Mostrando recursos 1 - 20 de 355

  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.

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