1.
$L^p$ improving multilinear Radon-like transforms - Stovall
,
Betsy
We characterize (up to endpoints) the $k$-tuples $(p_1,\ldots,p_k)$ for
which certain $k$-linear generalized Radon transforms map the product
$L^{p_1} \times \cdots \times L^{p_k}$ boundedly into $\mathbb{R}$.
This generalizes a result of Tao and Wright.
2.
Product kernels adapted to curves in the space - Casarino
,
Valentina; Ciatti
,
Paolo; Secco
,
Silvia
We establish $L^p$-boundedness for a class of operators that are given by
convolution with product kernels adapted to curves in the space. The $L^p$
bounds follow from the decomposition of the adapted kernel into a sum of
two kernels with singularities concentrated respectively on a coordinate
plane and along the curve.
The proof of the $L^p$-estimates for the two corresponding operators involves
Fourier analysis techniques and some algebraic tools, namely the Bernstein-Sato
polynomials.
As an application, we show that these bounds can be exploited in the study of
$L^p-L^q$ estimates for analytic families of fractional operators along curves
in the space.
3.
Sub-Riemannian geometry of parallelizable spheres - Godoy Molina
,
Mauricio; Markina
,
Irina
The first aim of the present paper is to compare various sub-Riemannian
structures over the three dimensional sphere $S^3$ originating from
different constructions. Namely, we describe the sub-Riemannian geometry
of $S^3$ arising through its right action as a Lie group over itself, the
one inherited from the natural complex structure of the open unit ball in
$\mathbb{C}^2$ and the geometry that appears when it is considered as a
principal $S^1$-bundle via the Hopf map. The main result of this comparison
is that in fact those three structures coincide.
We present two bracket generating distributions for the seven dimensional
sphere $S^7$ of step 2 with ranks 6 and 4. The...
4.
Closed ideals of $A^\infty$ and a famous problem of Grothendieck - Patel
,
S. R.
Using Fréchet algebraic technique, we show the existence of a
nuclear Fréchet space without basis, thus providing yet another
proof (of a different flavor) of a negative answer to a well known
problem of Grothendieck from 1955. Using Fefferman's construction (which
is based on complex-variable technique) of a $C^\infty$-function on
the unit circle with certain properties, we give much simpler, transparent,
and "natural" examples of restriction spaces without bases of nuclear
Fréchet spaces of $C^\infty$-functions; these latter spaces,
being classical objects of study, have attracted some attention because
of their relevance to the theories of PDE and complex dynamical systems,
and harmonic analysis. In particular, the restriction space $A^\infty(E)$,
being a...
5.
Geometric-arithmetic averaging of dyadic weights - Pipher
,
Jill; Ward
,
Lesley A.; Xiao
,
Xiao
The theory of Muckenhoupt's weight functions arises in many areas of
analysis, for example in connection with bounds for singular integrals
and maximal functions on weighted spaces. We prove that a certain
averaging process gives a method for constructing $A_p$ weights from
a measurably varying family of dyadic $A_p$ weights. This averaging
process is suggested by the relationship between the $A_p$ weight class
and the space of functions of bounded mean oscillation. The same averaging
process also constructs weights satisfying reverse Hölder ($RH_p$)
conditions from families of dyadic $RH_p$ weights, and extends to the
polydisc as well.
6.
On the interplay between Lorentzian Causality and Finsler metrics of Randers type - Caponio
,
Erasmo; Javaloyes
,
Miguel Ángel; Sánchez
,
Miguel
We obtain some results in both Lorentz and Finsler geometries, by using a
correspondence between the conformal structure (Causality) of standard
stationary spacetimes on $M = \mathbb{R} \times S$ and Randers metrics on $S$.
In particular:
(1) For stationary spacetimes: we give a simple characterization of when
$\mathbb{R} \times S$ is causally continuous or globally hyperbolic (including
in the latter case, when $S$ is a Cauchy hypersurface), in terms of an
associated Randers metric. Consequences for the computability of Cauchy
developments are also derived.
(2) For Finsler geometry: Causality suggests that the role of completeness in
many results of Riemannian Geometry (geodesic connectedness by minimizing geodesics,
Bonnet-Myers, Synge theorems) is played...
7.
Steiner and Schwarz symmetrization in warped products and fiber bundles with density - Morgan
,
Frank; Howe
,
Sean; Harman
,
Nate
We provide very general symmetrization theorems in arbitrary dimension
and codimension, in products, warped products, and certain fiber bundles
such as lens spaces, including Steiner, Schwarz, and spherical symmetrization
and admitting density.
8.
Nonnegative solutions of the heat equation on rotationally symmetric Riemannian
manifolds and semismall perturbations - Murata
,
Minoru
Let $M$ be a rotationally symmetric Riemannian manifold, and $\Delta$ be the
Laplace-Beltrami operator on $M$. We establish a necessary and sufficient condition
for the constant function 1 to be a semismall perturbation of $-\Delta +1$ on $M$,
and give optimal sufficient conditions for uniqueness of nonnegative solutions of
the Cauchy problem to the heat equation. As an application, we determine the
structure of all nonnegative solutions to the heat equation on $M\times(0,T)$.
9.
Auslander bounds and homological conjectures - Wei
,
Jiaqun
Inspired by recent works on rings satisfying Auslander's conjecture, we study
invariants, called Auslander bounds, and prove that they have strong relations
to some homological conjectures.
10.
Harmonic polynomials and tangent measures of harmonic measure - Badger
,
Matthew
We show that on an NTA domain if each tangent measure to harmonic measure at a
point is a polynomial harmonic measure then the associated polynomials are
homogeneous. Geometric information for solutions of a two-phase free boundary
problem studied by Kenig and Toro is derived.
11.
Regularity, local behavior and partial uniqueness for self-similar profiles of
Smoluchowski's coagulation equation - Cañizo
,
José A.; Mischler
,
Stéphane
We consider Smoluchowski's equation with a homogeneous kernel of the
form $a(x,y) = x^\alpha y ^\beta + x^\beta y^\alpha$ with
$-1 < \alpha \leq \beta < 1$ and $\lambda := \alpha + \beta \in (-1,1)$.
We first show that self-similar solutions of this equation
are infinitely differentiable and prove sharp results on the
behavior of self-similar profiles at $y = 0$ in the case $\alpha < 0$.
We also give some partial uniqueness results for self-similar
profiles: in the case $\alpha = 0$ we prove that two profiles with
the same mass and moment of order $\lambda$ are necessarily equal,
while in the case $\alpha < 0$ we prove...
12.
Partial regularity for subquadratic parabolic systems by $\mathcal{A}$-caloric approximation - Scheven
,
Christoph
We establish a partial regularity result for weak solutions of nonsingular
parabolic systems with subquadratic growth of the type
$$
\partial_t u - \mathrm{div} a(x,t,u,Du) = B(x,t,u,Du),
$$
where the structure function $a$ satisfies ellipticity and growth
conditions with growth rate $\frac{2n}{n+2} < p < 2$.
We prove Hölder continuity of the spatial gradient of solutions
away from a negligible set. The proof is based on a variant of a
harmonic type approximation lemma adapted to parabolic systems with
subquadratic growth.
13.
Finiteness of endomorphism algebras of CM modular abelian varieties - González
,
Josep
Let $A_f$ be the abelian variety attached by Shimura to a normalized newform
$f\in S_2(\Gamma_1(N))^{\operatorname{new}}$. We prove that for any integer
$n > 1$ the set of pairs of endomorphism algebras
$\big( \operatorname{End}_{\overline{\mathbb{Q}}}(A_f) \otimes \mathbb{Q},
\operatorname{End}_\mathbb{Q}(A_f) \otimes \mathbb{Q} \big)$
obtained from all normalized newforms $f$ with complex multiplication such
that $\dim A_f=n$ is finite. We determine that this set has exactly 83 pairs
for the particular case $n=2$ and show all of them. We also discuss a conjecture
related to the finiteness of the set of number fields
$\operatorname{End}_\mathbb{Q}(A_f) \otimes \mathbb{Q}$ for the non-CM case.
14.
Majorizing measures and proportional subsets of bounded orthonormal systems - Guédon
,
Olivier; Mendelson
,
Shahar; Pajor
,
Alain; Tomczak-Jaegermann
,
Nicole
In this article we prove that for any orthonormal system
$(\varphi_j)_{j=1}^n \subset L_2$ that is bounded in $L_{\infty}$, and
any $1 < k < n$, there exists a subset $I$ of cardinality greater
than $n-k$ such that on $\mathrm{span}\{\varphi_i\}_{i \in I}$, the $L_1$ norm
and the $L_2$ norm are equivalent up to a factor $\mu (\log
\mu)^{5/2}$, where $\mu = \sqrt{n/k} \sqrt{\log k}$. The proof is
based on a new estimate of the supremum of an empirical process on
the unit ball of a Banach space with a good modulus of convexity,
via the use of majorizing measures.
15.
Harnack inequality for hypoelliptic ultraparabolic equations
with a singular lower order term - Polidoro
,
Sergio; Ragusa
,
Maria Alessandra
We prove a Harnack inequality for the positive solutions of ultraparabolic
equations of the type
$$
\mathcal {L}_0 u + \mathcal {V} u = 0,
$$
where $\mathcal {L}_0$ is a linear second order hypoelliptic operator and
$\mathcal {V}$ belongs to a class of functions of Stummel-Kato type.
We also obtain the existence of a Green function and an uniqueness result
for the Cauchy-Dirichlet problem.
16.
Almost classical solutions of Hamilton-Jacobi equations - Deville
,
Robert; Jaramillo
,
Jesús A.
We study the existence of everywhere differentiable functions
which are almost everywhere solutions of quite general
Hamilton-Jacobi equations on open subsets of $\mathbb R^d$ or on
$d$-dimensional manifolds whenever $d\geq 2$. In particular,
when $M$ is a Riemannian manifold, we prove the existence of a
differentiable function $u$ on $M$ which satisfies the Eikonal
equation $\Vert \nabla u(x) \Vert_{x}=1$ almost everywhere on $M$.
17.
A note on boundaries of open polynomial images of $\mathbb R^2$ - Ueno
,
Carlos
We construct a family of polynomial maps $\mathbb{R}^2\rightarrow\mathbb{R}^2$
such that their images are open semialgebraic sets whose topological exteriors
have arbitrarily many connected components, which are parametric semilines.
18.
Homology exponents for $H$-spaces - Clément
,
Alain; Scherer
,
Jérôme
We say that a space $X$ admits a \emph{homology exponent} if there
exists an exponent for the torsion subgroup of $H^*(X;\mathbb Z)$.
Our main result states that if an $H$-space of finite type admits
a homology exponent, then either it is, up to $2$-completion,
a product of spaces of the form $B\mathbb Z/2^r$, $S^1$,
$\mathbb C P^\infty$, and $K(\mathbb Z,3)$, or it has infinitely
many non-trivial homotopy groups and $k$-invariants. Relying on
recent advances in the theory of $H$-spaces, we then show that
simply connected $H$-spaces whose mod $2$ cohomology is finitely
generated as an algebra over the Steenrod algebra do not have
homology exponents, except products of mod $2$...
19.
Tropical resultants for curves and stable intersection - Tabera
,
Luis Felipe
We introduce the notion of resultant of two planar curves in the
tropical geometry framework. We prove that the tropicalization of
the algebraic resultant can be used to compute the stable
intersection of two tropical plane curves. It is shown that, for two
generic preimages of the curves to an algebraic framework, their
intersection projects exactly onto the stable intersection of the
curves. It is also given sufficient conditions for such a generality
in terms of the residual coefficients of the algebraic coefficients
of defining equations of the curves.
20.
Reflections of regular maps and Riemann surfaces - Melekoğlu
,
Adnan; Singerman
,
David
A compact Riemann surface of genus $g$ is called an M-surface if
it admits an anti-conformal involution that fixes $g+1$ simple
closed curves, the maximum number by Harnack's Theorem. Underlying
every map on an orientable surface there is a Riemann surface and
so the conclusions of Harnack's theorem still apply. Here we show
that for each genus $g ϯ 1$ there is a unique M-surface of genus $g$
that underlies a regular map, and we prove a similar result for
Riemann surfaces admitting anti-conformal involutions that fix $g$
curves.
1.
