We consider Hausdorff and quasi-Hausdorff matrices as operators on classical spaces of analytic functions such as the Hardy and the Bergman spaces, the Dirichlet space, the Bloch spaces and $\BMOA$. When the generating sequence of the matrix is the moment sequence of a measure $\mu$, we find the conditions on $\mu$ which are equivalent to the boundedness of the matrix on the various spaces.

The following problem has been suggested by Paul Tur\' an. Let $\Omega$ be a symmetric convex body in the Euclidean space $\mathbb R^d$ or in the torus $\TT^d$. Then, what is the largest possible value of the integral of positive definite functions that are supported in $\Omega$ and normalized with the value $1$ at the origin? From this, Arestov, Berdysheva and Berens arrived at the analogous pointwise extremal problem for intervals in $\RR$. That is, under the same conditions and normalizations, the supremum of possible function values at $z$ is to be found for any given point $z\in\Omega$. However, it turns out that the problem for the real line has already been solved by Boas and Kac, who gave several proofs and also mentioned possible extensions to $\RR^d$ and to non-convex domains as well. Here we present another approach to the problem, giving the solution in $\RR^d$ and for several cases in~$\TT^d$. Actually, we elaborate on the fact that the problem is essentially one-dimensional and investigate non-convex open domains as well. We show that the extremal problems are equivalent to some more familiar ones concerning trigonometric polynomials, and thus find the extremal values for a few cases. An analysis of the relationship between the problem for $\RR^d$ and that for $\TT^d$ is given, showing that the former case is just the limiting case of the latter. Thus the hierarchy of difficulty is established, so that extremal problems for trigonometric polynomials gain renewed recognition.

We prove $L^p(\mathbb T^2)$ boundedness, $1
Categories:42B20, 42B08

We consider the Neumann problem for the Schr\"odinger equations $-\Delta u+Vu=0$, with singular nonnegative potentials $V$ belonging to the reverse H\"older class $\B_n$, in a connected Lipschitz domain $\Omega\subset\mathbf{R}^n$. Given boundary data $g$ in $H^p$ or $L^p$ for $1-\epsilon
Keywords:Neumann problem, Schrödinger equation, Lipschitz, domain, reverse Hölder class, $H^p$ space
Categories:42B20, 35J10

We continue our investigation in [RST] of a martingale formed by picking a measurable set $A$ in a compact group $G$, taking random rotates of $A$, and considering measures of the resulting intersections, suitably normalized. Here we concentrate on the inverse problem of recognizing $A$ from a small amount of data from this martingale. This leads to problems in harmonic analysis on $G$, including an analysis of integrals of products of Gegenbauer polynomials.

In this paper we estimate the $(L^p-L^2)$-norm of the complex harmonic projectors $\pi_{\ell\ell'}$, $1\le p\le 2$, uniformly with respect to the indexes $\ell,\ell'$. We provide sharp estimates both for the projectors $\pi_{\ell\ell'}$, when $\ell,\ell'$ belong to a proper angular sector in $\mathbb{N} \times \mathbb{N}$, and for the projectors $\pi_{\ell 0}$ and $\pi_{0 \ell}$. The proof is based on an extension of a complex interpolation argument by C.~Sogge. In the appendix, we prove in a direct way the uniform boundedness of a particular zonal kernel in the $L^1$ norm on the unit sphere of $\mathbb{R}^{2n}$.

Eventual positivity problems for real convergent Maclaurin series lead to density questions for sets of harmonic functions. These are solved for large classes of series, and in so doing, asymptotic estimates are obtained for the values of the series near the radius of convergence and for the coefficients of convolution powers.

It is well known that many polynomials which solve extremal problems on a single interval as the Chebyshev or the Bernstein-Szeg\"o polynomials can be represented by trigonometric functions and their inverses. On two intervals one has elliptic instead of trigonometric functions. In this paper we show that the counterparts of the Chebyshev and Bernstein-Szeg\"o polynomials for several intervals can be represented with the help of automorphic functions, so-called Schottky-Burnside functions. Based on this representation and using the Schottky-Burnside automorphic functions as a tool several extremal properties of such polynomials as orthogonality properties, extremal properties with respect to the maximum norm, behaviour of zeros and recurrence coefficients {\it etc.} are derived.

We study the singular integral operator $$ T_{\Omega,\alpha}f(x) = \pv \int_{R^n} b(|y|) \Omega(y') |y|^{-n-\alpha} f(x-y)\,dy, $$ defined on all test functions $f$,where $b$ is a bounded function, $\alpha\geq 0$, $\Omega(y')$ is an integrable function on the unit sphere $S^{n-1}$ satisfying certain cancellation conditions. We prove that, for $1
Categories:42B20, 42B25, 42B15

Let $X$ be a complex Banach space and let $B_p(X)$ denote the vector-valued Bergman space on the unit disc for $1\le p<\infty$. A sequence $(T_n)_n$ of bounded operators between two Banach spaces $X$ and $Y$ defines a multiplier between $B_p(X)$ and $B_q(Y)$ (resp.\ $B_p(X)$ and $\ell_q(Y)$) if for any function $f(z) = \sum_{n=0}^\infty x_n z^n$ in $B_p(X)$ we have that $g(z) = \sum_{n=0}^\infty T_n (x_n) z^n$ belongs to $B_q(Y)$ (resp.\ $\bigl( T_n (x_n) \bigr)_n \in \ell_q(Y)$). Several results on these multipliers are obtained, some of them depending upon the Fourier or Rademacher type of the spaces $X$ and $Y$. New properties defined by the vector-valued version of certain inequalities for Taylor coefficients of functions in $B_p(X)$ are introduced.

We consider a unitary representation of a discrete countable abelian group on a separable Hilbert space which is associated to a cyclic generalized frame multiresolution analysis. We extend Robertson's theorem to apply to frames generated by the action of the group. Within this setup we use Stone's theorem and the theory of projection valued measures to analyze wandering frame collections. This yields a functional analytic method of constructing a wavelet from a generalized frame multi\-resolution analysis in terms of the frame scaling vectors. We then explicitly apply our results to the action of the integers given by translations on $L^2({\mathbb R})$.

We prove that the operators $\int_{\mathbb{R}_+^2} e^{ix^a \cdot y^b} \varphi (x,y) f(y)\, dy$ map $L^p(\mathbb{R}^2)$ into itself for $p \in J =\bigl[\frac{a_l+b_l}{a_l+(\frac{b_l r}{2})},\frac{a_l+b_l} {a_l(1-\frac{r}{2})}\bigr]$ if $a_l,b_l\ge 1$ and $\varphi(x,y)=|x-y|^{-r}$, $0\le r <2$, the result is sharp. Generalizations to dimensions $d>2$ are indicated.

We study when the spaces of Lorentz multipliers from $L^{p,t} \rightarrow L^{p,s}$ are distinct. Our main interest is the case when $s
Keywords:multipliers, convolution operators, Lorentz spaces, Lorentz-improving multipliers
Categories:43A22, 42A45, 46E30

$H^p$ estimate for the multilinear operators which are finite sums of pointwise products of singular integrals and fractional integrals is given. An application to Sobolev space and some examples are also given.

Using Local Residues and the Duality Principle a multidimensional variation of the completeness theorems by T.~Carleman and A.~F.~Leontiev is proven for the space of holomorphic functions defined on a suitable open strip $T_{\alpha}\subset {\bf C}^2$. The completeness theorem is a direct consequence of the Cauchy Residue Theorem in a torus. With suitable modifications the same result holds in ${\bf C}^n$.

We consider the analogue of the $\Lambda(p)-$problem for subsets of the Legendre polynomials or more general ultraspherical polynomials. We obtain the ``best possible'' result that if $2
Categories:42C10, 33C45, 46B07

We obtain necessary and sufficient conditions for mean convergence of Lagrange interpolation at zeros of orthogonal polynomials for weights on $[-1,1]$, such as \[ w(x)=\exp \bigl(-(1-x^{2})^{-\alpha }\bigr),\quad \alpha >0 \] or \[ w(x)=\exp \bigl(-\exp _{k}(1-x^{2})^{-\alpha }\bigr),\quad k\geq 1, \ \alpha >0, \] where $\exp_{k}=\exp \Bigl(\exp \bigl(\cdots\exp (\ )\cdots\bigr)\Bigr)$ denotes the $k$-th iterated exponential.

We define Hardy spaces of conjugate systems of temperature functions on ${\bbd R}_{+}^{n+1}$. We show that their boundary distributions are the same as the boundary distributions of the usual Hardy spaces of conjugate systems of harmonic functions.

Given function $\Omega$ on ${\Bbb R^n}$, we define the fractional maximal operator and the fractional integral operator by $$ M_{\Omega,\alpha}\,f(x)=\sup_{r>0}\frac 1{r^{n-\alpha}} \int_{|\,y|1)$, homogeneous of degree zero.

In this paper we give a characterization of the pairs of weights $(\w,v)$ such that $T$ maps $L^p(v)$ into $L^q(\w)$, where $T$ is a general one-sided operator that includes as a particular case the Weyl fractional integral. As an application we solve the following problem: given a weight $v$, when is there a nontrivial weight $\w$ such that $T$ maps $L^p(v)$ into $L^q(\w )$?

For a truncated matrix moment problem, we describe in detail the set of positive definite matrices of measures $\mu$ in $V_{2n}$ (this is the set of solutions of the problem of degree $2n$) for which the polynomials up to degree $n$ are dense in the corresponding space ${\cal L}^2(\mu)$. These matrices of measures are exactly the extremal measures of the set $V_n$.

Based on the theory of spherical harmonics for measures invariant under a finite reflection group developed by Dunkl recently, we study orthogonal polynomials with respect to the weight functions $|x_1|^{\alpha_1}\cdots |x_d|^{\alpha_d}$ on the unit sphere $S^{d-1}$ in $\RR^d$. The results include explicit formulae for orthonormal polynomials, reproducing and Poisson kernel, as well as intertwining operator.