For general hyperbolic metric spaces, we introduce a new notion of a dual system (extending the influential notion from the context of normed linear spaces) that allows for a uniform study of different notions of duality for these nonlinear spaces. Using this abstract notion of duality, we lift various notions from convex analysis into this nonlinear setting, including Fréchet differentiability and Bregman distances. Further, we introduce a notion of a monotone operator relative to a given dual system and, using the new Fréchet derivatives, we study corresponding resolvents relative to a given gradient, generalizing the seminal notion of Eckstein from the linear setting. These resolvents are then related to corresponding notions of Bregman nonexpansive mappings which are introduced relative to this generalization of the classical Bregman distance and we prove a convergence result of an analogue of the proximal point algorithm. For that, using methods from proof mining, we even provide quantitative results on its convergence in very general settings.
We fully characterize well-posedness in vector-valued Lebesgue–Bochner spaces of a non-local in time Moore–Gibson–Thompson equation in terms of the R-boundedness of the associated operator-valued symbol by using operator valued Fourier multipliers techniques. From our results, we are able to ensure Lp well-posedness when dealing with selfadjoint operators on Hilbert spaces or sectorial ones. This allows to include several classes of operators other than the Laplacian.
We construct invariant quasimorphisms for groups acting on the circle. Furthermore, we provide a criterion for the non-extendablity of the resulting quasimorphisms and an explicit formula which relates the values of our quasimorphisms to those of the Poincaré translation number. By using them, we show that the stable commutator length sclG and the stable mixed commutator length sclG,N are not bi-Lipschitzly equivalent for the surface group G = π_1(Σ_ℓ) of genus at least 2 and its commutator subgroup N = [π_1(Σ_ℓ), π_1(Σ_ℓ)] . We also show the non-equivalence for a pair (G, N) such that G is the fundamental group of a 3-dimensional closed hyperbolic mapping torus. These pairs serve as the first family of examples of such (G, N) in which G is finitely generated.
Let Ω be a homogeneous function of degree zero which has vanishing moment of order one, A be a function on ℝd such that ∇A ∈ BMO(ℝd), and TΩ, A be the operator defined by T_Ω, Af(x) = p. v.∫_ℝ^dΩ(x-y)/|x-y|^d+1 (A(x) - A(y) - ∇ A(y)(x-y)) f(y) dy. In this paper, bilinear sparse dominations are established for the operator TΩ,A and its adjoint operator when Ω∈ L^∞(𝕊^d-1) . As applications, some quantitative weighted estimates for TΩ,A are given.
We precisely identify the extent of determinacy provable in Zermelo–Fraenkel set theory with the Axiom of Choice from the hypothesis that x# exists for every x ∈ ℝ, isolating an optimal strengthening of the Martin–Harrington Determinacy Transfer Theorem in a strong sense.
Let {k∈ℤ^n:1 ≤∏_j=1^n|k_j|^γ_j≤ R^γ_1+…+γ_n}, where γ1,…,γn > 0, be a “hyperbolic crosses” dilated homothetically as R → +∞. In our study, their Lebesgue constants are, as expected, always of power growth Rp, p > 0, maybe up to a logarithmic factor. What turned out to be surprising is that contrary to the expected p=n-1 2 in any case, p may become, for an appropriate choice of γ1,…,γn, an arbitrary number larger than that fraction. In many cases, the estimates of the Lebesgue constants are sharp in the sense that those from above and from below differ from one another only by coefficients.
In this paper, we study blow-up criteria, blow-up rates, global existences and a priori estimates for solutions to nonlinear parabolic problems involving the fractional Laplacian. We provide certain criteria to determine when the blow-up occurs; and in the case it occurs, we obtain almost optimal blow-up rates. We present conditions for the existence of global solutions and derive uniform a priori bounds for such solutions. Unlike parabolic equations involving the regular Laplacian, there have been very few such results for fractional parabolic problems. To circumvent the difficulty caused by the non-locality of the fractional Laplacian, we develop a direct method of blowing-up and re-scaling to derive the blow-up rate. As applications, we also obtain the blow-up rate for indefinite fractional parabolic equations. This new method is not only applicable to various nonlocal problems but also to local ones.
The reflection of stationary subsets of P_ω_1(H) for all sets H ⊇ ω1, which we denote by SR_ω_1 , is known to imply that λω = λ for all regular cardinals λ ≥ ω2. In particular, it implies 2ω ≤ ω2 and the Singular Cardinal Hypothesis. For a regular cardinal κ ≥ ω2, the reflection of stationary subsets of P_κ(H) for all H ⊇ κ is inconsistent with ZFC. But its restriction to stationary sets consisting of internally approachable sets, which we denote by SRκ ↾ IA, is consistent with ZFC. In this paper, we study consequences of SRκ ↾ IA on cardinal arithmetic. We prove that SRκ ↾ IA does not give any bound on 2μ for any regular uncountable cardinal μ, while it implies λω = λ for all regular cardinals λ ≥ κ+. We also prove that SRκ ↾ IA>ω does not give any bound on 2ω and does not imply the Singular Cardinal Hypothesis, where SRκ ↾ IA>ω denotes the reflection of stationary subsets of P_κ(H) consisting of internally approachable sets of uncountable cofinalities.
The development of a substantial body of work on the subject of uniqueness of unconditional structure in Banach and p-Banach spaces sprang from the 1985 celebrated Memoir [12] by Bourgain et al., where the authors aimed at classifying all Banach spaces with that property. One of the most striking results from that paper was that the 2-convexified Tsirelson space, T^(2) , had a unique unconditional basis (up to equivalence and permutation). Forty years later, many of the questions raised in the Memoir remain open but there has been a considerable effort in advancing a topic that had received relatively little attention until then. Continuing in the spirit of the program set in the Memoir, in this note we show that the direct sum of infinitely many copies of T^(2) for 0 < p < 1, denoted ℓ_p(T^(2)) , has a unique unconditional basis, and that the same property holds for ℓ_p((T^(2))^*) . Our results and methods are relevant in applications since they permit us to reprove the uniqueness of the (discrete) lattice structure induced by an unconditional basis in other spaces.
We introduce the notion of cover time to dynamical systems. This quantifies the rate at which orbits become dense in the state space and can be viewed as a global, rather than the more standard local, notion of recurrence for a system. Using transfer operator tools for systems with holes and inducing techniques, we obtain an asymptotic formula for the expected cover time in terms of the decay rate of the measure of the ball of minimum measure. We apply this to a wide class of uniformly hyperbolic and non-uniformly hyperbolic interval maps, including the Gauss map and Manneville–Pomeau maps.
In this paper, we are concerned with the pseudo-relativistic double phase problems. We first establish various maximal principles for nonlocal operators. As applications, combined with the direct method of moving planes, we prove the monotonicity and symmetry of positive solutions to pseudo-relativistic double phase problems with Hartree nonlinearities. Additionally, we also prove the monotonicity of positive bounded solutions to pseudo-relativistic double phase Dirichlet problems on half-spaces ℝ + . We believe that the various new ideas and techniques we used here would be very useful to deal with other nonlocal double phase problems.
We introduce a way of measuring nonconvexity of a metric space. We apply it to define a broad generalization and refinement of the classical curvature of a curve. We also use it to introduce a natural new notion of a fractal set.
Given a Banach space X and d ∈ ℕ, we construct a metric space 𝕍_X^d with the property that every d-homogeneous polynomial defined on X factors through a Lipschitz map on it. We prove that the metric on 𝕍_X^d is independent (up to a constant) of the norm of the tensor space in which it is embedded. We apply this fact to prove that a homogeneous polynomial is Lipschitz q-summing as a polynomial if and only if its associated Lipschitz map is Lipschitz q-summing. This result generalizes the already known theorem for linear operators
We consider an algebraic cycle on the triple product of the prime level modular curve X0(p) with origins in work of Darmon and Rotger. It is defined over the quadratic extension of ℚ ramified only at p whose associated quadratic character χ is the Legendre symbol at p. We prove that it is null-homologous and describe actions of various groups on it. For any three normalized cuspidal eigenforms f1, f2, f3 of weight 2 and level Γ0)(p), we prove that the global root number of the twisted triple product L-function L(f1 ⊗ f2 ⊗ f3 ⊗ χ, s) is −1. Assuming conjectures of Beilinson and Bloch, and guided by the Gross–Zagier philosophy, this suggests that the Darmon–Rotger cycle could be non-torsion, although we do not currently have a proof of this.
Recently, Glasner, Lin and Meyerovitch gave a first example of a partial invariant order on a certain group that cannot be invariantly extended to an invariant random total order. Using their result as a starting point we prove that any invariant random partial order on a countable group could be invariantly extended to an invariant random total order iff the group is amenable.
We prove a boundary version of the strong form of the Ahlfors-Schwarz lemma with optimal error term. This result provides nonlinear extensions of the boundary Schwarz lemma of Burns and Krantz to the class of negatively curved conformal pseudometrics defined on arbitary hyperbolic domains in the complex plane. Based on a new boundary Harnack inequality for solutions of the Gauss curvature equation, we also establish a sharp rigidity result for conformal metrics with isolated singularities. In the particular case of constant negative curvature this strengthens classical results of Nitsche and Heins about Liouville's equation $\Delta u=e^u$.
Let R=𝕂[x_1,…,x_n] and let a_1,…,a_m be homogeneous ideals satisfying certain properties, which include a description of the Noetherian symbolic Rees algebra. We give a solution to a question of Harbourne and Huneke for this set of ideals. We also compute the Waldschmidt constant and resurgence and show that it exhibits a stronger version of the Chudnovsky and Demailly-type bounds. We further show that these properties are satisfied for classical varieties such as the generic determinantal ideals, minors of generic symmetric matrices, generic extended Hankel matrices, and ideals of pfaffians of skew-symmetric matrices.
We give a new geometric proof of a theorem of Heuer showing that, in the presence of letter-quasimorphisms (which are analogues of real-valued quasimorphisms with image in free groups), and in particular in RAAGs, there is a sharp lower bound of 1/2 for stable commutator length. Our approach is to show that letter-quasimorphisms give rise to negatively curved angle structures on admissible surfaces. This generalises Duncan and Howie's proof of the 1/2-lower bound in free groups, and can also be seen as a version of Bavard Duality for letter-quasimorphisms.
In this article we prove that, over complete manifolds of dimension n with vanishing curvature at infinity, the essential spectrum of the Hodge Laplacian on differential k-forms is a connected interval for 0 ≤ k ≤ n. The main idea is to show that large balls of these manifolds, which capture their spectrum, are close in the Gromov–Hausdorff sense to product manifolds. We achieve this by carefully describing the collapsed limits of these balls. Then, via a new generalized version of the classical Weyl criterion, we demonstrate that very rough test forms that we get from the ε-approximation maps can be used to show that the essential spectrum is a connected interval. We also prove that, under a weaker condition where the Ricci curvature is asymptotically nonnegative, the essential spectrum on k-forms is [0, ∞), but only for 0 ≤ k ≤ q and n − q ≤ k ≤ n for some integer q ≥ 1 which depends on the structure of the manifolds at infinity. Our results can also be generalized to Schrödinger operators with double-well potential.
