Analysis & PDE最新文献

We establish a sharp comparison inequality between the negative moments and the second moment of the magnitude of sums of independent random vectors uniform on three-dimensional Euclidean spheres. This provides a probabilistic extension of the Oleszkiewicz–Pełczyński polydisc slicing result. The Haagerup-type phase transition occurs exactly when the $p$-norm recovers volume, in contrast to the real case. We also obtain partial results in higher dimensions.

We study polarised algebraic degenerations of Calabi–Yau manifolds. We prove a uniform Skoda-type estimate and a uniform $L^{\infty }$-estimate for the Calabi–Yau Kähler potentials.

Motivated by the study of small amplitude nonlinear waves in the anti-de Sitter spacetime and in particular the conjectured existence of periodic in time solutions to the Einstein equations, we construct families of arbitrary small time-periodic solutions to the conformal cubic wave equation and the spherically symmetric Yang–Mills equations on the Einstein cylinder $ℝ�\times {\mathbb{𝕊}}^3$. For the conformal cubic wave equation, we consider both spherically symmetric solutions and complex-valued aspherical solutions with an ansatz relying on the Hopf fibration of the $3$-sphere. In all three cases, the equations reduce to $1$+$1$ semilinear wave equations.

Our proof relies on a theorem of Bambusi–Paleari for which the main assumption is the existence of a seed solution, given by a nondegenerate zero of a nonlinear operator associated with the resonant system. For the problems that we consider, such seed solutions are simply given by the mode solutions of the linearized equations. Provided that the Fourier coefficients of the systems can be computed, the nondegeneracy conditions then amount to solving infinite dimensional linear systems. Since the eigenfunctions for all three cases studied are given by Jacobi polynomials, we derive the different Fourier and resonant systems using linearization and connection formulas as well as integral transformation of Jacobi polynomials.

In the Yang–Mills case, the original version of the theorem of Bambusi–Paleari is not applicable because the nonlinearity of smallest degree is nonresonant. The resonant terms are then provided by the next order nonlinear terms with an extra correction due to backreaction terms of the smallest degree of nonlinearity, and we prove an analogous theorem in this setting.

We prove the strong unique continuation property for the differential inequality

with $V$ contained in weak spaces. In particular, we establish the strong unique continuation property for $V�\in L_t^{\infty }{L}_x^{[t]d∕2,\infty }$, which has been left open since the works of Escauriaza (2000) and Escauriaza and Vega (2001). Our results are consequences of the Carleman estimates for the heat operator in the Lorentz spaces.

We consider the long-time behavior of a fast, charged particle interacting with an initially spatially homogeneous background plasma. The background is modeled by the screened Vlasov–Poisson equations, whereas the interaction potential of the point charge is assumed to be smooth. We rigorously prove the validity of the stopping power theory in physics, which predicts a decrease of the velocity $V�(t)$ of the point charge given by $\dot{V}�\sim -|V{|}^{-3}V$, a formula that goes back to Bohr (1915). Our result holds for all initial velocities larger than a threshold value that is larger than the velocity of all background particles and remains valid until the particle slows down to the threshold velocity or the time is exponentially long compared to the velocity of the point charge.

The long-time behavior of this coupled system is related to the question of Landau damping, which has remained open in this setting so far. Contrary to other results in nonlinear Landau damping, the long-time behavior of the system is driven by the nontrivial electric field of the plasma, and the damping only occurs in regions that the point charge has already passed.

Let $\mathcal{𝒫}$ be a countable multiset of primes and let $G�={\oplus <!--FUNCTION\; APPLICATION-->�}_{p\in P}\mathbb{Z}∕p\mathbb{Z}$. We study the universal characteristic factors associated with the Gowers–Host–Kra seminorms for the group $G$. We show that the universal characteristic factor of order $�<�k�+1$ is a factor of an inverse limit of finite-dimensional$k$-stepnilpotent homogeneous spaces. The latter is a counterpart of a $k$-step nilsystem where the homogeneous group is not necessarily a Lie group. As an application of our structure theorem we derive an alternative proof for the $L^2$-convergence of multiple ergodic averages associated with $k$-term arithmetic progressions in $G$ and derive a formula for the limit in the special case where the underlying space is a nilpotent homogeneous system. Our results provide a counterpart of the structure theorem of Host and Kra (2005) and Ziegler (2007) concerning $\mathbb{Z}$-actions and generalize the results of Bergelson, Tao and Ziegler (2011, 2015) concerning ${\mathbb{𝔽}}_p^{\omega }$-actions. This is also the first instance of studying the Host–Kra factors of nonfinitely generated groups of unbounded torsion.

We investigate $L_p$-estimates for balanced averages of Fourier truncations in group algebras, in terms of “differential operators” acting on them. Our results extend a fundamental inequality of Naor for the hypercube (with profound consequences in metric geometry) to discrete groups. Different inequalities are established in terms of “directional derivatives” which are constructed via affine representations determined by the Fourier truncations. Our proofs rely on the Banach ${X<!--FUNCTION\; APPLICATION-->}_p$ nature of noncommutative $L_p$-spaces and dimension-free estimates for noncommutative Riesz transforms. In the particular case of free groups we use an alternative approach based on free Hilbert transforms.

The well-known theorem of Shalom–Vaserstein and Ershov–Jaikin-Zapirain states that the group ${\mathrm{EL}<!--FUNCTION\; APPLICATION-->}_n(\mathcal{ℛ})$, generated by elementary matrices over a finitely generated commutative ring $\mathcal{ℛ}$, has Kazhdan’s property (T) as soon as $n�\ge 3$. This is no longer true if the ring $\mathcal{ℛ}$ is replaced by a commutative rng (a ring but without the identity) due to nilpotent quotients ${\mathrm{EL}<!--FUNCTION\; APPLICATION-->}_n(\mathcal{ℛ}∕{\mathcal{ℛ}}^k)$. We prove that even in such a case the group ${\mathrm{EL}<!--FUNCTION\; APPLICATION-->}_n(\mathcal{ℛ})$ satisfies a certain property that can substitute property (T), provided that $n$ is large enough.

We study minimal graphs with linear growth on complete manifolds $M^m$ with $\mathrm{Ric}<!--FUNCTION\; APPLICATION-->�\ge 0$. Under the further assumption that the $(m-2)$-th Ricci curvature in radial direction is bounded below by $Cr{(x)}^{-2}$, we prove that any such graph, if nonconstant, forces tangent cones at infinity of $M$ to split off a line. Note that $M$ is not required to have Euclidean volume growth. We also show that $M$ may not split off any line. Our result parallels that obtained by Cheeger, Colding and Minicozzi for harmonic functions. The core of the paper is a new refinement of Korevaar’s gradient estimate for minimal graphs, together with heat equation techniques.