Journal d'Analyse Mathématique最新文献

The trigonometric monomial cos(〈k, x〉)on (mathbb{T}^{d}), a harmonic polynomial (p:mathbb{S}^{d-1}rightarrowmathbb{R}) of degree k and a Laplacian eigenfunction −Δf = k²f have a root in each ball of radius ∼ ∥k∥⁻¹ or ∼ k⁻¹, respectively. We extend this to linear combinations and show that for any trigonometric polynomials on (mathbb{T}^{d}), any polynomial p ∈ ℝ[x₁,…,x_d] restricted to (mathbb{S}^{d-1}) and any linear combination of global Laplacian eigenfunctions on ℝ^d with d ∈ {2, 3} the same property holds for any ball whose radius is given by the sum of the inverse constituent frequencies. We also refine the fact that an eigenfunction −Δφ = λφ in Ω ⊂ ℝⁿ has a root in each B(x, α_nλ^−1/2) ball: the positive and negative mass in each B(x, β_nλ^−1/2) ball cancel when integrated against ∥x − y∥²⁻ⁿ.

Necessary and sufficient conditions are obtained for injectivity of the shifted Funk–Radon transform associated with k-dimensional totally geodesic submanifolds of the unit sphere Sⁿ in ℝⁿ⁺¹. This result generalizes the well known statement for the spherical means on Sⁿ and is formulated in terms of zeros of Jacobi polynomials. The relevant harmonic analysis is developed, including a new concept of induced Stiefel (or Grassmannian) harmonics, the Funk–Hecke type theorems, addition formula, and multipliers. Some perspectives and conjectures are discussed.

A function f is arc-smooth if the composite f ◦ c with every smooth curve c in its domain of definition is smooth. On open sets in smooth manifolds the arc-smooth functions are precisely the smooth functions by a classical theorem of Boman. Recently, we extended this result to certain tame closed sets (namely, Hölder sets and simple fat subanalytic sets). In this paper we link, in a precise way, the cuspidality of the (boundary of the) set to the loss of regularity, i.e., how many derivatives of f ◦ c are needed in order to determine the derivatives of f. We also discuss how flatness of f ◦ c affects flatness of f. Besides Hölder sets and subanalytic sets we treat sets that are definable in certain polynomially bounded o-minimal expansions of the real field.

Double forms are sections of the vector bundles (Lambda^{k}T^{ast}{cal{M}}otimesLambda^{m}T^{ast}cal{M}), where in this work ((cal{M},frak{g})) is a compact Riemannian manifold with boundary. We study graded second-order differential operators on double forms, which are used in physical applications. A combination of these operators yields a fourth-order operator, which we call a double bilaplacian. We establish the regular ellipticity of the double bilaplacian for several sets of boundary conditions. Under additional conditions, we obtain a Hodge-like decomposition for double forms, whose components are images of the second-order operators, along with a biharmonic element. This analysis lays foundations for resolving several topics in incompatible elasticity, most prominently the existence of stress potentials and Saint-Venant compatibility.

We prove an abstract theorem which provides multiple critical points for locally Lipschtiz functionals under the presence of symmetry. The abstract result is applied to find multiple solutions in H¹₀ (Ω) for the critical semi-linear elliptic equation − Δu = f(x, u) + ∣u∣^4/(N−2)u, where f is a discontinuous perturbation and Ω ⊂ ℝ^N is a bounded smooth domain.

In this article we study local and global properties of positive solutions of − Δ_mu = ∣u∣^p−1u+M∣∇u∣^q in a domain Ω of ℝ^N, with m > 1, p, q > 0 and M ∈ ℝ. Following some ideas used in [7, 8], and by using a direct Bernstein method combined with Keller–Osserman’s estimate, we obtain several a priori estimates as well as Liouville type theorems. Moreover, we prove a local Harnack inequality with the help of Serrin’s classical results.

We examine the limiting behavior of multiple ergodic averages associated with arithmetic progressions whose differences are elements of a fixed integer sequence. For each ℓ, we give necessary and sufficient conditions under which averages of length ℓ of the aforementioned form have the same limit as averages of ℓ-term arithmetic progressions. As a corollary, we derive a sufficient condition for the presence of arithmetic progressions with length ℓ+1 and restricted differences in dense subsets of integers. These results are a consequence of the following general theorem: in order to verify that a multiple ergodic average is controlled by the degree d Gowers–Host–Kra seminorm, it suffices to show that it is controlled by some Gowers–Host–Kra seminorm, and that the degree d control follows whenever we have degree d + 1 control. The proof relies on an elementary inverse theorem for the Gowers–Host–Kra seminorms involving dual functions, combined with novel estimates on averages of seminorms of dual functions. We use these estimates to obtain a higher order variant of the degree lowering argument previously used to cover averages that converge to the product of integrals.

We study perturbations of non-recurrent parameters in the exponential family. It is shown that the set of such parameters has Lebesgue measure zero. This particularly implies that the set of escaping parameters has Lebesgue measure zero, which complements a result of Qiu from 1994. Moreover, we show that non-recurrent parameters can be approximated by hyperbolic ones.

We study the norm of point evaluation at the origin in the Paley–Wiener space PW^p for 0 < p < ∞, i.e., we search for the smallest positive constant C, called ({mathscr{C}}_{p}), such that the inequality (vert f(0)vert ^{p}leq C Vert fVert _{p}^{p}) holds for every f in PW^p. We present evidence and prove several results supporting the following monotonicity conjecture: The function (p mapsto {mathscr{C}}_{p}/p) is strictly decreasing on the half-line (0, ∞). Our main result implies that ({mathscr{C}}_{p} < p/2) for 2 < p < ∞, and we verify numerically that ({mathscr{C}}_{p} > p/2) for 1 ≤ p < 2. We also estimate the asymptotic behavior of ({mathscr{C}}_{p}) as p → ∞ and as p → 0⁺. Our approach is based on expressing ({mathscr{C}}_{p}) as the solution of an extremal problem. Extremal functions exist for all 0 < p < ∞; they are real entire functions with only real zeros, and the extremal functions are known to be unique for 1 ≤ p < ∞. Following work of Hörmander and Bernhardsson, we rely on certain orthogonality relations associated with the zeros of extremal functions, along with certain integral formulas representing respectively extremal functions and general functions at the origin. We also use precise numerical estimates for the largest eigenvalue of the Landau–Pollak–Slepian operator of time-frequency concentration. A number of qualitative and quantitative results on the distribution of the zeros of extremal functions are established. In the range 1 < p < ∞, the orthogonality relations associated with the zeros of the extremal function are linked to a de Branges space. We state a number of conjectures and further open problems pertaining to ({mathscr{C}}_{p}) and the extremal functions.

We study how the spectral properties of ergodic Schrödinger operators are reflected in the asymptotic properties of its periodic approximation as the period tends to infinity. The first property we address is the asymptotics of the bandwidths on the logarithmic scale, which quantifies the sensitivity of the finite volume restriction to the boundary conditions. We show that the bandwidths can always be bounded from below in terms of the Lyapunov exponent. Under an additional assumption satisfied by i.i.d. potentials, we also prove a matching upper bound. Finally, we provide an additional assumption which is also satisfied in the i.i.d. case, under which the corresponding eigenvectors are exponentially localised with a localisation centre independent of the Floquet number.