tau} \) is the oriented Lebesgue measure on \( \tau ,\mathcal{F}\) is the set of the nodes, \( a,b:\Gamma\rightarrow\mathbb{C},g:\Gamma\times\Gamma \rightarrow \mathbb{C} \) are a bounded functions with oscillating discontinuities at the nodes only.

We give applications of such operators to the Dirichlet and Neumann problems with boundary function in \( L^{p(.)}(\Gamma,w) \) for domains with boundaries having a finite set of oscillating singularities.

Vladimir Rabinovich

A Note on Boundedness of Operators in Grand Grand Morrey Spaces

Abstract

In this note we introduce grand grand Morrey spaces, in the spirit of the grand Lebesgue spaces.We prove a kind of reduction lemma which is applicable to a variety of operators to reduce their boundedness in grand grand Morrey spaces to the corresponding boundedness in Morrey spaces. As a result of this application, we obtain the boundedness of the Hardy-Littlewood maximal operator and Calderón–Zygmund operators in the framework of grand grand Morrey spaces.

Humberto Rafeiro

Operational Calculus for Bessel’s Fractional Equation

Abstract

This paper is intended to investigate a fractional differential Bessel’s equation of order 2α with \( \alpha \in]0,1] \) involving the Riemann–Liouville derivative. We seek a possible solution in terms of power series by using operational approach for the Laplace and Mellin transform. A recurrence relation for coefficients is obtained. The existence and uniqueness of solutions is discussed via Banach fixed point theorem.

M. M. Rodrigues, N. Vieira, S. Yakubovich

The Dirichlet Problem for Elliptic Equations with VMO Coefficients in Generalized Morrey Spaces

Abstract

We consider the Dirichlet problem in a bounded smooth domain \( \Omega \subset \mathbb{R}^{n} \) for linear uniformly elliptic equation \( \mathfrak{L}u(x)=f(x) \) with VMO principal coefficients. Its unique strong solvability is proved in [5] and [6]. Our aim is to show that for every f belonging to the generalized Morrey space \( L^{p,\omega}(\Omega),p \in (1,\infty),\omega:\mathbb{R}^{n}\times\mathbb{R}_{+}\rightarrow \mathbb{R}_{+} \rm {the\; operator} \mathfrak{L}:W^{2,p,\omega}\cap W_{0}^{1,p}(\Omega)\rightarrow L^{p,\omega}(\Omega) \) is bijective and the estimate \( \parallel D^{2}u \parallel _{L^{p,\omega}(\Omega)}\leq C(\parallel {f} \parallel_{L^{p,\omega}(\Omega)}+ \parallel{u}\parallel_ {L^{p,\omega}(\Omega)}) \) holds.

Lubomira G. Softova

Riesz-Thorin-Stein-Weiss Interpolation Theorem in a Lebesgue-Morrey Setting

Abstract

We prove an analogue of Riesz-Thorin-Stein-Weiss interpolation theorem in the weighted Lebesgue-Morrey setting (a generalization of Campanato– Murthy interpolation theorem to the case of weighed spaces).

Salaudin M. Umarkhadzhiev