485 research outputs found

### On zero sets in the Dirichlet space

We study the zeros sets of functions in the Dirichlet space. Using Carleson
formula for Dirichlet integral, we obtain some new families of zero sets. We
also show that any closed subset of E \subset \TT with logarithmic capacity
zero is the accumulation points of the zeros of a function in the Dirichlet
space. The zeros satisfy a growth restriction which depends on $E$.Comment: Journal of Geometric Analysis (2011

### Another extension of the disc algebra

We identify the complex plane C with the open unit disc D={z:|z|<1} by the
homeomorphism z --> z/(1+|z|). This leads to a compactification $\bar{C}$ of C,
homeomorphic to the closed unit disc. The Euclidean metric on the closed unit
disc induces a metric d on $\bar{C}$. We identify all uniform limits of
polynomials on $\bar{D}$ with respect to the metric d. The class of the above
limits is an extension of the disc algebra and it is denoted by $\bar{A}(D)$.
We study properties of the elements of $\bar{A}(D)$ and topological properties
of the class $\bar{A}(D)$ endowed with its natural topology. The class
$\bar{A}(D)$ is different and, from the geometric point of view, richer than
the class $\tilde{A}(D)$ introduced in Nestoridis (2010), Arxiv:1009.5364, on
the basis of the chordal metric.Comment: 14 page

### The Vector Valued Quartile Operator

Certain vector-valued inequalities are shown to hold for a Walsh analog of
the bilinear Hilbert transform. These extensions are phrased in terms of a
recent notion of quartile type of a UMD (Unconditional Martingale Differences)
Banach space X. Every known UMD Banach space has finite quartile type, and it
was recently shown that the Walsh analog of Carleson's Theorem holds under a
closely related assumption of finite tile type. For a Walsh model of the
bilinear Hilbert transform however, the quartile type should be sufficiently
close to that of a Hilbert space for our results to hold. A full set of
inequalities is quantified in terms of quartile type.Comment: 32 pages, 5 figures, incorporates referee's report, to appear in
Collect. Mat

### Pointwise convergence of vector-valued Fourier series

We prove a vector-valued version of Carleson's theorem: Let Y=[X,H]_t be a
complex interpolation space between a UMD space X and a Hilbert space H. For
p\in(1,\infty) and f\in L^p(T;Y), the partial sums of the Fourier series of f
converge to f pointwise almost everywhere. Apparently, all known examples of
UMD spaces are of this intermediate form Y=[X,H]_t. In particular, we answer
affirmatively a question of Rubio de Francia on the pointwise convergence of
Fourier series of Schatten class valued functions.Comment: 26 page

### On the structure of covariant phase observables

We study the mathematical structure of covariant phase observables. Such an
observable can alternatively be expressed as a phase matrix, as a sequence of
unit vectors, as a sequence of phase states, or as an equivalent class of
covariant trace-preserving operations. Covariant generalized operator measures
are defined by structure matrices which form a W*-algebra with phase matrices
as its subset. The properties of the Radon-Nikodym derivatives of phase
probability measures are studied.Comment: 11 page

### Local dynamics for fibered holomorphic transformations

Fibered holomorphic dynamics are skew-product transformations over an
irrational rotation, whose fibers are holomorphic functions. In this paper we
study such a dynamics on a neighborhood of an invariant curve. We obtain some
results analogous to the results in the non fibered case

### Hastings-Levitov aggregation in the small-particle limit

We establish some scaling limits for a model of planar aggregation. The model is described by the composition of a sequence of independent and identically distributed random conformal maps, each corresponding to the addition of one particle. We study the limit of small particle size and rapid aggregation. The process of growing clusters converges, in the sense of Caratheodory, to an inflating disc. A more refined analysis reveals, within the cluster, a tree structure of branching fingers, whose radial component increases deterministically with time. The arguments of any finite sample of fingers, tracked inwards, perform coalescing Brownian motions. The arguments of any finite sample of gaps between the fingers, tracked outwards, also perform coalescing Brownian motions. These properties are closely related to the evolution of harmonic measure on the boundary of the cluster, which is shown to converge to the Brownian web

### Quantitative estimates of discrete harmonic measures

A theorem of Bourgain states that the harmonic measure for a domain in $\R^d$
is supported on a set of Hausdorff dimension strictly less than $d$
\cite{Bourgain}. We apply Bourgain's method to the discrete case, i.e., to the
distribution of the first entrance point of a random walk into a subset of $\Z
^d$, $d\geq 2$. By refining the argument, we prove that for all \b>0 there
exists \rho (d,\b)N(d,\b), any $x \in
\Z^d$, and any $A\subset \{1,..., n\}^d$ | \{y\in\Z^d\colon \nu_{A,x}(y)
\geq n^{-\b} \}| \leq n^{\rho(d,\b)}, where $\nu_{A,x} (y)$ denotes the
probability that $y$ is the first entrance point of the simple random walk
starting at $x$ into $A$. Furthermore, $\rho$ must converge to $d$ as \b \to
\infty.Comment: 16 pages, 2 figures. Part (B) of the theorem is ne

### Divided Differences & Restriction Operator on Paley-Wiener Spaces $PW_{tau}^{p}$ for $N-$Carleson Sequences

For a sequence of complex numbers $\Lambda$ we consider the restriction
operator $R_{\Lambda}$ defined on Paley-Wiener spaces $PW_{\tau}^{p}$
($1<p<\infty$). Lyubarskii and Seip gave necessary and sufficient conditions on
$\Lambda$ for $R_{\Lambda}$ to be an isomorphism between $PW_{\tau}^{p}$ and a
certain weighted $l^{p}$ space. The Carleson condition appears to be necessary.
We extend their result to $N-$Carleson sequences (finite unions of $N$ disjoint
Carleson sequences). More precisely, we give necessary and sufficient
conditions for $R_{\Lambda}$ to be an isomorphism between $PW_{\tau}^{p}$ and
an appropriate sequence space involving divided differences

### dbar-equations, integrable deformations of quasiconformal mappings and Whitham hierarchy

It is shown that the dispersionless scalar integrable hierarchies and, in
general, the universal Whitham hierarchy are nothing but classes of integrable
deformations of quasiconformal mappings on the plane. Examples of deformations
of quasiconformal mappings associated with explicit solutions of the
dispersionless KP hierarchy are presented.Comment: 13 pages, LaTe

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