Abstracts

We establish criteria for boundedness for some classes of integral opearators with logarithmic singularities in weighted Lebesgue spaces for cases $1\lt p \leq q \lt \infty$ and $1\lt q \lt p \lt \infty$. As corollaries some corresponding new Hardy inequalities are pointed out.

Given a domain $D$ in $\mathbb{C}^n$ and $K$ a compact subset of $D$, the set $\mathcal{A}_K^D$ of all restrictions of functions holomorphic on $D$ the modulus of which is bounded by $1$ is a compact subset of the Banach space $C(K)$ of continuous functions on $K$. The sequence $(d_m(\mathcal{A}_K^D))_{m\in \mathbb{N}}$ of Kolmogorov $m$-widths of $\mathcal{A}_K^D$ provides a measure of the degree of compactness of the set $\mathcal{A}_K^D$ in $C(K)$ and the study of its asymptotics has a long history, essentially going back to Kolmogorov's work on $\epsilon$-entropy of compact sets in the 1950s. In the 1980s Zakharyuta showed that for suitable $D$ and $K$ the asymptotics \begin{equation} \label{eq:KP} \lim_{m\to \infty}\frac{- \log d_m(\mathcal{A}_K^D)}{m^{1/n}} = 2\pi \left ( \frac{n!}{C(K,D)}\right ) ^{1/n}\,, \end{equation} where $C(K,D)$ is the Bedford-Taylor relative capacity of $K$ in $D$ is implied by a conjecture, now known as Zakharyuta's Conjecture, concerning the approximability of the regularised relative extremal function of $K$ and $D$ by certain pluricomplex Green functions. Zakharyuta's Conjecture was proved by Nivoche in 2004 thus settling (\ref{eq:KP}) at the same time.

In this talk I will outline a new approach, developed together with Stéphanie Nivoche, for the proof of the asymptotics (\ref{eq:KP}) with $D$ strictly hyperconvex and $K$ of non-zero Lebesgue measure which does not rely on Zakharyuta's Conjecture. The approach is more direct, proceeding instead in a two-pronged fashion, by establishing sharp upper and lower bounds for the Kolmogorov widths. The lower bounds follow from concentration results of independent interest for the eigenvalues of a certain family of Toeplitz operators, while the upper bounds follow from an application of the Bergman-Weil formula coupled with exhaustion arguments using special holomorphic polyhedra.

Many problems in applied mathematics and engineering can be formulated as Fredholm integral equations of the first kind: \begin{equation} Kf(x)=\int_{a}^{b}k(x,y)f(y)dy =g(x), \label{eq:1}\end{equation} where the kernel $k(.,.)$ and the right-hand side $g$ are smooth real-valued functions.

The determination of the solution $f$ of ($\ref{eq:1}$) is an ill-posed problem in the sense of Hadamard; in the sense that the solution (if it exists) does not depend continuously on the data.

In this study one of the new techniques is used to solve numerical problems involving integral equations known as regularized sinc-collocation method. This method has been shown to be a powerful numerical tool for finding accurate solutions. So, in this talk, some properties of the regularized sinc-collocation method required for our subsequent development are given and are utilized to reduce integral equation of the first kind to some algebraic equations. Then by a theorem we show error in the approximation of the solution decays at an exponential rate. Finally, numerical examples are included to demonstrate the validity and applicability of the technique.

We discuss, in the context of inverse linear problems in Hilbert space, the notion of the associated infinite-dimensional Krylov subspace and we produce necessary and sufficient conditions for the Krylov-solvability of the considered inverse problem. A linear inverse problem $Af = g$ for $A$ a bounded linear operator on Hilbert space $\mathcal{H}$, $g \in \mathrm{ran}A$, and $f$ a solution; has associated Krylov space \[\mathcal{K}(A,g) := \mathrm{span}\{A^ng \,|\, n \in \mathbb{N}_0\}\,.\] Krylov-solvability refers to the existence of a solution $f$ in the closure of $\mathcal{K}(A,g)$. Some aspects specifically concerning Krylov-solvability for self-adjoint operators are presented, along with operator theoretic constructions for more general classes of operator $A$. The presentation is based on theoretical results together with a series of model examples.

We are interested in the solution to a boundary integral formulation of a diffusion problem in 3D involving material coefficients that are piecewise constant with respect to a partition of space into Lipschitz subdo-mains. We allow in particular the presence of material junctions where three subdomains or more are adjacent. We will discuss boundary integral formulations of the second kind adapted to such problems, exhibiting a natural generalization of the Neumann-Poincaré operator that arises in single interface problems. We will also comment on the continuity properties of this operator, and show how it can be used to impose transmission conditions through interfaces of a geometric partition of the space involving arbitrary junctions.

Real-valued pluriharmonic functions appear naturally as the real and imaginary parts of holomorphic functions in several complex variables. This gives a tight connection between Bergman- and Segal-Bargmann spaces of holomorphic and of pluriharmonic functions. We will use this connection to show how results concerning Toeplitz operators on holomorphic function spaces can be, more or less easily, transfered to results concerning Toeplitz operators on pluriharmonic function spaces. In particular, but not exclusively, we will be concerned with results on deformation quantization of pluriharmonic Toeplitz operators.

The Neumann-Poincaré operator is a boundary-integral operator associated with harmonic layer potentials. We prove the existence of eigenvalues within the essential spectrum for the Neumann-Poincaré operator for certain Lipschitz curves in the plane with reflectional symmetry, when considered in the functional space in which it is self-adjoint. The proof combines the compactness of the Neumann-Poincar\'e operator for curves of class $C^{2,\alpha}$ with the essential spectrum generated by a corner. Eigenvalues corresponding to even (odd) eigenfunctions are proved to lie within the essential spectrum of the odd (even) component of the operator when a $C^{2,\alpha}$ curve is perturbed by inserting a small corner.

Hankel operators can be defined in two ways: either as infinite matrices of the form ${a(j+k)}$ or as integral operators on $L_2(\mathbb R_+)$ with the integral kernels of the form $a(x+y)$. We will consider a class of maps from integral Hankel operators to Hankel matrices, which we call restriction maps. In the simplest case, such a map is simply a restriction of the integral kernel onto integers. More generally, it is given by an averaging of the kernel with a sufficiently regular weight function. In this talk, we will describe the boundedness of certain restriction maps with respect to the operator norm and the Schatten norms. If time permits, we will also discuss the boundedness of a converse operation, an extension of a matrix to an integral kernel.

This is joint work with Alexander Pushnitski.

The infinite Hilbert matrix $\mathcal{H} = \left(\frac{1}{j+k+1}\right)_{j, k = 0}^\infty$ can be interpreted as a linear operator on spaces of analytic functions in the open unit disc of the complex plane by its action on their Taylor coefficients. The boundedness of $\mathcal{H}$ on the Hardy spaces $H^p$ for $1 \lt p \lt \infty$ and Bergman spaces $A^p$ for $2 \lt p \lt \infty$ was established by Diamantopoulos and Siskakis. The exact value of the norm of $\mathcal{H}$ acting on the Bergman spaces $A^p$ for $4 \le p \lt \infty$ was shown to be $\frac{\pi}{\sin(\frac{2\pi}{p})}$ by Dostaniç, Jevtiç and Vukotiç in 2008. The case $2 \lt p \lt 4$ was an open problem until in 2018 it was shown by Božin and Karapetroviç that the norm has the same value also on the scale $2 \lt p \lt 4$. In this talk, we review some of the old results and consider the still partly open problem regarding the value of the norm on weighted Bergman spaces. The talk is based on a joint work with Lindström and Wikman (Åbo Akademi).

Distributed-order fractional non-local integral operators have been introduced and studied by Caputo at the end of the 20th century. They generalize fractional order derivatives/integrals in the sense that such operators are defined by a weighted integral of different orders of differentiation over a certain range. The subject of distributed-order non-local operators is currently under strong development due to its applications in modeling some complex real world phenomena. Fractional optimal control theory deals with the optimization of a performance integral functional subject to a fractional control system. One of the most important results in classical and fractional optimal control is the Pontryagin Maximum Principle, which gives a necessary optimality condition that every solution to the optimization problem must verify. In our work, we extend the fractional optimal control theory by considering dynamical systems constraints depending on distributed-order fractional derivatives. Precisely, we prove a version of Pontryagin’s maximum principle and a sufficient optimality condition under appropriate convexity assumptions.

This research is partially supported by FCT within the R&D unit UID/MAT/04106/2019 (CIDMA).

A recent generalization of the theory of fractional calculus is to allow the fractional order of the derivatives to be dependent on time. As a result, introducing numerical methods for finding approximations of variable-order fractional integrals and derivatives of a given function is an important feature in presenting new numerical methods for solving variable-order fractional differential equations. In this talk, we consider the left Riemann--Liouville fractional integral operator of order $\alpha(t)$ of a given function $y$ on $[0,\tau]$ which is defined by \[ _0I_t^{\alpha(t)}y(t)=\frac{1}{\Gamma{(\alpha(t))}}\int_0^t(t-s)^{\alpha(t)-1}y(s)ds,\quad t>0, \] where $\Gamma(\cdot)$ is the Euler gamma function. By considering the Newton--Cotes equal distance points and a suitable set of basis functions, an approximation of the aforementioned integral is given. We find an error for this approximation.

In recent years, there has been much interest in limit operators. For example in the most easy case, given a bounded linear operator $A$ on $\ell^2(\mathbb{Z})$, we can think of $A$ a an bi-infinite matrix (with respect to the canonical basis in $\ell^2(\mathbb{Z})$). Then a limit operator of $A$ is given by a strong limit of shifts of the matrix $A$ along the main diagonal with respect to a seqeuence tending to $\pm\infty$. One can then characterize Fredholmness and spectral properties of $A$ by studying its limit operators. In this talk we explain how to generalizes the above method to rather general metric measure spaces of bounded geometry.

This is a joint work with Raffael Hagger.

Embedded eigenvalues often can be constructed by exploiting symmetries of an operator with continuous spectrum.  The symmetries induce multiple components of the continuous spectrum, and then appropriately chosen local defects create spectrally embedded eigenstates. It turns out that embedded eigenvalues due to local defects occur even in the absence of operator symmetries. The mechanism for periodic operators is the reducibility of the Fermi surface, which is a richer phenomenon than operator symmetry. I will show how this works for multi-layer graph operators; and I am interested in this question for the Neumann-Poincaré operator, building on recent work with Wei Li on embedded eigenvalues for the NP operator for symmetric domains.

The objective of this talk is to present some of the Rubia de Francia type extrapolation results in the framework of generalized weighted grand Lebesgue spaces defined on $\Omega\subseteq \mathbb R^n$ with $|\Omega|<\infty.$ Here we shall be discussing the diagonal and off-diagonal cases, as well as their applications to study the boundedness of  fractional Reisz potetial operator and fractional Maximal operator.