IWOTA 2019

International Workshop on

Operator Theory and its Applications

IWOTA 2019

International Workshop on

Operator Theory and its Applications

July 22-26, 2019

Instituto Superior Técnico

Lisbon, Portugal

Instituto Superior Técnico

Lisbon, Portugal

- Shoshana Abramovich

Department of Mathematics, University of Haifa, Haifa, Israel ## Heinz Inequality, Superquadracity and Rearrangements

The purpose of this talk is to present several refinements and extensions of the well known Heinz and Jensen Inequalities. We use rearrangements techniques of the set (

**x**) involved in these inequalities and the superquadracity properties of the functions involved.- Alexandre Almeida

CIDMA, University of Aveiro ## Classical operators on different vanishing Morrey spaces

Certain vanishing properties defining new subspaces of Morrey spaces were recently introduced to describe the closure of nice functions in Morrey norm. This talk aims to show that those properties are preserved under the action of various classical operators of harmonic analysis, such as maximal, singular, potential and Hardy operators. This is based on joint work with A. Alabalik and S. Samko.

- Rizwan Anjum

Abdus Salam school of mathematical sciences and national center for mathematics Pakistan ## Rearrangement-Invariant Function and Interpolation Spaces

In recent years, there has emerged a new field of study in functional analysis: the theory of interpolation spaces. Interpolation theory has been applied to other branches of analysis, so it has also attracted considerable interest itself. Function spaces play an important role in both classical and modern mathematics. They are also useful tools for the study of ordinary and partial differential equations. The decreasing rearrangement received its first systematic treatment in the 1934 book of G. H. Hardy, J. E. Littlewood, and G. Polya. The theory of rearrangement-invariant function spaces is based on the decreasing rearrangement of a measurable function. In the talk, I will present an introduction to this theory with some examples. I will also discuss the close relation between rearrangement-invariant function spaces and interpolation spaces

- Yonggeun Cho

Chonbuk National University ## A Sobolev estimate for the adjoint restriction operator

In this talk we consider the adjoint restriction estimate for hypersurface under additional regularity assumption. We will see an optimal $H^s$-$L^q$ estimate and its mixed norm generalizations. As an application we provide some weighted Strichartz estimates for the propagator $\varphi\to e^{it(-\Delta)^{ \alpha/2}}\varphi$, $\alpha>0$. This talk is based on the joint work with Z. Guo and S. Lee.

- Stepanov Vladimir D

Computing Center, Far Eastern Branch of Russian Academy of Sciences; Steklov Mathematical Institute ## Characterization of associate function spaces

We analyse the problem of characterization of function spaces associated to a given function spaces. The situation is rather different for an ideal and non-ideal function spaces. We provide several examples of such a characterization including the weighted Sobolev space of the first order on the real line.

- Douadi Drihem

Department of Mathematics, M ## Spline representations of Triebel-Lizorkin spaces with general weights

In this talk we present new Triebel-Lizorkin type spaces of variable smoothness. Here the smoothness lies in a new weighted class which, in some particular cases, is just the Muckenhoupt class. We give some equivalents norms especially their characterizations via oscillations. The box spline and tensor-product B-spline are suitable to obtain stable representations of these function spaces.

- Javier Duoandikoetxea

University of the Basque Country UPV/EHU ## Boundedness of operators on weighted Morrey spaces

There are several definitions for weighted Morrey spaces. From the assumptions of the extrapolation theorem for weights (that is, Lebesgue weighted estimates with $A_p$ weights) we get estimates for different weighted Morrey norms, sometimes with restrictions in terms of the reverse Hölder inequalties satisfied by the weight. These results can be applied to obtain the boundedness on the corresponding weighted Morrey spaces of a variety of operators, together with the definition of the operator by embedding. For a fixed value of p the weighted inequalities go beyond the $A_p$ class and we get sharp general results for power weights. All the results are joint work with Marcel Rosenthal.

- Oscar Mauricio Guzmán Fonseca

Universidad Nacional de Colombia ## Weighted norm inequalities for the bilinear maximal operator on variable Lebesgue spaces

We extend the theory of weighted norm inequalities on variable Lebesgue spaces to the case of bilinear operators. We introduce a bilinear version of the variable $A_p(\cdot)$ condition, and show that it is necessary and sufficient for the bilinear maximal operator to satisfy a weighted norm inequality. Our work generalizes the linear results of Cruz-Uribe, Fiorenza and Neugebauer in the variable Lebesgue spaces and the bilinear results of Lerner

*et al*. in the classical Lebesgue spaces. As an application we prove weighted norm inequalities for bilinear singular integral operators in the variable Lebesgue spaces.- Ahmet Turan Gurkanli

Istanbul Arel University, Turkey ## New Properties of Grand Amalgam Spaces

In [14] (see arXiv:1811.08629v1[Math.FA] 21.Nov,2018), a new family called grand amalgam space $W(L^{p),\theta}, L^{q),\theta})$ of amalgam spaces was defined and investigated properties of these spaces. The present paper is a sequel to my work [14]. In this paper, notations are included in Section 1. In Section 2, we introduce another equivalent but discrete definition of grand amalgam space and study properties of these spaces. In Section 3, we determine necessary and sufficient conditions on a locally compact Abelian group G for the grand amalgam spaces $W(L^{p),\theta}, L^{q),\theta})(G)$ to be an algebra under convolution.

- Dorothee Haroske

Friedrich Schiller University Jena ## Compact embeddings of Smoothness Morrey spaces on bounded domains

We study compact embeddings between smoothness Morrey spaces on bounded domains and characterise its entropy numbers. In case of approximation numbers we also have some first results. We discover a new phenomenon in situations where the influence of the Morrey parameters is rather strong. Our argument relies on wavelet decomposition techniques of the function spaces and a careful study of the related sequence space setting, as well as sharp embeddings and interpolation. This also covers a parallel approach for Besov-type and Triebel-Lizorkin type spaces on domains. It is joint work with Leszek Skrzypczak (Poznan).

- Peter Hästö

University of Turku ## Maximal regularity for local minimizers of non-autonomous functionals

In this talk, I present a new and comprehensive approach to the regularity theory of minimizers of the non-autonomous functional\[u\mapsto \int_\Omega \phi(x,|Du|)\, dx\]where $\phi$ satisfies a $(p,q)$-growth condition. In contrast to earlier results over the past 30 years, we treat the variability of $\phi(x,t)$ with one unified condition, rather than separately in the $x$- and $t$-directions. Thus our results cover, to the best of our knowledge, all special cases of $(p,q)$-growth that have been studied in hundreds of papers since the 1980s, as well as many other cases. Furthermore, our condition is essentially optimal for maximal regularity in the non-autonomous case.

This is joint work with Jihoon Ok.

- Anders Holmbom

Mid Sweden University ## On connections between some concepts of convergence with multiple scales

We discuss the relation between the notions of two-scale convergence and unfolding including unfolding in a setting with an adjoint unfolding operator. They are traced back to a more general concept named operator-governed two-scale convergence. We introduce the fundamental concepts of two-scale convergence and unfolding in their simplest formulation based on functions bounded in $L^2$-spaces. Then we extend the discussion to also involve functions bounded in the corresponding Sobolev-spaces.

- Pankaj Jain

South Asian University ## Bilinear Hardy-type inequalities: continuous, discrete and quantum versions

The continuous form of the bilinear Hardy operatot is the following: \[H_2(f,g)(x)=\left(\int_0^x f\right)\left(\int_0^x g\right).\] In this talk, we shall discuss the weight characterization of the bilinear Hardy inequality \[\begin{split}& \left( \int_0^\infty[H_2(f,g)(x)]^q w(x)\,dx \right)^{1/q}\\ \le & \left( \int_0^\infty f^{p_1}(x) v_1(x)\,dx \right)^{1/p_1} \left( \int_0^\infty g^{p_2}(x) v_2(x)\,dx \right)^{1/p_2}\end{split}\] for various choices of the indices $p_1,p_2,q$. We shall also discuss the discrete version of the above inequality. Moreover, as an application of the discrete bilinear Hardy inequality, we derive the corresponding inequality in the framework of $q$-calculus. Finally, inequality of the type (1) will also be discussed for bilinear Hardy-Steklov operator.

- Haifa Bin Jebreen

King Saud University ## Fuzzy twisted sums of quasi-normed spaces

We introduce, in this work, the notion of fuzzy twisted sums of two quasi-Banach spaces $Y$ and $Z$ based on the notion of twisted sums, defined by Kalton and Peck (1978), as the space $X$ in the exact sequence $0 → Y → X → Z → 0$. They proved that a twisted sum of two quasi-normed spaces $Y$ and $Z$ can be defined by endowing the product space $Y×Z$ with the quasi-norm $‖(y,z)‖_F=‖y-F(z)‖+‖z‖$, where $F:Z→Y$ is a quasi-linear map. We examine the analogous properties of the twisted sums in the case of the fuzzy twisted sums $N((y,z),t)=t/(t+‖(y,z)‖_F )$ , such as triviality of a fuzzy twisted sum, equivalence of two fuzzy twisted sums. We also discuss the one to one correspondence between twisted sums and fuzzy quasilinear maps.

### References

- C. Alegre, S. Romaguera, On the uniform boundedness theorem in fuzzy quasi-normed spaces. Fuzzy Sets and Systems, 282 (2016) 143-153.
- S. M. Assev, Quasilinear operators and their application in the theory ofmultivalued mappings, Proc. Steklov Inst. Math. 2 (1986) 23–52.
- T. Bag, S. K. Samanta, Operator’s fuzzy norm and some properties. Fuzzy information and engineering, 7(2), (2015)151-164.
- F. Cabello Sanchez and J.M.F. Castillo, Duality and twisted sums of Banach spaces, Journal of Functional Analysis 175 (2000) 1–16.
- F. Cabello Sanchez and J. M. F. Castillo, Uniform boundedness and twisted sums of Banach spaces, Houston J. Math., 30 (2) (2004) 523–536.
- J. M. F. Castillo and M. Gonzalez, Three-space problems in Banach space theory, Springer Lecture Notes in Mathematics 1667 (1997).
- N. J. Kalton, and N. T. Peck, Twisted sums of sequence spaces and the three-space problem, Trans. Amer. Math. Soc. 255 (1979), 1–30
- S. Nadaban, Fuzzy continuous mappings in fuzzy normed linear spaces. International Journal of Computers Communications & Control, 10(6) (2015) 74-82.
- M. A. Rojas-Medar, M. D. Jiménez-Gamero, Y. Chalco-Cano, & A. J. Viera-Brandão, Fuzzy quasilinear spaces and applications. Fuzzy Sets and Systems, 152(2) (2005)173-190.

- Saikat Kanjilal

South Asian University, New Delhi, India ## Weighted bilinear inequalities of Hardy-type

The weight characterization of weighted bilinear Hardy inequality \begin{align} \label{H2} \left(\int_a^b\left( \int_a^xf(t)\, dt\right)^q \left( \int_a^xg(t)\, dt\right)^q u(x)dx\right)^\frac{1}{q}&\leq C\left(\int_a^b f^{p_1}(x)v_1(x)\, dt\right)^\frac{1}{p_1} \nonumber \\ & \quad \quad \times \left(\int_a^b g^{p_2}(x)v_2(x)\right)^\frac{1}{p_2} \end{align} has been obtained by Cañestro

*et al*. [1] and Křepela [5]. The aim in this presentation is to provide several scales of weight characterizations for ($\ref{H2}$) along with the weight characterizations of the higher dimensional and discrete version of the inequality ($\ref{H2}$), see [2, 3, 4].- M. I. Aguilar Cañestro, P. Ortega Salvador, C. Ramirez Torreblanca,
*Weighted bilinear Hardy inequalities*, J. Math. Anal. Appl., 387 (1), (2012), 320-334. - P. Jain, S. Kanjilal and L. -E. Persson,
*Hardy-type inequalities over balls in $\mathbb{R}^N$ for some bilinear and iterated operators*, J. Inequl. Spl. Func., to appear. - P. Jain, S. Kanjilal, V. D. Stepanov and G. E. Shambilova,
*Bilinear Hardy-type inequalities in discrete and q-calculus frameworks*, submitted. - S. Kanjilal, L.-E. Persson, G. E. Shambilova,
*Equivalent integral conditions related to bilinear Hardy-type inequalities*, Math. Inequal. Appl., to appear. - M. Křepela,
*Iterating bilinear Hardy inequalities*, Proc. Edin. Math. Soc., 60 (4), (2017), 955-971.

- M. I. Aguilar Cañestro, P. Ortega Salvador, C. Ramirez Torreblanca,
- Alexey Karapetyants

Southern Federal University ## A class of Hausdorff-Berezin operators on the unit disc

We introduce and study a class of Hausdorff-Berezin operators on the unit disc based on Haar measure (that is, the Möbius invariant area measure). We discuss certain algebraic properties of these operators and obtain boundedness conditions for them. We also reformulate the obtained results in terms of ordinary area measure. Joint work with Profs. K. Zhu and S. Samko.

- Oznur Ozkan Kilic

Baskent University ## On a subclass of the Janowski $\lambda$-spirallike functions of complex order

We consider a subclass of the Janowski-spirallike functions of complex order. The aim is to obtain some subordination results belonging to this class.

### References

- F. M. Al-Oboudi, M. M. Haidan, Spirallike function of complex order, J. Natur. Geom., 19 (2000), 53-72.
- M. K. Aouf, F. M. Al-Oboudi, M. M. Haidan, On some results for spiral-like and -Robertson functions of complex order, Publications De Lnstitut Mathematique, Nouvella serie, tome 75 (91) (2005), 93-98.
- Y. Polatoglu, A. Sen, Some results on subclasses of Janowski-spirallike functions of complex order, General Math., 15, 2-3 (2007), 88-97.

- Sebastian Król

Nicolaus Copernicus University, Toruń, Poland ## The identification of trace spaces via boundedness properties of the Hardy operator

Motivated by recent applications of weighted maximal regularity we study the initial values spaces corresponding to the first-order Cauchy problems. We show equivalence of the trace method and the K-method in this general setting, identify real interpolation spaces between a Banach space and the domain of a sectorial operator, and reprove an extension of Dore's theorem on the boundedness of holomorphic functional calculus. The boundedness properties of the Hardy operator on various function spaces are a crucial ingredient in the proofs of these results.

The talk is based on the joint work with Ralph Chill.

- Staffan Lundberg

UiT The Arctic University of Norway, Narvik ## Mixed Hardy-type operators in anisotropic Morrey spaces and some applications

We find conditions for the boundedness of mixed Hardy-type operators in weighted anisotropic Morrey spaces. Relation of mixed Hardy operators to certain degenerate hyperbolic partial differential equations is also discussed.

- Katsuo Matsuoka

College of Economics, Nihon University ## Generalized fractional integrals on $B_{\sigma}$-function spaces

Let $I_{\alpha}$ and ${\tilde I}_{\alpha}$ be a fractional integral and a modified fractional integral, respectively. Then, the following boundedness results are well-known: $I_{\alpha}$ is bounded from Morrey space $L_{p,\lambda}({\mathbb R}^n)$ to $L_{q,\mu}({\mathbb R}^n)$ and bounded from central Morrey space $B^{p,\lambda}({\mathbb R}^n)$ to $B^{q,\mu}({\mathbb R}^n)$; ${\tilde I}_{\alpha}$ is bounded from $L_{p,\lambda}({\mathbb R}^n)$ to Campanato space ${\mathcal L}_{q,\mu}({\mathbb R}^n)$ and bounded from $B^{p,\lambda}({\mathbb R}^n)$ to $\lambda$-central mean oscillation space ${\mathrm{CMO}}^{q,\mu}({\mathbb R}^n)$. In this talk, we will extend the boundedness results for ${\tilde I}_{\alpha}$ to the results for generalized fractional integral ${\tilde I}_{\alpha,d}$, using $B_{\sigma}$-function space.

- Susana D Moura

CMUC, University of Coimbra, Portugal ## Some embeddings of Morrey spaces with critical smoothness

We study embeddings of Besov-Morrey spaces ${\mathcal N}_{u,p,q}^s(\mathbb{R}^d)$ and of Triebel-Lizorkin-Morrey spaces ${\mathcal E}_{u,p,q}^s(\mathbb{R}^d)$ in the limiting cases when the smoothness $s$ equals $s_0=d\max(1/u-p/u,0)$ or $s_{\infty}=d/u$, which is related to the embeddings in $L_1^{\rm loc}(\mathbb{R}^d)$ or in $L_{\infty}(\mathbb{R}^d)$, respectively. When $s=s_0$ we characterise the embeddings in $L_1^{\rm loc}(\mathbb{R}^d)$ and when $s=s_{\infty}$ we obtain embeddings into Orlicz-Morrey spaces of exponential type and into generalised Morrey spaces.

This is joint work with Dorothee Haroske (Jena) and Leszek Skrzypczak (Poznan).

- Júlio Neves

CMUC, University of Coimbra ## Besov spaces involving only slowly varying smoothness, embeddings and optimality

We deal with (local) optimal embeddings of Besov spaces $B^{0,b}_{p,r}$ involving only a slowly varying smoothness $b$. Our target spaces are outside of the scale of Lorentz-Karamata spaces and are related with small Lebesgue spaces.

This is joint work with B. Opic, Charles University (Czech Republic).

- Liudmila Novikova

SFedU ## Bifurcations of fixed points of smooth mappings in Banach spaces

There are studied problems of branching of fixed points of nonlinear operators in a Banach space $U$.

By means of Poincare normalizing transformations there are given conditions for an analytical (in the Fréchet sense) nonlinear operator $N_\delta: U\oplus U$ to have a non-trivial fixed point for small $\delta: u(\delta)=N_\delta u(\delta)$, at a distance of order $o(\delta)$ from the fixed point $0=N_\delta(0)$.

- Ryskul Oinarov

L.N.Gumiliov Eurasian National University ## Oscillation and non-oscillation conditions of a class of fourth order differential equation

Let $I=(0,\infty)$ and $u$ be a continuous nonnegative function. Let positive functions $v$, $r$ be sufficiently times continuously differentiable on the interval $I$, and functions $v^{-1}=\frac{1}{v}$, $r^{-1}=\frac{1}{r}$ be integrable on the interval $(0,a)$ for any $a\gt 0$.

Consider the following fourth order differential equation \begin{equation}\label{eq1.1} D_{r}^2(v(t)D_{r}^2y(t))-u(t)y(t)=0,\,\,\,t\in{I}, \end{equation}where $D_{r}^2y(t)=\frac{d}{dt}r(t)\frac{dy(t)}{dt}$. For convenience, we assume that $D_{r}^1y(t)=r(t)\frac{dy(t)}{dt}$.

The solution of the equation (\ref{eq1.1}) is a function $y:~I\to R$ continuously differentiable with the functions $D_{r}^1y(t),~vD_{r}^2y$ and $r\frac{d}{dt}vD_{r}^2y$ on the interval $I$ that satisfies the equation (\ref{eq1.1}) for all $t\in{I}$.

The equation ($\ref{eq1.1}$) is called oscillatory [1, p.69] if for any $T\gt 0$ there exists a (non-trivial) solution of this equation, having more than one double zero to the right of $T$. Otherwise, the equation ($\ref{eq1.1}$) is called non-oscillatory.

In the mathematical literature in this area, most of the works is devoted to the oscillatory properties of linear, semilinear and nonlinear second-order differential equations (see, for example, [2] and references cited therein). However, such studies for fourth and higher order equations are relatively rare due to the fact that not all methods for the investigation of second-order equations can be extended to higher order equations (see, [3]).

The main aim of this paper is to study oscillatory or non-oscillatory properties of the equation ($\ref {eq1.1}$) in the neighborhood of infinity in more general situations than in the works [2, 3, 4] published in recent years.

From a joint paper with A. Zh. Adiyeva.

### References

- I.M. Glazman,
*Direct methods of qualitative analysis of singular differtial operators*, Jerusalem, 1965. - O. Dosly, P. Rehak,
*Half-linear differential equations*, North Holland Mathematics Studies, Vol. 202, Elsevier, Amsterdam 2005. - O. Dosly, V. Ruzicka,
*Nonoscillation criteria and energy functional for even-order half-linear two-term differential equations*, Electronic J. Dif. Equ., 2016, Vol. 2016, No. 95, 1-17. - R. Oinarov, S.Y. Rakhimova,
*Oscillation and nonoscillatorion of two terms linear and half-linear equations of higher order*, Electronic J. Qual. Theory Differ. Equ., 2010, Vol. 2010, No. 49, 1-15.

- I.M. Glazman,
- Kordan Ospanov

L.N. Gumiliov Eurasian National University ## $L_p$-theory of a second order differential operator with unbounded drift

It is known that in stochastic processes is widely used the following generalized Ornstein-Uhlenbeck operator $$Au=-\Delta u+ \nabla u \cdot b(x) + c(x)u,$$ where $x\in R^n$. The vector-valued function $b$ is called a drift. If $b=0$, then $A$ is the Schrodinger operator, which has been systematically studied for a long time. If $b\neq 0$ and it is unbounded, then the properties of $A$ are significally differ. For example, the well-posedness and spectral properties of $A$ depend on the relationship between the growth at infinity of $|b|$ and the norm of $c$. For more details, we refer to papers of A. Lunardi and V. Vespri (1997), G. Metafune (2001) and P. J. Rabier (2005).

In this talk we discuss the $L_p -$ maximal regularity estimate and discreteness spectrum criterion for a following one-dimentional operator $$Ly=-y''+r(x)y'+q(x)y,$$ where $1\lt p \lt \infty$ and $r$ is a continuously differentiable function. We assume the growth of $r$ not depends on $q$. To get the desired result, we first give sufficient conditions for the bounded invertibility of $L$.

- Bae Jun Park

Korea Institute for Advanced Study ## Equivalences of (quasi-)norms in certain vector-valued function spaces and their applications to multilinear operators

In this talk we will study some (quasi-)norm equivalences, involving $L^p(l^q)$ norm, in a certain vector-valued function space and extend the equivalences to $p = \infty$ and $0 \lt q \lt ∞$ in the scale of Triebel-Lizorkin spaces. As an immediate consequence of our results, BMO norm of $f$ can be written as $L^\infty(l^2)$ norm of a variant of $f$. We will also discuss some applications to multilinear Hormander multiplier and multilinear pseudo-differential operator of type $(1,1)$.

- Tao Qian

Macau University of Science and Technology ## Results and extreme examples of Blaschke products and impacts to operator theory and signal analysis

The talk will show some new results and constructed extreme examples of Blaschke products in relation to their angular derivatives (boundary phase derivatives). Impacts of the results and examples to operator theory and signal analysis are presented.

- Stefan Samko

Universidade do Algarve, Portugal ## Boundedness of fractional operators from variable exponent Morrey spaces to BMO or VMO in the limiting case

We prove the boundedness of the fractional integration operator of variable order in the limiting Sobolev-Adams case from variable exponent Morrey space into BMO, on bounded open sets. We also show the boundedness from variable exponent vanishing Morrey spaces into VMO in the same Sobolev-Adams limiting case. The results are new even when in the case of constant parameters.

- Igor Sheipak

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics ## On Hölder exponents of fractal functions. Application to some spectral problems.

We consider some class of fractal continuous non-decreasing functions on interval $[0,1]$. The corresponding measure $\rho$ is defined as $\rho:=dP$. The main property of function $P$ from this class is following: there exists the set $A$ of full measure ($\rho(A)=\rho[0;1]$) such that the Hölder exponent of the restriction of function $P$ on $A$ coincides with the Hausdorff dimension of the support of measure $\rho$.

We also consider spectral problem $$-y''=\lambda \rho y,\quad y(0)=y(1)=0$$ with mentioned measure $\rho$ as the weight. The asymptotics of the counting function $N(\lambda)$ of the eigenvalues of the problem can be established via Hölder exponents of the function $P$.

We obtaine the full description of the Hölder exponents of the continuous affine self-similar functions through self-similarity parameters.

The example of not affine self-similar function (Minkowski function) is also considered.

- Harpal Singh

UiT, The Arctic University of Norway ## Recent Trends in Signal Processing Techniques for Structural Health Monitoring

Signal processing is the most crucial component for the vibration-based structural health monitoring (SHM). The signal processing techniques convert time-based signals to the frequency-domain signals which give more detailed information about the non-stationary signals. The main objective of the signal processing techniques we discuss in this lecture is to extract minute changes in the vibration signals with the perspective to detect, locate and quantify the damage in the actual structure (e.g. bridge, dams or tunnels).In this lecture, I present some different signal processing techniques and their applications in SHM with concrete examples. In particular, some very new signal processing techniques that can play an important role for future research are presented. The major challenge for damage detection is the analysis of a large amount of noisy data collected by various sensors. The new techniques should be able to handle the noisy sensor data effectively and be computationally efficient.

- George Tephnadze

The University of Georgia, School of Science and Technology ## Convergence and summability of the one- and two-dimensional Vilenkin-Fourier series in the martingale Hardy spaces

The classical theory of Fourier series deals with decomposition of a function into sinusoidal waves. Unlike these continuous waves the Walsh functions are rectangular waves. Such waves have already been used frequently in the theory of signal transmission, multiplexing, filtering, image enhancement, codic theory, digital signal processing and pattern recognition. The development of the theory of Vilenkin-Fourier series has been strongly influenced by the classical theory of trigonometric series. Because of this it is inevitable to compare results of Vilenkin series to those on trigonometric series. There are many similarities between these theories, but there exist differences also. Much of these can be explained by modern abstract harmonic analysis, which studies orthonormal systems from the point of view of the structure of a topological group.

This lecture is devoted to review theory of martingale Hardy spaces. We present central theorem about atomic decomposition of these spaces and show how this result can be used to derive necessary and sufficient conditions for the modulus of continuity such that partial sums and some classical summability methods with respect to one- and two-dimensional Walsh-Fourier series converge in norm. Moreover, we also present some strong convergence theorems of partial sums and some classical summability methods of the one- and two-dimensional Walsh system.

**References**- I. Blahota, L.E. Persson, G. Tephnadze, On the Norlund means of Vilenkin-Fourier series, Czech. Math J., 65, 4 (2015), 983-1002.
- D. Lukkassen, L.E. Persson, G. Tephnadze, G. Tutberidze, Some inequalities related to strong convergence of Riesz logarithmic means of Vilenkin-Fourier series, (submitted).
- N. Memic, L. E. Persson, G. Tephnadze, A note on the maximal operators of Vilenkin-Nörlund means with non-increasing coefficients, Stud. Sci. Math. Hung., 53, 4, (2016) 545-556.
- L. E. Persson, G. Tephnadze, A sharp boundedness result concerning some maximal operators of Vilenkin-Fejér means, Mediterr. J. Math., 13, 4 (2016) 1841-1853.
- I. Blahota, K. Nagy, L. E. Persson, G. Tephnadze, A sharp boundedness result concerning some maximal operators of partial sums with respect to Vilenkin systems, Georgian Math., J., DOI: https://doi.org/10.1515/gmj-2018-0045.

- Francesco Tschinke

Palermo University ## Distributions frames and bases

Distribution valued functions are considered and some conditions are investigated for them to constitute a

*continuous basis*for the smallest space $\mathcal D$ of a rigged Hilbert space $\mathcal D\hookrightarrow\mathcal H\hookrightarrow\mathcal D^\times$.This analysis requires suitable extensions of familiar notions as those of frames, Riesz bases and orthonormal bases. Furthermore, given a pair of distributions functions, the notion of

*compatible pairs*is introduced, and it is shown that it is a generalization of*reproducing pairs*, introduced in [1, 2] and studied in [3, 4]. This work come from a joint venture with C. Trapani and S. Triolo and me [5].### References

- M. Speckbacher and P. Balazs,
*Reproducing pairs and the continuous nonstationary Gabor transform on LCA groups*. J. Phys. A: Math. Theor.,**48**(2015) 395201. - M. Speckbacher and P. Balazs,
*Reproducing pairs and Gabor systems at critical density*, J. Math. Anal. Appl.**455**(2017), 1072-1087. - J.-P. Antoine, M. Speckbacher and C. Trapani,
*Reproducing pairs of measurable functions*, Acta Applicandae Math.**150**(2017). - J.-P. Antoine and C. Trapani,
*Reproducing pairs of measurable functions and partial inner product spaces*, Advances in Operator Theory**2**(2017) 126-146. - C. Trapani, S. Triolo, F. Tschinke,
*Distributions frames and basis*, J. Fourier Anal. Appl. https://doi.org/10.1007/s00041-018-09659-5

- M. Speckbacher and P. Balazs,
- Rumen Uluchev

Faculty of Mathematics and Informatics, Sofia University St. Kliment Ohridski ## On Hardy $L_2$ Inequality in Certain Finite Dimensional Spaces

Let ${\cal P}_n$ be the set of real-valued algebraic polynomials of degree not exceeding $n$ and let ${\cal H}_n:=\{f\,:\,f(x)=e^{-x/2}\,p(x),\ p\in {\cal P}_n\}$.

We examine the best (i.e. the smallest possible) constant $c_n$ in the $L_2$ Hardy inequality $$\int_{0}^{\infty} \Bigg(\frac{1}{x} \int_{0}^{x}f(t)\,dt\Bigg)^2 dx \le c_n \int_{0}^{\infty}f^2(x)\,dx,\qquad f\in {\cal H}_n.$$

Our main result is two-sided estimate for $c_n$ of the form $$4-\frac{c}{\ln n}\lt c_n \lt 4-\frac{c}{\ln^2 n},\qquad c\gt 0.$$ It confirms the expected $\lim_{n\to\infty}c_n=4$, showing however that the convergence speed is rather slow.

We prove essentially the same two-sided estimates for the sharp constant $d_n$ in the discrete Hardy inequality for finite sequences

\[ \sum_{k=1}^n \Big(\frac{1}{k}\sum_{j=1}^k a_j\Big)^2 \le d_n \sum_{k=1}^n a_k^2. \]

The talk is based on a joint work with Dimitar K. Dimitrov (State University of Sao Paulo UNESP, Brazil), Ivan Gadjev and Geno Nikolov (Sofia University St. Kliment Ohridski, Bulgaria).- Salaudin Umarkhadzhiev

Kh. Ibragimov Complex Institute of the Russian Academy of Sciences ## Grand Lebesgue spaces on sets of infinite measure

We study operators of harmonic analysis in grand Lebesgue spaces on sets of infinite measure. To define such grand spaces, we introduce the notion of so called grandizer in terms of integrability of which we get a criterion for grand Lebesgue space to extend the classical Lebesgue space. We obtain a general statement for the boundedness of a class of sublinear operators, which is applied to a number of concrete operators of harmonic analysis such as maximal, potential, Hardy operators and others.

- Elena Ushakova

Computing Center of FEB RAS, Khabarovsk, Russia; Steklov Mathematical Institute of RAS, Moscow, Russia ## Images of integration operators in weighted function spaces

Images of integration operators of natural orders are considered as elements of Besov and Triebel-Lizorkin spaces with Muckenhoupt weights on $\mathbb{R}^N$. We study connections between norms of images and pre-images of these operators. The results connect estimates for entropy numbers of embedding operators with the same characteristics of the integration operators in weighted Besov and Triebel-Lizorkin spaces.

The research was supported by the Russian Science Foundation (project 19-11-00087).

- Ferenc Weisz

Department of Numerical Analysis, Eötvös Loránd University ## Variable Hardy spaces and Fourier analysis

Let $p(\cdot): \mathbb{R}^n\to(0,\infty)$ be a variable exponent function satisfying the globally log-Hölder condition. We introduce the variable Hardy spaces $H_{p(\cdot)}(\mathbb{R})$ and $H_{p(\cdot)}[0,1)$ and give their atomic decompositions. It is proved that the maximal operator of the Fejér means of the Fourier transforms and Walsh-Fourier series is bounded on these spaces. This implies some norm and almost everywhere convergence results for the Fejér-means, amongst others the generalization of the well known Lebesgue's theorem.

- Umme Zainab

Aligarh Muslim University India ## Lacunary Generating Function for 2-variable and 3-variable Hermite Polynomial: Umbral and Operational Formalism

In this paper, we employ an umbral formalism to reformulate the theory of 3-variable Hermite polynomials. The 3-variable associated the umbral Hermite polynomials and the 3-variable 2-parameters Hermite polynomials are introduced by exploiting the umbral formalism. Further Triple lacunary generating function for 2-variable Hermite polynomials are found. Also the series definition and operational definitions are obtained. Further, the higher order 3-variables 2-parameters Hermite polynomials are introduced and studied by using the techniques of umbral calculus. some concluding remarks are also given.