### Session S05 - Advances on Spaces of Non-absolutely Integrable Functions and Related Applications

## Talks

Monday, July 12, 12:00 ~ 12:30 UTC-3

## Sobolev Orthogonal polynomials on the Sierpinski gasket ($SG$)

### Kasso Okoudjou

#### Tufts University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk I will introduce a theory of Sobolev orthogonal polynomials on $SG$. These orthogonal polynomials arise through the Gram-Schmidt orthogonalisation process applied on the set of monomials on $SG$ using several notions of a Sobolev inner products. After establishing some recurrence relations for these orthogonal polynomials, we give estimates for their $L^2, L^{\infty}$ and Sobolev norms, and study their asymptotic behaviour.

Joint work with Q. Jiang, T. Lan, R. Strichartz, S. Sule, S. Venkat, and and X. Wang.

Monday, July 12, 12:35 ~ 13:05 UTC-3

## Correct Hilbert Space for the Feymann formulation of quantum mechanics

### Tepper Gill

#### HowardUniversity, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

The Feynman formulation of quantum mechanics created two mathematical problems, which are dual of each other. The first problem was recognized immediately because there existed no theory that could provide meaning for the path integral suggested by Feynman. Approaches to this problem generated extensive research with inconclusive limited outcomes and has lost favor in the mathematics community. However, in the physics community, the study and applications of path integrals has expanded to include quantum liquids, quantum gravity and condensed matter physics to name a few. The dual problem is that the Feynman path integral cannot be formulated on $L^2[\mathbb{R}^n]$, the standard space for quantum mechanics.

The purpose of this talk is to introduce the Kuelbs-Steadman Hilbert space, $KS^2[\mathbb{R}^n]$. This separable Hilbert space contains all non-absolutely integrable functions, the space of distributions $\mathcal{D}'$ and $L^2[\mathbb{R}^n]$ as a continuous embedding. Showing that both the convolution and Fourier transform extend to bounded linear operators on $KS^2[\mathbb{R}^n]$ is sufficient for the Feynman formulation of quantum mechanics and the construction of the path integral in the manner suggested by Feynman

Monday, July 12, 13:10 ~ 13:40 UTC-3

## On the equivalence between Weak BMO and the space of derivatives of the Zygmund class

### Eddy Kwessi

#### Trinity University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this paper, we will discuss the space of functions of weak bounded mean oscillation. In particular, we will show that this space is the dual space of the special atom space, whose dual space was already known to be the space of derivative of functions (in the sense of distribution) belonging to the Zygmund class of functions. We show in particular that this proves that the Hardy space $H^1$ strictly contains the space special atom space.

Monday, July 12, 13:45 ~ 14:15 UTC-3

## Haar, Wavelets System, Multi-Resolutions and The special atom space in Higher Dimensions and its Analytic Characterizations

### Geraldo De Souza

#### Auburn University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this presentation, we will explore the special atom spaces introduced by De Souza in 1980 in his Ph.D thesis. The impetus of this exploration to extend to higher dimension the definition originally proposed by De Souza. This leads to a natural “new” definitions of Haar System and Wavelets in Higher dimensions, even though these definitions has been discussed by numerous authors in the literature, the definitions proposed do not always seems natural extension of one dimensions case and often are unnecessarily cumbersome and difficulty to follows. Also, we explore the analytic characterizations of these special atom spaces in Higher Dimensions, which answer a question proposed by Brett Wick on the spaces of certain analytic functions spaces in the bidisc. We also have some applications of Haar-Wavelets and a Multi-resolution analysis in 2-dimensions

Joint work with Eddy Kwessi (Trinity University, USA).

Monday, July 12, 14:20 ~ 14:50 UTC-3

## A Constructive Definition of the Fourier Transform on a Separable Banach Space

### Timothy Myers

#### Howard University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Gill and Myers proved that every separable Banach space, denoted $\mathcal{B}$, has an isomorphic, isometric embedding in $\mathbb{R}^{\infty}=\mathbb{R}\times\mathbb{R}\times\cdots$ . They used this result and a method due to Yamasaki to construct a sigma-finite Lebesgue measure $\lambda_{\mathcal{B}}$ for $\mathcal{B}$\ and defined the associated integral $\int_{\mathcal{B}}\cdot d\lambda_{\mathcal{B}}$ in a way that equals a limit of finite-dimensional Lebesgue integrals.

The objective of this talk is to develop a constructive definition of the Fourier transform on $L^1[\mathcal{B}]$. Our approach is constructive in the sense that this Fourier transform is defined as an integral on $\mathcal{B}$, which, by the aforementioned definition, equals a limit of Lebesgue integrals on Euclidean space as the dimension $n\to\infty$. Because this transform has all of the familiar properties, we will use these to derive the solution to the diffuse equation on $\mathcal{B}$.

Monday, July 12, 14:55 ~ 15:25 UTC-3

## An application of a theorem of Kuelbs' to a characterization of complemented subspaces of a separable Banach space

### Douglas Mupasiri

#### University of Northern Iowa, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we apply a theorem of Kuelbs' on continuously and densely embedding separable Banach spaces in Hilbert spaces to obtain a necessary and sufficient condition for a closed, proper subspace of a separable Banach space to be complemented. To motivate our approach we first show that $c_0$ is dense in $\ell_\infty$ with respect to the topology of a Hilbert space, $\mathcal{H}$, containing $\ell_\infty$. We use this fact to give an alternative proof to the well-known result, due to R. S. Phillips (1940), that $c_0$ is not complemented in $\ell_{\infty}$. We apply the idea of our proof of this result to show that if $A$ is a separable Banach space and $B$ is a closed subspace of $A$, then $B$ is complemented in $A$ if and only if $B$ is not dense in $A$ with respect to the topology of the (separable) Kuelbs Hilbert space determined by $A$

Joint work with Tepper Gill (Howard University, USA).

Monday, July 12, 15:30 ~ 16:00 UTC-3

## A note on localizable Hardy-Morrey spaces and Fourier transform decay

### Marcelo De Almeida

#### Univeridad Federal de Sergipe, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we reintroduce the localizable Hardy-Morrey spaces establishing an atomic decomposition and decay estimates for Fourier transform motivated by works of D. Goldberg (1979) and J. Hounie and R. Rapp (2009). As application , we study the continuity of pseudodifferential operators in the class $OpS^{0}_{1,\delta}(\mathbb{R}^n)$ for $0 \leq \delta<1$ on localizable Hardy-Morrey spaces {as well as its optimality}.

Joint work with Tiago Picon USP - Ribeir\~ao Preto.

Monday, July 12, 16:05 ~ 16:35 UTC-3

## Fourier Analysis within a frame of the generalized integral

### Maria Guadalupe Morales Macia

#### Masaryk University, Czech Republic - This email address is being protected from spambots. You need JavaScript enabled to view it.

This talk is devoted to present new fundamental properties of the Fourier transform dened over spaces of non-absolutely integrable functions. For example, we consider functions in classical subspaces, as non-absolutely integrable functions in $L^p(\mathbb{R})$, to obtain e.g. continuity, differentiability, and asymptotic behavior of the Fourier transform of such functions.

Monday, July 12, 16:40 ~ 17:10 UTC-3

## The L p primitive integral on the real line

### Erik Talvila

#### University of the Fraser Valley, Canada - The $L^p$ primitive integral on the real line

For each $1\leq p<\infty$ a space of integrable Schwartz distributions, $L{\!}'^{\,p}$, is defined by taking the distributional derivative of all functions in $L^p$. Here, $L^p$ is with respect to Lebesgue measure on the real line. If $f\in L{\!}'^{\,p}$ such that $f$ is the distributional derivative of $F\in L^p$ then the integral is defined as $\int^\infty_{-\infty} fG=-\int^\infty_{-\infty} F(x)g(x)\,dx$, where $g\in L^q$, $G(x)= \int_0^x g(t)\,dt$ and $1/p+1/q=1$. A norm is $\lVert f\rVert'_p=\lVert F\rVert_p$. The spaces $L{\!}'^{\,p}$ and $L^p$ are isometrically isomorphic. Functions and distributions in $L{\!}'^{\,p}$ share many properties with functions in $L^p$. For example, $L{\!}'^{\,p}$ is reflexive, its dual space is identified with $L^q$, and there is a type of H\"older inequality. The $L^p$ primitive integral is able to integrate some functions with local singularities that are not locally integrable in the Lebesgue or Henstock--Kurzweil sense. Some applications are considered.

Monday, July 12, 17:15 ~ 17:45 UTC-3

## On Existence and Approximation of Solutions of Hammerstein Integral Equations

### Maaruf Minjibir

#### Bayero University, Nigeria - This email address is being protected from spambots. You need JavaScript enabled to view it.

Integral equations of Hammerstein type are intimately connected to Semilinear Elliptic Boundary Value Problems. In addition to the fact that such problems can be reduced to Hammerstein equations, the operator theoretic form of Hammerstein Equation is of interest in the theory of optimal control systems and in automation and network theory. Our talk concerns this form of Hammerstein equation. In particular, we shall discuss a little on the ex- istence of solutions, and dwell more on approximating such solutions since having a closed-form solutions is next to impossible due to the nonlinear- ity of the equation. We shall also touch on the multi-valued version of the Hammerstein equation, which, itself, nds applications, e.g., in the theory of thermostats.

Monday, July 12, 17:50 ~ 18:20 UTC-3

## On the use of path integration to solve Schrodinger equation}

### Marcia Federson

#### Universidade de São Paulo, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we recall the main ideas involved in the Feynman integral in order to address some issues which cause non absolute integration to come into scene. Then, we explain the foundations of the Henstock path integral and give an overview of the latest achievements

Joint work with Felipe Federson and Everaldo Bonotto (Universidade de São Paulo, Brazil).

Monday, July 12, 18:25 ~ 18:55 UTC-3

## Measure functional differential equations with state-dependent delay

### Jaqueline Mesquita

#### Universidade de Brazilia, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we will introduce the measure FDEs with state-dependent delays and their relation with generalized ODEs. Also, we will prove some applications.

Joint work with Hernán Henríquez (Universidad de Santiago de Chile, Chile) and Henrique dos Reis (Universidade de Brasília, Brazil).

Monday, July 12, 19:00 ~ 19:30 UTC-3

## Exponential Dichotomy for generalized ordinary differential equations

### Everaldo Bonotto

#### Universidade de São Paulo, Brazil

This talk deals with the theory of exponential dichotomy in the context of generalized ordinary differential equations. We study conditions for the existence of exponential dichotomies and bounded solutions. We apply the results for a class of measure differential equations

Monday, July 12, 19:35 ~ 20:05 UTC-3

## On the periodic solutions of measure functional differential equations

### Suzete Afonso

#### Universidade Estadual Paulista, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we present new contributions concerning periodic solutions of a class of measure functional differential equations connected to the non-absolute integration theory.

Joint work with Márcia Ritchielle (Universidade de São Paulo, Brazil) e Everaldo Bonotto (Universidade de São Paulo, Brazil).