# Analysis

### Previous Lectures

Euclidean lattice points counting problems, the primordial example of which is the Gauss circle problem, are an important topic in classical analysis. Their non-Euclidean analogs in irreducible symmetric spaces (such as hyperbolic spaces and the space of positive-definite symmetric matrices) are equally significant, and we will present an approach to establishing such results in considerable generality. Our method is based on dynamical arguments together with representation theory and non-commutative harmonic analysis, and produces the current best error estimate in the higher rank case. We will describe some of the remarkably diverse applications of lattice point counting problems, as time permits.

In this talk we discuss asymptotic relations between sharp constants of approximation theory

in a general setting. We first present a general model that includes a circle of problems of

finding sharp or asymptotically sharp constants in some areas of univariate and multivariate

approximation theory, such as inequalities for approximating elements, approximation of individual

elements, and approximation on classes of elements. Next we discuss sufficient conditions that

imply limit inequalities and equalities between various sharp constants. Finally, we present

applications of these results to sharp constants in Bernstein-V. A. Markov type inequalities of

different metrics for univariate and multivariate trigonometric and algebraic polynomials and

entire functions of exponential type.

I will discuss how large the Hausdorff dimension of a set $E\subset{\mathbb R}^d$ needs to be

to ensure that it contains vertices of an equilateral triangle. An argument due to Chan, Laba

and Pramanik (2013) implies that a Salem set of large Hausdorff dimension contains equilateral

triangles. We prove that, without assuming the set is Salem, this result still holds in dimensions

four and higher. In ${\mathbb R}^2$, there exists a set of Hausdorff dimension $2$ containing no

equilateral triangle (Maga, 2010).

I will also introduce some interesting parallels between the triangle problem in Euclidean space

and its counter-part in vector spaces over finite fields. It is a joint work with Alex Iosevich.

Tame dynamical systems were introduced by A. K\"{o}hler in 1995 and their theory was

developed during last decade in a series of works by several authors. Connections to

other areas of mathematics like: Banach spaces, model theory, tilings, cut and project

schemes were established. A metric dynamical $G$-system $X$ is tame if every element

$p \in E(X)$ of the enveloping semigroup $E(X)$ is a limit of a sequence of elements

from $G$. In a recent joint work with Eli Glasner we study the following general question:

which finite coloring $G \to \{0, \dots ,d\}$ of a discrete countable group $G$ defines a

tame minimal symbolic system $X \subset \{0, \dots ,d\}^G$. Any Sturmian bisequence

$\Z \to \{0,1\}$ on the integers is an important prototype.

As closely related directions we study cutting coding functions coming from circularly ordered

systems. As well as generalized Helly's sequential compactness type theorems about families

with bounded total variation. We show that circularly ordered dynamical systems are tame and

that several Sturmian like symbolic $G$-systems are circularly ordered.

Linear algebra over a field have been studied for centuries. In many branches of math one

faces matrices over a ring, these came e.g. as "matrices of functions" or "matrices depending

on parameters". Linear algebra over a (commutative, associative) ring is infinitely more

complicated. Yet, some particular questions can be solved.

I will speak about two problems: block-diagonalization (block-diagonal reduction) of matrices

and stability of matrices under perturbations by higher-order-terms.

Finitely additive measures on convex convex sets are called valuations. Valuations continuous in

the Hausdorff metric are of special interest and have been studied in convexity for a long time.

In this talk I will present a non-traditional method of constructing continuous valuations using

various Monge-Ampere (MA) operators, namely the classical complex MA operator and introduced by

the speaker quaternionic MA operators (if time permits, I will briefly discuss also octonionic case).

In several aspects analytic properties of the latter are very similar to the properties of the former,

but the geometric meaning is different. The construction of the quaternionic MA operator uses

non-commutative determinants.

By a crystalline measure in R^d one means a measure whose support and

spectrum are both discrete closed sets. I will survey the subject and

discuss recent results obtained jointly with Alexander Olevskii.

Differential inequalities and their connection to normality (and quasi normality) have been studied since Marty’s Theorem in 1935. We discuss when these inequalities imply some degree of normality, and present a new result, joint with S. Nevo and J. Grahl.

Following Marty's Theorem we present recent results about differential inequalities that imply (or not) some degree of normality. We deal with inequalities with reversed sign of inequality than that in Marty's Theorem, i.e. $|f^(k)(z)|> h(|f(z))$.

Following Marty's Theorem we present recent results about differential inequalities that imply (or not) some degree of normality. We deal with inequalities with reversed sign of inequality than that in Marty's Theorem, i.e. $|f^(k)(z)|> h(|f(z))$.

Earlier and recent one-dimensional estimates and asymptotic relations for the

cosine and sine Fourier transform of a function of bounded variation are refined

in such a way that become applicable for obtaining multidimensional asymptotic

relations for the Fourier transform of a function with bounded Hardy variation.

Selfadjoint extensions of a closed symmetric operator A in a Hilbert

space with equal deficiency indices were described by in the 30s by

J. von Neumann. Another approach, based on the notion of abstract boundary

triple originates to the works of J.W. Calkin and was developed by M.I. Visik,

G.Grubb, F.S.Rofe-Beketov, M.L.Gorbachuck, A.N.Kochubei and others.

By Calkin's approach all selfadjoint extensions of the symmetric operator A can

be parametrized via "multivalued" selfadjoint operators in an auxiliary Hilbert spaces.

Spectral properties of these extensions can be characterized in terms of the abstract

Weyl function, associated to the boundary triple. In the present talk some recent

developments in the theory of boundary triples will be presented. Applications to

boundary value problems for Laplacian operator in bounded domains with smooth and

rough boundaries will be discussed.

A Muskat problem describes an evolution of the interface $\Gamma (t)\subset{\mathbb R}^{2}$ between two immiscible fluids, occupying regions $\Omega _1$ and $\Omega _2$ in a Hele-Shaw cell. The interface evolves due to the presence of sinks and sources located in $\Omega _j$, $j=1,2$.

The case where one of the fluids is effectively inviscid, that is, it has a constant pressure, is called

one-phase problem. This case has been studied extensively. Much less progress has been made for the two-phase problem, the Muskat problem.

The main difficulty of the two-phase problem is the fact that the pressure on the interface, separating the fluids, is unknown. In this talk we introduce a notion of a two-phase mother body (the terminology comes from the potential theory) as a union of two distributions $\mu _j$ with integrable densities of sinks and sources, allowing to control the evolution of the interface, such that $\rm{supp}\, \mu _j \subset\Omega _j$. We use the Schwarz function approach and the introduced two-phase mother body to find the evolution of the curve $\Gamma (t)$ as well as two harmonic functions $p_j$, the pressures, defined almost everywhere in $\Omega_j$ and satisfied prescribed boundary conditions on $\Gamma (t)$.

A bounded set O in R^d is called spectral if the space L^2(O) admits an orthogonal basis consisting of

exponential functions. In 1974 Fuglede conjectured that spectral sets can be characterized geometrically

by their ability to tile the space by translations. Although since then spectral sets have been intensively

studied, the connection between spectrality and tiling is still unresolved in many aspects.

I will focus on cylindric sets and discuss a new result, joint with Nir Lev, on the spectrality of such sets.

Since also the tiling analogue of the result holds, it provides a further evidence of the strong connection

between these two properties.

In this talk we survey several recent results establishing a pointwise domination of Calder\'on-Zygmund

operators by sparse operators defined by

$${\mathcal A}_{\mathcal S}f(x)=\sum_{Q\in {\mathcal S}}\Big(\frac{1}{|Q|}\int_Qf\Big)\chi_{Q}(x),$$

where ${\mathcal S}$ is a sparse family of cubes from ${\mathbb R}^n$.

In particular, we present a simple proof of M. Lacey's theorem about Calder\'on-Zygmund operators

with Dini-continuous kernels in its quantitative form obtained by T. Hyt\"onen-L. Roncal-O. Tapiola.

Evans and Portilheiro introduced in 2004 the functional framework that allows to tackle

the problem of a forward-backward diffusion equation with a cubic-like diffusion function,

that is classically ill-posed. The key is to consider its ``entropy'' formulation

determined by considering the equation as the singular limit of a third-order

pseudo-parabolic equation. Obtaining numerical simulations is not easy, since

the ill-posedness related to the negativity of the diffusion coefficient induces

severe oscillations. However, we showed that, in 1D, the regularization offered by

the basic Euler in time-centered finite differences in space renders a fairly

good numerical solution, except for the fact that the entropy condition is

violated. We thus proposed an adapted entropic scheme in 1D. The finite volume framework

has since allowed us to prove new properties of the problem.

The theory of real Hardy spaces has been applied to the study of partial

differential equations in many different contexts. In the 1990's, one of main results

in this direction was the div-curl lemma of Coifman, Lions, Meyer and Semmes. We

discuss some variants of this lemma in the context of the local Hardy spaces of Goldberg,

and of weighted Hardy spaces. This is joint work with Der-Chen Chang and Hong Yue.

For the nondegenerate Beltrami equations in the quasidisks and, in particular, in smooth

Jordan domains, we prove the existence of regular solutions of the Riemann–Hilbert problem

with coefficients of bounded variation and boundary data that are measurable with respect

to the absolute harmonic measure (logarithmic capacity).

For a system of polynomials orthonormal with weight on a curve in the complex plane,

the problem of sharp estimates of these polynomials is of considerable importance.

We discuss known conditions and inequalities and present certain refinements of them.

For functions $f(x_{1},x_{2})=f_{0}\big(\max\{|x_{1}|,|x_{2}|\}\big)$ from

$L_{1}(\mathbb{R}^{2})$, sufficient and necessary conditions for the belonging of their Fourier transform

$\widehat{f}$ to $L_{1}(\mathbb{R}^{2})$ as well as of a function $t\cdot \sup\limits_{y_{1}^{2}+y_{2}^{2}\geq

t^{2}}\big|\widehat{f}(y_{1},y_{2})\big|$ to $L_{1}(\mathbb{R}^{1}_{+})$. As for the positivity of $\widehat{f}$ on

$\mathbb{R}^{2}$, it is completely reduced to the same question on $\mathbb{R}^{1}$ for a function

$f_{1}(x)=|x|f_{0}\big(|x|\big)+\int\limits_{|x|}^{\infty}f_{0}(t)dt$.

It is quite obvious that one should expect that the degree of constrained approximation

be worse than the degree of unconstrained approximation. However, it turns out that in certain cases

we can deduce the behavior of the degrees of the former from information about the latter.

Let $E_n(f)$ denote the degree of approximation of $f\in C[-1,1]$,

by algebraic polynomials of degree $<n$, and assume that we know

that for some $\alpha>0$ and $\Cal N\ge1$,

$$n^\alpha E_n(f)\leq1,\quad n\geq\Cal N.$$

Suppose that $f\in C[-1,1]$, changes its monotonicity or convexity $s\ge0$ times in $[-1,1]$ ($s=0$ means that $f$

is monotone or convex, respectively). We are interested in what may be said about its degree of

approximation by polynomials of degree $<n$ that are comonotone or coconvex with

$f$. Specifically, if $f$ changes its monotonicity or convexity at

$Y_s:=\{y_1,\dots,y_s\}$ ($Y_0=\emptyset$) and the degrees of comonotone and coconvex approximation

are denoted by $E^{(q)}_n(f,Y_s)$, $q=1,2$, respectively. We investigate when can one say that

$$n^\alpha E^{(q)}_n(f,Y_s)\le c(\alpha,s,\Cal N),\quad n\ge\Cal N^*,$$

for some $\Cal N^*$. Clearly, $\Cal N^*$, if it exists at all (we prove it

always does), depends on $\alpha$, $s$ and $\Cal N$. However, it turns

out that for certain values of $\alpha$, $s$ and $\Cal N$, $\Cal N^*$ depends also

on $Y_s$, and in some cases even on $f$ itself, and this dependence is essential.

Recent decades brought increasing interest in Finsler spaces $(M,F)$,

especially, in extrinsic geometry of their hypersurfaces.

Randers metrics (i.e., $F=\alpha+\beta$, $\alpha$ being the norm of a Riemannian structure

and $\beta$ a 1-form of $\alpha$-norm smaller than $1$ on~$M$),

appeared in Zermelo's control problem, are of special interest.

After a short survey of above, we will discuss

Integral formulae, which provide obstructions for existence of foliations

(or compact leaves of them) with given geometric properties.

The first known Integral formula (by G.\,Reeb) for codimension-1 foliated closed manifolds tells us that

the total mean curvature $H$ of the leaves is zero (thus, either $H\equiv0$ or $H(x)H(y)<0$ for some $x,y\in M$).

Using a unit normal to the leaves of a codimension-one foliated $(M,F)$,

we define a new Riemannian metric $g$ on $M$, which for Randers case depends nicely on $(\alpha,\beta)$.

For that $g$ we derive several geometric invariants of a foliation in terms of $F$;

then express them in terms of invariants of $\alpha$ and~$\beta$.

Using our results \cite{rw2} for Riemannian case, we present new Integral formulae

for codimension-one foliated $(M, F)$ and $(M, \alpha+\beta)$.

Some of them generalize Reeb's formula.

The study of aperiodic point sets in Euclidean space is a classical topic in harmonic analysis,

combinatorics and geometry. Aperiodic point sets in R^3 are models for quasi-crystals, and in

this context it is of interest to study their diffraction measure, i.e. the way they scatter an

incoming laser or x-ray beam. By a classical theorem of Meyer, every sufficiently regular

aperiodic point set in a Euclidean space is a shadow of a periodic one in a larger locally

compact abelian group. The diffraction of these "model sets" can be computed in terms of a

certain group of irrational rotations of an associated torus.

In this talk, I will review the classical theory of diffraction of Euclidean model sets and then

explain how the theory generalizes to model sets in arbitrary (non-abelian) locally compact groups.

We will explain the construction of new examples of different flavours, and how the classical

theory has to be modified in order to accomodate these new examples. We will focus on the case

of model sets in groups admitting a Gelfand pair, since for these the (spherical) diffraction

theory is particularly accessible.

No previous knowledge of model sets or diffraction theory is assumed.

This is based on joint work with Michael Bjorklund and Felix Pogorzelski.

We consider classes of mappings (with controlled moduli) whose $p$-module of the families of curves/surfaces is restricted by integrals containing measurable functions and arbitrary admissible metrics. In the talk we discuss various properties of mappings with controlled moduli including their differential features (Lusin's $N-$ and $N^{-1}$-conditions, Jacobian bounds, estimates for distortion dilatations, H\"older/logarithmically H\"older continuity) and the topological structure (openness, discreteness, invertibility, finiteness of the multiplicity function). This allows us to investigate the interconnection between mappings of bounded and finite distortion defined analytically and mapping with controlled moduli having no analytic assumptions.

Let $\Omega$ be an open 0-symmetric subset of $\mathbb R^d$ which contains 0 and

f a continuous positive definite function vanishing off O, that is,

supp f is contained in the closure of $\Omega$. The problem is to approximate

f by a continuous positive definite function F supported in $\Omega$. We prove

this when 1. d=1. 2 $\Omega$ is strictly star-shaped 3. f is a radial function.

We also consider the following problem: Given a measure $\mu$

supported in $\Omega$, does there exist an extremal function for the problem

$\sup \int f d\mu$, where the sup is taken over the cone of continuous

positive definite functions f supported in $\Omega$ with f(0)=1?

A function f on the real line is said to tile by translates

along a discrete set $\Lambda$ if the sum of all the functions

f(x-\lambda), $\lambda \in \Lambda$, is equal to one identically.

Which functions can tile by translates, and what can be said

about the translation set $\Lambda$? I will survey the subject and

discuss some recent results joint with Mihail Kolountzakis.

New relations between the Fourier transform of a function of bounded

variation and the Hilbert transform of its derivative are revealed.

The main result is an asymptotic formula for the {\bf cosine} Fourier

transform. Such relations have previously been known only for the sine

Fourier transform. Interrelations of various function spaces are studied

in this context, first of all of two types of Hardy spaces. The obtained

results are used for proving completely new results on the integrability

of trigonometric series.

Consider a polynomial map $f: C^n\to C^n$, vanishing at some point $z_0$ in $C^n$. In differential equations, such points are called

equilibria of the vector field $z' = f(z)$, or their singular points. The question is "how singular". Can we quantify the singularity of $f$ at $z_0$?

Attempting only to demystify the problem, in this presentation we make an effort to quantify singularity in the sense of differential equations

and also discuss connections of this theory to analysis, topology and commutative algebra.

It is now more than 52 years since Studia Mathematica received Alberto

Calder\'on's very remarkable paper about his theory of complex

interpolation spaces. And one of the questions which Calder\'on

implicitly asked in that paper, by solving it in a significant special

case, is apparently still open today:

DOES COMPLEX INTERPOLATION PRESERVE THE COMPACTNESS OF AN OPERATOR?

After briefly surveying attempts to solve this question over several

decades, I will also report on a few new partial answers obtained

recently, some of them (arXiv:1411.0171) jointly with Richard

Rochberg. Among other things there is an interplay with Jaak Peetre's

"plus-minus" interpolation method, (arXiv:1502.00986) a method which

probably deserves to be better known. Banach lattices and UMD spaces

also have some roles to play.

Several distinguished mathematicians have expressed the belief that

that the general answer to this question will ultimately turn out to be

negative. Among other things, I will try to hint at where a counterexample

might perhaps be hiding. You are all warmly invited to seek it out,

or prove that it does not exist.

A fairly recent survey which discusses this question is available at

arXiv:1410.4527.

We present a new construction of Hardy spaces on the Klein-Dirac

quadric; we show that the quadric is obtained as a complexification of the

unit ball in R^n. We introduce also Hardy spaces on complexified

multidimensional annulus.

We show some natural properties of these Hardy spaces, in particular,

Cauchy type formula, and Brothers Riesz type theorem.

We prove applications to the multidimensional Moment problem,

multidimensional Interpolation theory, and Cubature formulas.

We provide restricted negative answers to the Royden-Sullivan problem

whether any Teichm\"{u}ller space of dimension greater than $1$

is biholomorphically equivalent to bounded domain in a complex Banach

space. The only known result here is Tukia's theorem of 1977 that there is

a real analytic homeomorphism of the universal Teichm\"{u}ller

space onto a convex domain in some Banach space.

We prove:

(a) Any Teichm\"{u}ller space $\mathbf T(0,n)$ of the punctured spheres

(the surfaces of genus zero) with sufficiently large number of punctures

$(n \ge n_0 > 4)$ cannot be mapped biholomorphically onto a bounded

convex domain in $\mathbf C^{n-3}$.

(b) The universal Teichm\"{u}ller space is not biholomorphically equivalent

to a bounded convex domain in uniformly convex Banach space, in

particular, to convex domain in the Hilbert space.

The proofs involve the existence of conformally rigid domains established

by Thurston and some interpolation results for bounded univalent functions.

The class of fractals referred to are those which may be specified by a finite system of affine transformations,

assuming contractive scaling; and their corresponding selfsimilar measures, $\mu$. They include standard Cantor

spaces such as the middle third, and the planar Sierpinski caskets in various forms, and their corresponding

selfsimilar measures, but the class is more general than this; including fractals realized in $\mathbb R^d$, for

$d > 2$.

In part 1, we motivate the need for wavelets in the harmonic analysis of these selfsimilar measures $\mu$. While

classes of the Hilbert spaces $L^2(\mu)$ have Fourier bases, it is known (the speaker and Pedersen) that many do

not, for example the middle third Cantor can have no more than two orthogonal Fourier frequencies.

In part 2 of the talk, we outline a construction by the speaker and Dutkay to the effect that all the affine systems

do have wavelet bases; this entails what we call thin Cantor spaces.

In the problem of summability at a point at which the derivative of indefinite

integral exists for Fourier series and Fourier integrals of integrable functions

a new sufficient condition is obtained. In the case of "arithmetic means" the

corresponding condition is also necessary.

Exact rates of approximation by the classical Gauss-Weierstrass, Bochner-Riesz,

and Marcinkiewicz-Riesz means, as well as by non-classical Bernstein-Stechkin means

are found.

These problems are related to the representability of a function as an absolutely

convergent Fourier integral. For this, new conditions are obtained, while for radial functions

even a criterion.

For each positive integer $m$ and each $p>2$ we characterize bounded simply connected

Sobolev $W^m_p$-extension domains $\Omega$ in $R^2$. Our criterion is expressed in terms of

certain intrinsic subhyperbolic metrics in $\Omega$. Its proof is based on a series of results related

to the existence of special chains of squares joining given points $x$ and $y$ in $\Omega$.

An important geometrical ingredient for obtaining these results is a new ''Square Separation Theorem''.

It states that under certain natural assumptions on the relative positions of a point $x$ and a square

$S\subset\Omega$ there exists a similar square $Q\subset\Omega$ which touches $S$ and has

the property that $x$ and $S$ belong to distinct connected components of $\Omega\setminus Q$.

This is a joint work with Nahum Zobin.

The triply truncated solutions of the first Painlev\'e equation were specified by Boutroux

in his famous paper of 1913 as those having no poles (of large modulus) except in one sector

of angle $2\pi/5$. There are five such solutions and each of them can be obtained from any

other one by applying a certain symmetry transformation. One of these solutions is real on

the real axis. We will discuss a characteristic property of this solution (discovered by Prof.

Joshi, and Prof. Kitaev), different from the asymptotic description given by Boutroux.

For $\lambda\in (0,1)$, the Bernoulli convolution measure $\nu_\lambda$ may be defined as the distribution

of the random series $\sum_{n=0}^\infty \pm \lambda^n$, where the signs are chosen independently with equal

probabilities. For $\lambda =1/3$, this is the familiar Cantor-Lebesgue measure (up to a linear change of variable).

The Fourier transform of $\nu_\lambda$ has an infinite product formula:

$$\widehat{\nu}_\lambda(t) = \prod_{n=0}^\infty \cos(2\pi \lam^n t).$$

The properties of $\nu_\lambda$ and their Fourier transforms have been studied since the 1930's by many mathematicians,

among them Jessen, Wintner, Erd\H{o}s, Salem, Kahane, Garcia. In particular, it was proved by Erd\H{o}s and Salem that

$\widehat{\nu}_\lambda(t)$ does not vanish at infinity (i.e. $\nu_\lambda$ is not a Rajchman measure) if and only if

$1/\lambda$ is a Pisot number (an algebraic integer greater than one with all conjugates inside the unit circle).

However, very little is known about the rate of decay, especially for specific $\lambda$, as opposed to "typical" ones.

In this talk I will survey known results and open problems in this direction. Recently in a joint work with A. Bufetov

we proved that if $1/\lam$ is an algebraic integer with at least one conjugate outside of the unit circle, then the

Fourier transform of $\nu_\lam$ has at least a logarithmic decay rate at infinity.

This is a joint work with Walter Bergweiler.

We construct differential equations of the form w"+Aw=0, where $A$

is an entire function of finite order, with the property that two

linearly independent solutions have finite exponent of convergence

of zeros. This solves a problem proposed by Bank and Laine in 1982.

In recent years the study of quaternionic linear spaces has been widely developed

by mathematicians and has been widely used by physicists. At the same

time it turns out that some basic and fundamental properties of those spaces

are not treated properly and this requires to develop the corresponding theory.

In this talk we will analyze certain peculiarities of the situation via the notion

of quaternionic extension of real and complex linear spaces as well as using the

notion of internal quaternionization. We will see, for example, how the norms of

some operators behave when they are “quaternionically extended”.

Hecke, Ostrowski and Kesten characterized the intervals on the circle

for which the ergodic sums of their indicator function, under an

irrational rotation, stay at a bounded distance from their integral

with respect to the Lebesgue measure on the circle.

In this talk I will discuss this phenomenon in multi-dimensional setting.

Based on joint work with Sigrid Grepstad.

Topological methods based on the usage of degree theory have proved

themselves to be an important tool for qualitative studying of solutions to

nonlinear differential systems (including such problems as existence,

uniqueness, multiplicity, bifurcation, etc.).

During the last twenty years the equivariant degree theory emerged in Non-

linear Analysis. In short, the equivariant degree is a topological tool

allowing “counting” orbits of solutions to symmetric equations in the same

way as the usual Brouwer degree does, but according to their symmetry

properties. This method is an alternative and/or complement to the

equivariant singularity theory developed by M. Golubitsky et al., as well as

to a variety of methods rooted in Morse theory/Lusternik–Schnirelman theory.

In fact, the equivariant degree has different faces reflecting a diversity of

symmetric equations related to applications. In the two talks, I will discuss

three variants of the equivariant degree: (i) non-parameter equivariant

degree, (ii) twisted equivariant degree with one parameter, and (iii)

gradient equivariant degree. Each of the three variants of equivariant degree

will be illustrated by appropriate examples of applications: (i) boundary

value problems for vector symmetric pendulum equation, (ii) Hopf bifurcation

in symmetric neural networks (simulation of legged locomotion), and (iii)

bifurcation of relative equilibria in Lennard-Jones three-body problem.

The talk is addressed to a general audience, without any special knowledge

of the subject.

In the talk, which is joint work with Alexander Olevskii, I will present our study of the

relationship between the existence of arithmetic progressions with specified lengths and

step sizes and lower Riesz bounds of complex exponentials indexed by a set of integers

$\Lambda$ on subsets of the circle.

The following problems (or a part of them) will be discussed.

1. Generalization of the Abel-Poisson summation method.

2. Generalization of the Riemann-Lebesgue lemma.

3. Strengthening of the Hardy-McGehee-Pigno-Smith inequality.

4. Generalization of the Euler-Maclaurin formula.

5. Absolute convergence of grouped Fourier series.

6. Comparison of linear differential operators with constant coefficients.

7. Positive definite functions and splines.

8. Strong converse theorems in approximation theory. Bernstein-Stechkin polynomials.

We study the influence the angular distribution of zeroes of the Taylor

series with pseudo-random and random coefficients, and show that the

distribution of zeroes is governed by certain autocorrelations of the

coefficients. Using this guiding principle, we consider several examples

of random and pseudo-random sequences $\xi$ and, in particular, answer

some questions posed by Chen and Littlewood in 1967.

As a by-product we show that if $\xi$ is a stationary random

integer-valued sequence, then either it is periodic, or its spectral

measure has no gaps in its support. The same conclusion is true if $\xi$

is a complex-valued stationary ergodic sequence that takes values from a

uniformly discrete set (joint work with Alexander Borichev and Alon Nishry).

Nodal sets are zero loci of Laplace eigenfunctions (e.f.). Study of nodal sets is important

for understanding wave processes. The geometry of a single nodal set may be very complicated

and hardly can be well understood. More realistic might be describing geometry of sets which

are nodal for a large family of e.f. (the condition of simultaneous vanishing, resonanse, of

a large packet of e.f., on a large set, is overdetermined and hence may be expected to occur

only for exclusive sets).

Indeed, it was proved that common nodal curves for large, in different senses, families

of e.f. in $\mathbb R^2$ are straight lines (non-periodic case: Quinto and the speaker, ’96; periodic

case: Bourgain and Rudnick, ’11). It was conjectured that in a Euclidean space of

arbitrary dimension, common nodal hypersurfaces for large families of e.f. are cones, more precisely,

are translates of zero sets of harmonic homogeneous polynomials.

The talk will be devoted to a recent result confirming the conjecture for ruled hypersurfaces

in $\mathbb R^3$. Relation to the injectivity problem for the spherical Radon transform will be explained.

We study the influence the angular distribution of zeroes of the Taylor

series with pseudo-random and random coefficients, and show that the

distribution of zeroes is governed by certain autocorrelations of the

coefficients. Using this guiding principle, we consider several examples

of random and pseudo-random sequences $\xi$ and, in particular, answer

some questions posed by Chen and Littlewood in 1967.

As a by-product we show that if $\xi$ is a stationary random

integer-valued sequence, then either it is periodic, or its spectral

measure has no gaps in its support. The same conclusion is true if $\xi$

is a complex-valued stationary ergodic sequence that takes values from a

uniformly discrete set (joint work with Alexander Borichev and Alon Nishry).

The talk is will be devoted to two questions in the theory of quasianalytic

Carleman classes. The first one is how to describe the image of a quasianalytic

Carleman class under Borel's map $f\to\{f^{(n)}(0)/n!\}_{n\geq 0}$ ?

The second one is how to sum the formal Taylor series of functions in quasianalytic Carleman classes? In the talk, I will present a method of Beurling that gives a solution to both of the problems for some quasianalytic Carleman classes. If time permits, I will also discuss the image problem in some non-quasianalytic classes.

We will begin by introducing the notion of hypergroup, give some examples,

and describe the convolution of measures on a hypergroup. After a review of

some basic operator space theory, we shall describe how to extend the notion

of convolution to the space of completely bounded multilinear forms on a cartesian

product of spaces of continuous functions on hypergroups, thus making that space

into a Banach algebra. When the hypergroups are commutative, we introduce and study

a notion of Fourier transform in this setting.

The talk is devoted to applications of the limit operators to the study of

essential spectra and exponential decay of eigenfunctions of the discrete

spectra for Schr\"{o}dinger and Dirac operators for wide classes of

potentials. Outline of the talk:

1) Fredholm property and location of the essential spectrum of systems of

partial differential operators with variable bounded coefficients;

2) Exponential estimates of solutions of systems of partial differential

operators with variable bounded coefficients;

3) Location of the essential spectrum of Schr\"{o}dinger and Dirac operators

and exponential estimates of eigenfunctions of the discrete spectrum.

Two types of optimal estimates for derivatives of analytic functions

with bounded real part are considered. The first of them is a pointwise

inequality for derivatives of analytic functions in the complement

of a convex closed domain in ${\mathbb C}$. The second type of inequalities

is a limit relation for derivatives of analytic functions in an arbitrary proper

subdomain of ${\mathbb C}$. Optimal estimates for derivatives of a vector

field with bounded harmonic components as well as optimal estimates for the

divergence of an elastic displacement field and pressure in a fluid in

subdomains of ${\mathbb R}^n$ are discussed.

Let $D$ be a domain of $\R^d$; we say that $L^2(D)$ has an exponential basis if there exists

sequence of functions ${\mathcal B}=\{ e^{2\pi i \langle s_m x\rangle}\}_{ m \in Z^d}$,

with $s_m\in\R^d$, with the following property: every function in $L^2(D)$ can be written in

a unique way as $\sum_{m\in\Z^d} c_m e^{ 2\pi i \langle s_m, x\rangle} $, with $c_m \in \C$.

For example, $\{ e^{2\pi i mx}\} _{m \in Z}$ is an exponential basis of $L^2(0, 1 )$.

Exponential bases are very useful in the application, especially when they are orthogonal; however,

the existence or non-existence of exponential bases is proved only on very special domains of $\R^d$.

In particular, it is not known whether the unit ball in $\R^2$ has an exponential basis or not.

An important property of exponential bases is their stability. That is, if $\{ e^{2\pi i \langle s_m,

x\rangle}\}_{ m \in Z^d}$ is an exponential basis of $L^2(D)$ and $\Delta=\{\delta_m\}_{ m \in Z^d} $ is

a sequence of sufficiently small real number, then also $\{ e^{2\pi i \langle s_m+\delta_m,

x\rangle}\}_{ m \in Z^d}$ is an exponential basis of $L^2(D)$. In this talk I will discuss the existence

and stability of exponential bases on special 2-dimensional domains called trapezoids.

I will also generalize a celebrate theorem by M. Kadec and obtain stability bounds for exponential bases on domains of $\R^d$.

The result that I will present in my talk are part of joint projects with my students A. Kumar and S. Pathak.

END_OF_ABSTRACT

If a function and its conjugate (in a special sense) both have

bounded variation, then their Fourier transforms are integrable.

This recent extension of a classical (for Fourier series) Hardy-

Littlewood theorem gives rise to new thoughts and results.

The aim of this talk is to present a simple two-sided estimate for the

operator norm of a finite Hankel matrix in terms of its standard

symbol. We will also discuss several reformulations and consequences

of this estimate, including the classical Fefferman's duality theorem

for the Hardy space $H^1$.

The aim of this talk is to present a simple two-sided estimate for the

operator norm of a finite Hankel matrix in terms of its standard

symbol. We will also discuss several reformulations and consequences

of this estimate, including the classical Fefferman's duality theorem

for the Hardy space $H^1$.

We consider the wave flow on a surface of constant negative curvature.

For short times, the propagation is approximated by the geodesic flow, with

errors controlled by the “semiclassical expansion” coming from geometric optics.

In negative curvature, this expansion is useful up the Ehrenfest time $|\log{\hbar}|$,

after which the error terms in the expansion become as large as the main term. It is

believed that the approximation of wave propagation by the geodesic flow should hold

for much larger times, perhaps all the way up to the Heisenberg time $1/\hbar$. However,

we show that this cannot hold in general, and exhibit explicit examples where the semiclassical

approximation breaks down at a constant multiple of Ehrenfest time. These examples come from

Eisenstein series on the modular surface, and are intimately tied to the arithmetic structure,

and highly non-generic. We will also discuss these non-generic features of the arithmetic setting,

and whether this breakdown at the Ehrenfest time is likely to be a more generic phenomenon or not.

Includes joint work with Roman Schubert.

The aim of this work is to establish a number of elementary

properties about the topology and algebra of real quadratic homogeneous

mapping and Ricatti type ODEs occurring in non-associative algebras.

We construct a series of examples of the quadratic vector field to

show the impact of their spectral properties into qualitative theory.

We present sharp Ul’yanov type inequalities for fractional moduli of smoothness and K-functionals for the values of the parameters: 0<p<1, p<q. We also provide a generalization of Kolyada's inequality and relations between fractional moduli of smoothness of a function and its derivatives in the spaces L_p, 0<p<1.

We present new sufficient conditions for Fourier multipliers. These conditions are given in terms of simultaneous

behavior of (quasi-)norms of a function in different Lebesgue and Besov spaces. We also provide some sufficient

conditions for the representation of a function as an absolute convergence Fourier integrals in terms of belonging

of a function simultaneously to several spaces of smooth functions.

We discuss sharp continuity and regularity results for solutions of the

polyharmonic equation in an arbitrary open set. The absence of

information about geometry of the domain puts the question of

regularity properties beyond the scope of applicability of the methods

devised previously, which typically rely on specific geometric assumptions.

Positive results have been available only when the domain is

sufficiently smooth, Lipschitz or diffeomorphic to a polyhedron.

The techniques developed recently allow to establish the

boundedness of derivatives of solutions to the Dirichlet problem for the

polyharmonic equation under no restrictions on the underlying domain

and to show that the order of the derivatives is maximal. An appropriate

notion of polyharmonic capacity is introduced which

allows one to describe the precise correlation between the smoothness of

solutions and the geometry of the domain.

We also study the 3D Lam\'e system and establish its weighted

positive definiteness for a certain range of elastic constants. By modifying

the general theory developed by Maz'ya (Duke, 2002), we then show, under the

assumption of weighted positive definiteness, that the divergence

of the classical Wiener integral for a boundary point guarantees the

continuity of solutions to the Lam\'e system at this point.

The talk is based on my joint work with S.Mayboroda (Minnesota) and Guo Luo (Caltech)

First-order systems of partial differential equations appear

in many areas of physics, from the Maxwell equations to the Dirac

operator.

The aim of the talk is to describe a general method for the study of

the spectral density of all such systems, connecting it to traces on the

(geometric-optical) "slowness surfaces" .

The Holder continuity of the spectral density leads to a derivation of

the limiting absorption principle and global spacetime estimates

(based on joint work with Tomio Umeda).

The beautiful and extensive Coifman-Meyer theory, developed in the 70s and 80s to study

singular multi-linear operators does not apply to many naturally arising operators with

positive kernels. We shall describe some elementary approaches to such operators and apply

them to some problems in geometric measure theory and classical harmonic analysis.

H. H. Bauschke and J. M. Borwein showed that in the space of all tuples of bounded,

closed and convex subsets of a Hilbert space with a nonempty intersection, a typical

tuple has the bounded linear regularity property. This property is important because

it leads to the convergence of infinite products of the corresponding nearest point

projections to a point in the intersection.

We show that the subset of all tuples possessing the bounded linear

regularity property has a porous complement. Moreover, our result is

established in all normed spaces and for tuples of closed and convex sets

which are not necessarily bounded. This is joint work with A. J. Zaslavski.

We derive an extension of the standard time-dependent WKB theory, which can be applied

to propagate coherent states and other strongly localized states for long times. It in

particular allows us to give a uniform description of the transformation from a localized

coherent state into a delocalized Lagrangian state, which takes place at the Ehrenfest time.

The main new ingredient is a metaplectic operator that is used to modify the initial state

in a way that the standard time-dependent WKB theory can then be applied for the propagation.

This is based on joint work with Raul Vallejos and Fabricio Toscano, but in this talk we will

focus on the special case of propagation on a manifold of negative curvature.

~We consider the generic discrete open mappings in ${\mathbb R}^n$ under which the perturbation of extremal lengths of curve collections is controlled integrally via $\int Q(x)\eta^p(|x-x_0|)

dm(x)$ with $n-1<p<n$, where $Q$ is a measurable function on ${\mathbb R}^n$ and $\int\limits_{r_1}^{r_2} \eta(r) dr \ge 1$ for any $\eta$ on a given interval $[r_1,r_2].$ The main results state that the family of all open discrete mappings of above type is normal under appropriate restrictions on the majorant $Q.$

We also provide conditions ensuring the local H\"older continuity

of such mappings with respect to euclidian distances (in the general case with respect to their

logarithms). The inequalities defining the continuity are sharp with respect to

the order.

This is a joint work with R. Salimov and E. Sevost'yanov.

It is well known that the geometric nature of semigroup trajectories essentially depends on the semigroup type.

In this work, we concentrate on parabolic type semigroups of holomorphic self-mappings of the open unit disk and of the right

half-plane, and study the structure of semigroup trajectories near the Denjoy--Wolff point. In particular, we find the limit

order of contact and the limit curvature of trajectories and their `closeness’, determine whether these trajectories have

asymptotes. For these purposes, we suggest that two terms in the asymptotic power expansion of semigroup generators are known.

Our methods are based on the asymptotic expansion of a semigroup that we find on the first step. Inter alia, this enable us

to establish a new rigidity property for semigroups of parabolic type.

The talk is based on a joint work with F. Jacobzon.

A generalization of Newton's attraction theorem will be discussed.

The same analytic method is applied for reconstruction in photoacoustic geometry.

We extend Caratheodory's generalization of Montel's fundamental normality test

to "wandering" exceptional functions (i.e. depending on the respective function in the

family under consideration), and we give a corresponding result on shared functions.

Furthermore, we prove that if we have a family of pairs (a,b) of functions meromorphic

in a domain such that a and b uniformly "stay away from each other " , then the families

of the functions a resp. b are normal. The proofs are based on a "simultaneous rescaling"

version of Zalcman's Lemma. We also introduce a somewhat "strange" result about some

sharing wandering values assumptions that imply normality.

Let $X$ be Banach space, $(\Omega,\Sigma)$ is a measure space, where $\Omega$

is a set and $\Sigma$ is a $\sigma$-algebra of subsets of $\Omega.$ If $m:\Sigma\rightarrow X$ is

a $\sigma$-additive $X$-valued measure, then the range of $m$ is the set $m(\Sigma)=\{m(A):

\ A\in\Sigma.\}$ The measure $m$ is {\it non-atomic} if for every set $A\in\Sigma$ with $m(A)>0,$

there exist $B\subset A,B\in\Sigma$ such that $m(B)\neq0$ and $m(A \backslash B)\neq0.$

$X$-valued measure we will call {\it Lyapunov measure} if the closure of its range is convex.

And Banach space $X$ is {\it Lyapunov space} if every $X$-valued non-atomic measure is Lyapunov.

Theorem. Let X be Banach space with unconditional basis, q-concave, $q<\infty$, and which doesn't contain isomorphic copy of $l_2.$

Then X is Lyapunov space.

I will present a quantum ergodicity theorem on large regular graphs.

This is a result of spatial equidistribution of most eigenfunctions of the discrete Laplacian

in the limit of large regular graphs. It is analogous to the quantum ergodicity theorem on

Riemannian manifolds, which is concerned with the eigenfunctions of the Laplace-Beltrami operator

in the high frequency limit. I will also talk about pseudo-differential calculus on regular graphs,

one of the tools constructed for the proof of the theorem.

This is a joint work with Nalini Anantharaman.

This presentation is devoted to the problem of

recovering a function from its spherical means with

centers located on an ellipsoid $\Sigma$ in the two- and

three-dimensional spaces. We will show how to

generalize methods for obtaining inverse

formulas for the case when $\Sigma$ is a sphere to the

case when $\Sigma$ is an ellipsoid.

This presentation is devoted to the problem of

recovering a function from its spherical means with

centers located on an ellipsoid $\Sigma$ in the two- and

three-dimensional spaces. We will show how to

generalize methods for obtaining inverse

formulas for the case when $\Sigma$ is a sphere to the

case when $\Sigma$ is an ellipsoid.

We discuss sharp continuity and regularity results for solutions of the

polyharmonic equation in an arbitrary open set. The absence of

information about geometry of the domain puts the question of

regularity properties beyond the scope of applicability of the methods

devised previously, which typically rely on specific geometric assumptions.

Positive results have been available only when the domain is

sufficiently smooth, Lipschitz or diffeomorphic to a polyhedron.

The techniques developed recently allow to establish the

boundedness of derivatives of solutions to the Dirichlet problem for the

polyharmonic equation under no restrictions on the underlying domain

and to show that the order of the derivatives is maximal. An appropriate

notion of polyharmonic capacity is introduced which

allows one to describe the precise correlation between the smoothness of

solutions and the geometry of the domain.

We also study the 3D Lam\'e system and establish its weighted

positive definiteness for a certain range of elastic constants. By modifying

the general theory developed by Maz'ya (Duke, 2002), we then show, under the

assumption of weighted positive definiteness, that the divergence

of the classical Wiener integral for a boundary point guarantees the

continuity of solutions to the Lam\'e system at this point.

The talk is based on my joint work with

S.Mayboroda (Minnesota) and Guo Luo (Caltech)

In this talk we will introduce a definition of Generalized Analytic Functions

(in the sense of Vekua), in elliptic complex numbers. One advantage of this definition

is that its Canonical Form is more general than Vekua's and in many cases the reduction

from the elliptic and linear partial differential equation of first order can be done

without solving an associated Beltrami equation. Finally, using techniques of Vekua we

will show that these functions satisfy a representation formula that generalizes the

Similarity Principle in the ordinary case.

We introduce a notion of r-convexity for subsets of the complex

plane. It is a pure geometric characteristic that generalizes the

usual notion of convexity. Next, we investigate subharmonic

functions that grow near the boundary in unbounded domains with

r-convex compact complement. We obtain the Blaschke-type bounds

for its Riesz measure and, in particular, for zeros of unbounded

analytic functions in unbounded domains. These results are based

on a certain estimates for Green functions on complements of some

neighborhoods of $r$-convex compact set. Also, we apply our

results in perturbation theory of linear operators in a Hilbert

space. More precisely, we find quantitative estimates for the rate

of condensation of the discrete spectrum of a perturbed operator

near its the essential spectrum.

Let S be a bounded, Riemann measurable set in R^d, and L

be a lattice. By a theorem of Fuglede, if S tiles R^d with

translation set L, then S has an orthogonal basis of exponentials.

We show that, under the more general condition that S multi-tiles

R^d with translation set L, S has a Riesz basis of exponentials. The

proof is based on Meyer's quasicrystals. This is a joint work with

Nir Lev.

A classical theorem due to Mattila says that if $A,B

\subset {\Bbb R}^d$ of Hausdorff dimension $s_A, s_B$, respectively,

with $s_A+s_B \ge d$, $s_B>\frac{d+1}{2}$ and $dim_{{\mathcal H}}(A

\times B)=s_A+s_B\ge d$, then

$$ dim_{{\mathcal H}}(A \cap (z+B)) \leq s_A+s_B-d$$ for almost every

$z \in {\Bbb R}^d$, in the sense of Lebesgue measure.

We obtain a variable coefficient variant of this result in which we

are able to replace the Hausdorff dimension with the upper Minkowski

dimension on the left-hand-side of the first inequality. This is

joint work with Alex Iosevich and Suresh Eswarathasan. Fourier

Integral Operator bounds and other techniques of harmonic analysis

play a crucial role in our investigation.

Recently S. Artstein-Avidan and V. Milman have developed an

abstract duality theory and proved the following remarkable

result: up to linear terms, the only fully order preserving

operator (namely, an invertible operator whose inverse also

preserves the pointwise order between functions) acting

on the class of lower semicontinuous proper convex functions

defined on R^n is the identity operator, and the only fully

order reversing operator acting on the same set is the

Fenchel conjugation (Legendre transform). We establish

a suitable extension of their result to infinite dimensional

Banach spaces.

This is a joint work with Alfredo N. Iusem and Benar F. Svaiter

The $A_2$ conjecture says that the $L^2(w)$ operator norm

of any Calder\'on-Zygmund operator is bounded linearly by the $A_2$

constant of the weight $w$.

This conjecture was completely solved in 2010 by T. Hyt\"onen.

The proof was based on a rather difficult representation of a general

Calder\'on-Zygmund operator in terms of the Haar shift operators.

In this talk we shall discuss a recent simpler proof completely

avoiding the notion of the Haar shift operator.

We consider an extremal configuration of points on a manifold, called Fekete points,

and study their equidistribution through their relation to Beurling-Landau theory

of sampling and interpolation (joint work with Joaquim Ortega-Cerda).

A classification of commuting rational functions, that is of rational

solutions of the functional equation A(X)=X(A), was obtained in the beginning of

the past century by Fatou, Julia, and Ritt. In the talk we will present a solution

of a more general problem of description of semiconjugate rational functions, that

is of rational solutions of the functional equation A(X)=X(B) in terms of groups

acting properly discontinuously on the Riemann sphere or complex plane.

I plan to discuss a recent

progress in Eckhoff Conjecture obtained via "Algebraic Sampling" approach, and a general

bound on sampling accuracy provided by a combination of Kolmogorov's entropy and

Johnson-Lindenstrauss dimensionality reduction. This is a joint work with D. Batenkov.

The Nitsche conjecture was solved recently by Iwaniec, Kovalov and Onninen and in the same

paper they pose the same problem from Teichmuller domain onto Teiuchmuller

domain. We present a solution to this problem.

We will discuss the case of surfaces of constant negative curvature; in particular,

we will explain how to construct examples of sufficiently weak quasimodes that do not

satisfy QUE, and show how they fit into the larger theory.

In this talk I will discuss some recent results obtained in

collaboration with G. Leoni on new characterizations of Sobolev spaces for

arbitrary open sets. The motivation for such a characterization stems from a

2001 paper of Bourgain, Brezis, and Mironescu that gives a related one for

smooth and bounded domains, and an open question on the extension of these

results to arbitrary open sets.

In 1996, Talagrand established a lower bound on the second-level Fourier

coefficients of a monotone Boolean function, in terms of its first-level coefficients.

This lower bound and its enhancements were used in various applications to

correlation inequalities, noise sensitivity, geometry, percolation, etc.

In this talk we present a new proof of Talagrand's inequality, which is somewhat

simpler than the original proof, and allows to generalize the result easily to

non-monotone functions (with influences replacing the first-level coefficients) and

to more general measures on the discrete cube. We then apply our proof to obtain

a quantitative version of a theorem of Benjamini-Kalai-Schramm on the relation

between influences and noise sensitivity.

Time permitting, we shall present recent results and open questions, related

to an application of Talagrand's lower bound to correlation inequalities.

The first part of the talk is joint work with Guy Kindler.

The talk consists of two parts.

1) Fourier multipliers and absolute convergence of Fourier integrals.

Based on the paper by Liflyand-Samko-Trigub "The Winer

algebra of absolutely convergent Fourier integrals: an overview",

Analysis and Math. Physics 2(2012), 1-68.

2) Comparison of linear differential operators with constant coefficients

by their norms in $L_p,$ $1\le p\le \infty$. In particular, three criteria

of comparison are obtained for functions on the circle, on the axis, and on

the half-axis, as well as one sharp inequality.

In the talk we first review the highly anisotropic Hardy spaces]. We then discuss a careful approximation

argument that is needed when analyzing dual spaces of Hardy spaces. One cannot assume that a linear

functional, uniformly bounded on all atoms, is automatically bounded

on spaces that have atomic representations (e.g. Hardy spaces).

We characterize the radial basis functions whose

scattered shifts form a fundamental system in

the space $L_{p}(\rrd)$. In particular, we show that for any even function $h$ from the space

$L_{2,{\rm loc}}(\rrd)$

the space formed by all possible linear combinations of

shifted radial basis functions $h(\|x+a\|)$, $a\in \rrd$, is dense in the

space $L_p(\rrd)$, $1\le p\le 2$, if

and only if the function $h$ is not a polynomial.

The classical Teichm\"uller-Wittich-Belinskii theorem implies

the conformality of a planar continuous mapping at a point under rather

general integral restrictions for the dilatation of this mapping

near the point.

This theorem is very rich in applications and has been generalized

by many authors in various directions (weak conformality, differentiability,

multidimensional analogs, etc.).

Certain complete generalizations are due to Reshetnyak and Bishop, Gutlyanskii, Martio, Vuorinen.

I will show in the talk that the assumptions under which the main results have been obtained,

can be essentially weakened and give much stronger estimate

for the limit of $|f(x)|/|x|$ as $x$ approaches $0$.

We essentially improve the underlying modular technique.

We study to which extent the Poisson summation formula determines the Fourier transform. The answer is positive under certain technical smoothness and rate of decay conditions. This study leads to a class of unitary operators on L^2 that satisfy a weighted form of the Poisson summation formula, which we explicitly diagonalize, with eigenvalues related to associated L-functions.

The Euler-Gauss linear transformation formula for the

hypergeometric function was extended by Goursat for the case of

logarithmic singularities. By replacing the perturbed Bessel

differential equation by a monodromic functional equation, and

studying this equation separately from the differential equation by

an appropriate Laplace-Borel technique, we associate with the latter

equation another monodromic relation in the dual complex plane. This

enables us to prove a duality theorem and to extend Goursat's

formula to much larger classes of functions.

Let $(M^{n+p},g)$ be a closed Riemannian manifold, and $\pi: M\to B$

a smooth fiber bundle with compact and orientable $p$-dimensional

fiber $F$. Denote by $D_F$ ($D$) the distribution tangent

(orthogonal, resp.) to fibers.

We discuss conformal flows of the metric restricted to $D$ with the

speed proportional to

(i) the divergence of the mean curvature vector $H$ of $D$,

(ii) the mixed scalar curvature $Sc_{mix}$ of the distributions.

(If $M$ is a surface, then $Sc_{mix}$ is the gaussian curvature $K$).

For (i), we show that the flow is equivalent to the heat flow of the

1-form dual to $H$, provided the initial 1-form is $D_F$-closed. We

use known long-time existence results for the heat flow to show that

our flow has a global solution $g_t$. It converges to a limiting

metric, for which $D$ is harmonic (i.e., $H=0$); actually under some

topological assumptions we can prescribe $H$.

For (ii) on a twisted product, we observe that $H$ satisfies the

Burgers type PDE, while the warping function satisfies the heat

equation; in this case the metrics $g_t$ converge to the product.

We consider illustrative examples of flows similar to (i) and (ii)

on a surface (of revolution), they yield convection-diffusion PDEs

for curvature of $D$-curves (parallels) and solutions -- non-linear

waves.

For $M$ with general $D$, we modify the flow (ii) with the help of a

measure of ``non-umbilicity" of $D_F$, and the integrability tensor

of $D$, while the fibers are totally geodesic. Let $\lambda_0$ be

the smallest eigenvalue of certain Schrödinger operator on the

fibers. We assume $H$ to be $D_F$-potential and show that

-- $H$ satisfies the forced Burgers type PDE;

-- the flow has a unique solution converging to a metric, for which

$Sc_{mix}\ge-n\lambda_0$,

and $H$ depends only on the $D$-conformal class of the initial metric.

-- if $D$ had constant rate of ``non-umbilicity" on fibers, then the

limiting metric

has the properties: $Sc_{mix}$ is quasi-positive, and $D$ is harmonic.

Define on $\, L^p({\mathbb R}^n),\, p\ge 1,$ moduli of smoothness of order

$\, r,\, r \in {\mathbb N},$ by

\[

\omega_r(t,f)_p:=\sup _{|h| <t} \| \Delta_h^rf\|_p\, ,\quad t>0,\; \;

\Delta_hf(\cdot)= f(\cdot +h)-f(\cdot),\; \Delta_h^r=\Delta_h \Delta^{r-1}_h .

\]

Trivially one has $\, \omega_r(t,f)_p \lesssim \omega_k(t,f)_p\, ,\;

k<r.$ Its converse is known as Marchaud inequality. M.F. Timan 1958

proved a sharpening of the converse, nowadays called

{\it sharp Marchaud inequality}, which in the present context takes the

form,

\[

\omega_k(t,f)_p \lesssim t^k \left( \int_{t}^{\infty} [s^{-k}

\omega_r(u,f)_p]^q \frac{du}{u} \right)^{1/q},\qquad t>0,\quad k<r.

\]

where $\, q:=\min (p,2),\, 1<p<\infty.$

Here we will show that the sharp Marchaud inequality as well as further

sharp inequalities for moduli of smoothness like Ulyanov and Kolyada type

ones are equivalent to (known) embeddings

between Besov and potential spaces.\\

To this end one has to make

use of moduli of smoothness of fractional order which can be

characterized by Peetre's (modified) $\, K$-functional, living on $\,

L^p$ and associated Riesz potential spaces. Limit cases of

the Holmstedt formula (connecting different $\, K$-functionals) show

that the embeddings imply the desired inequalities.

Conversely, the embeddings result from the inequalities for moduli of

smoothness by limit procedures.

- תאריך עדכון אחרון: 18/01/2012