# Showing papers in "Mathematische Annalen in 2020"

••

TL;DR: In this paper, the authors derived a sufficient criterion for homologically smooth graded gentle algebras to be derived equivalent, using numerical invariants generalizing those given by Avella et al. (J Pure Appl Algebra 212(1):228-243, 2008).

Abstract: Following the approach of Haiden–Katzarkov–Kontsevich (Publ Math Inst Hautes Etudes Sci 126:247–318, 2017), to any homologically smooth $$\mathbb {Z}$$-graded gentle algebra A we associate a triple $$(\Sigma _A, \Lambda _A; \eta _A)$$, where $$\Sigma _A$$ is an oriented smooth surface with non-empty boundary, $$\Lambda _A$$ is a set of stops on $$\partial \Sigma _A$$ and $$\eta _A$$ is a line field on $$\Sigma _A$$, such that the derived category of perfect dg-modules of A is equivalent to the partially wrapped Fukaya category of $$(\Sigma _A, \Lambda _A ;\eta _A)$$. Modifying arguments of Johnson and Kawazumi, we classify the orbit decomposition of the action of the (symplectic) mapping class group of $$\Sigma _A$$ on the homotopy classes of line fields. As a result we obtain a sufficient criterion for homologically smooth graded gentle algebras to be derived equivalent. Our criterion uses numerical invariants generalizing those given by Avella–Alaminos–Geiss in Avella et al. (J Pure Appl Algebra 212(1):228–243, 2008), as well as some other numerical invariants. As an application, we find many new cases when the AAG-invariants determine the derived Morita class. As another application, we establish some derived equivalences between the stacky nodal curves considered in Lekili and Polishchuk (J Topology 11:615–444, 2018)

73 citations

••

TL;DR: In this paper, the nonlinear elastic wave equations generalizing Gol’dberg's five constants model were considered and the boundary measurements were used to solve the inverse problem of determining elastic parameters from the displacement-to-traction map.

Abstract: We consider nonlinear elastic wave equations generalizing Gol’dberg’s five constants model. We analyze the nonlinear interaction of two distorted plane waves and characterize the possible nonlinear responses. Using the boundary measurements of the nonlinear responses, we solve the inverse problem of determining elastic parameters from the displacement-to-traction map.

45 citations

••

TL;DR: In this paper, a unified method to obtain unweighted and weighted estimates of linear and multilinear commutators with BMO functions is presented, which is amenable to a plethora of operators and functional settings.

Abstract: We present a unified method to obtain unweighted and weighted estimates of linear and multilinear commutators with BMO functions, that is amenable to a plethora of operators and functional settings. Our approach elaborates on a commonly used Cauchy integral trick, recovering many known results but yielding also numerous new ones. In particular, we solve a problem about the boundedness of the commutators of the bilinear Hilbert transform with functions in BMO.

34 citations

••

TL;DR: In this paper, it was shown that any skew-symmetrizable principal coefficients cluster algebra has the enough g-pairs property, which can be interpreted as a strong version of the sign-coherence of the G-matrices.

Abstract: In this paper, we introduce the enough g-pairs property for principal coefficients cluster algebras, which can be understood as a strong version of the sign-coherence of the G-matrices. Then we prove that any skew-symmetrizable principal coefficients cluster algebra has the enough g-pairs property. As applications, we prove some long standing conjectures in cluster algebras, including a conjecture on denominator vectors and a conjecture on exchange graphs (see Conjectures 1, 2 below). In addition, we give a criterion to distinguish whether particular cluster variables belong to one common cluster for any skew-symmetrizable cluster algebra. As a corollary, we prove a conclusion which was conjectured by Fomin et al., cf. (Acta Math 201(1):83–146, 2008, Conjecture 5.5).

33 citations

••

Fudan University

^{1}TL;DR: In this article, the authors studied self-expanding solutions for mean curvature flows and their relationship to minimal cones in Euclidean space and proved that the corresponding selfexpanding hypersurfaces are smooth, embedded, and have positive mean curvatures everywhere (see Theorem 1.1).

Abstract: In this paper, we study self-expanding solutions for mean curvature flows and their relationship to minimal cones in Euclidean space. Ilmanen (Lectures on mean curvature flow and related equations (Trieste Notes), 1995) proved the existence of self-expanding hypersurfaces with prescribed tangent cones at infinity. If the cone is $$C^{3,\alpha }$$-regular and mean convex (but not area-minimizing), we can prove that the corresponding self-expanding hypersurfaces are smooth, embedded, and have positive mean curvature everywhere (see Theorem 1.1). As a result, for regular minimal but not area-minimizing cones we can give an affirmative answer to a problem arisen by Lawson (Brothers, in Proc Sympos Pure Math 44:441–464, 1986).

29 citations

••

TL;DR: Recently, Brendle et al. as mentioned in this paper studied the locally constrained curvature flow of hypersurfaces in hyperbolic space, which was introduced by Brendle, Guan and Li.

Abstract: In this paper, we first study the locally constrained curvature flow of hypersurfaces in hyperbolic space, which was introduced by Brendle, Guan and Li (An inverse curvature type hypersurface flow in
$${\mathbb {H}}^{n+1}$$
, preprint). This flow preserves the mth quermassintegral and decreases
$$(m+1)$$
th quermassintegral, so the convergence of the flow yields sharp Alexandrov–Fenchel type inequalities in hyperbolic space. Some special cases have been studied in Brendle et al. In the first part of this paper, we show that h-convexity of the hypersurface is preserved along the flow and then the smooth convergence of the flow for h-convex hypersurfaces follows. We then apply this result to establish some new sharp geometric inequalities comparing the integral of kth Gauss–Bonnet curvature of a smooth h-convex hypersurface to its mth quermassintegral (for
$$0\le m\le 2k+1\le n$$
), and comparing the weighted integral of kth mean curvature to its mth quermassintegral (for
$$0\le m\le k\le n$$
). In particular, we give an affirmative answer to a conjecture proposed by Ge, Wang and Wu (Math Z 281, 257–297, 2015). In the second part of this paper, we introduce a new locally constrained curvature flow using the shifted principal curvatures. This is natural in the context of h-convexity. We prove the smooth convergence to a geodesic sphere of the flow for h-convex hypersurfaces, and provide a new proof of the geometric inequalities proved by Andrews, Chen and the third author of this paper in 2018. We also prove a family of new sharp inequalities involving the weighted integral of kth shifted mean curvature for h-convex hypersurfaces, which as application implies a higher order analogue of Brendle, Hung and Wang’s (Commun Pure Appl Math 69(1), 124–144, 2016) inequality.

26 citations

••

TL;DR: In this article, Chen et al. studied the linear stability with variable coefficients of the vortex sheets for the two-dimensional compressible elastic flows and found that elasticity generates notable stabilization effects, and there are additional stable subsonic regions compared with the isentropic Euler flows.

Abstract: The linear stability with variable coefficients of the vortex sheets for the two-dimensional compressible elastic flows is studied. As in our earlier work (Chen et al. in Adv Math 311:18–60, 2017b) on the linear stability with constant coefficients, the problem has a free boundary which is characteristic, and also the Kreiss–Lopatinskii condition is not uniformly satisfied. In addition, the roots of the Lopatinskii determinant of the para-linearized system may coincide with the poles of the system. Such a new collapsing phenomenon causes serious difficulties when applying the bicharacteristic extension method of Coulombel (SIAM J Math Anal 34(1):142–172, 2002; Ann Inst H Poincare Anal Non Lineaire 21(4):401–443, 2004) and Coulombel and Secchi (Indiana Univ Math J 53(4):941–1012, 2004). Motivated by our method introduced in the constant-coefficient case (Adv Math 311:18–60, 2017b), we perform an upper triangularization to the para-linearized system to separate the outgoing mode into a closed form where the outgoing mode only appears at the leading order. This procedure results in a gain of regularity for the outgoing mode, which allows us to overcome the loss of regularity of the characteristic components at the poles and hence to close all the energy estimates. We find that, analogous to the constant-coefficient case, elasticity generates notable stabilization effects, and there are additional stable subsonic regions compared with the isentropic Euler flows. Moreover, since our method does not rely on the construction of the characterisic curves, it can also be applied to other fluid models such as the non-isentropic Euler equations and the MHD equations.

21 citations

••

TL;DR: In this article, Wang et al. proved global-in-time Strichartz estimates without loss regularity for the wave equation associated with the operator LV and showed a Sobolev inequality and a boundedness of a generalized Riesz transform in this setting.

Abstract: Consider the metric cone X=C(Y)=(0,∞)r×Y with metric g=dr2+r2h where the cross section Y is a compact (n−1)-dimensional Riemannian manifold (Y, h). Let Δg be the positive Friedrichs extension Laplacian on X and let Δh be the positive Laplacian on Y, and consider the operator LV=Δg+V0r−2 where V0∈C∞(Y) such that Δh+V0+(n−2)2/4 is a strictly positive operator on L2(Y). In this paper, we prove global-in-time Strichartz estimates without loss regularity for the wave equation associated with the operator LV. It verifies a conjecture in Wang (Remark 2.4 in Ann Inst Fourier 56:1903–1945, 2006) for wave equation. The range of the admissible pair is sharp and the range is influenced by the smallest eigenvalue of Δh+V0+(n−2)2/4. To prove the result, we show a Sobolev inequality and a boundedness of a generalized Riesz transform in this setting. In addition, as an application, we study the well-posed theory and scattering theory for energy-critical wave equation with small data on this setting of dimension n≥3.

20 citations

••

TL;DR: In this paper, a general version of Siu's lemma for plurisubharmonic functions with nontrivial multiplier ideal sheaves was proved and used to prove an optimal $L 2 ) extension theorem and an optimal $$L 2 m extension theorem for Kahler fibrations.

Abstract: We prove a general version of Siu’s lemma for plurisubharmonic functions with nontrivial multiplier ideal sheaves and use it to prove an optimal $$L^2$$
extension theorem and an optimal $$L^{\frac{2}{m}}$$
extension theorem for Kahler fibrations. These results are used to prove the positivity of twisted relative pluricanonical bundles and their direct images for Kahler fibrations and to answer a comparison question about singular metrics of twisted pluricanonical bundles.

19 citations

••

TL;DR: In this paper, the authors interpret Lepingle's jump inequalities in terms of Banach spaces, which are real interpolation spaces, and prove endpoint jump estimates for vector-valued martingales and doubly stochastic operators.

Abstract: Jump inequalities are the $$r=2$$ endpoint of Lepingle’s inequality for r-variation of martingales. Extending earlier work by Pisier and Xu (Probab Theory Relat Fields 77(4):497–514, 1988) we interpret these inequalities in terms of Banach spaces which are real interpolation spaces. This interpretation is used to prove endpoint jump estimates for vector-valued martingales and doubly stochastic operators as well as to pass via sampling from $$\mathbb {R}^{d}$$ to $$\mathbb {Z}^{d}$$ for jump estimates for Fourier multipliers.

19 citations

••

TL;DR: In this paper, a decomposition of the Laplacian into a direct integral in terms of minimal forms was obtained, where fiber LaplACs (matrices) have the minimal number of coefficients depending on the quasimomentum and show that the number is an invariant of the periodic graph.

Abstract: We consider a Laplacian on periodic discrete graphs. Its spectrum consists of a finite number of bands. In a class of periodic 1-forms, i.e., functions defined on edges of the periodic graph, we introduce a subclass of minimal forms with a minimal number $${{\mathcal {I}}}$$
of edges in their supports on the period. We obtain a specific decomposition of the Laplacian into a direct integral in terms of minimal forms, where fiber Laplacians (matrices) have the minimal number $$2{{\mathcal {I}}}$$
of coefficients depending on the quasimomentum and show that the number $${{\mathcal {I}}}$$
is an invariant of the periodic graph. Using this decomposition, we estimate the position of each band, the Lebesgue measure of the Laplacian spectrum and the effective masses at the bottom of the spectrum in terms of the invariant $${{\mathcal {I}}}$$
and the minimal forms. In addition, we consider an inverse problem: we determine necessary and sufficient conditions for matrices depending on the quasimomentum on a finite graph to be fiber Laplacians. Moreover, similar results for Schrodinger operators with periodic potentials are obtained.

••

TL;DR: For convex co-compact hyperbolic surfaces, the dependence of the exponent in the fractal uncertainty principle of Bourgain-Dyatlov on the dimension of the limit set and the regularity constant of the regular set was shown in this paper.

Abstract: We prove an explicit formula for the dependence of the exponent $$\beta $$ in the fractal uncertainty principle of Bourgain–Dyatlov (Ann Math 187:1–43, 2018) on the dimension $$\delta $$ and on the regularity constant $$C_R$$ for the regular set. In particular, this implies an explicit essential spectral gap for convex co-compact hyperbolic surfaces when the Hausdorff dimension of the limit set is close to 1.

••

TL;DR: In this article, a p-adic triple product L-function is constructed that interpolates (square roots of) central critical L-values in the balanced region of affinoid algebras.

Abstract: We construct p-adic triple product L-functions that interpolate (square roots of) central critical L-values in the balanced region. Thus, our construction complements that of Harris and Tilouine. There are four central critical regions for the triple product L-functions and two opposite settings, according to the sign of the functional equation. In the first case, three of these regions are of interpolation, having positive sign; they are called the unbalanced regions and one gets three p-adic L-functions, one for each region of interpolation (this is the Harris-Tilouine setting). In the other setting there is only one region of interpolation, called the balanced region: we produce the corresponding p-adic L-function. Our triple product p-adic L-function arises as p-adic period integrals interpolating normalizations of the local archimedean period integrals. The latter encode information about classical representation theoretic branching laws. The main step in our construction of p-adic period integrals is showing that these branching laws vary in a p-adic analytic fashion. This relies crucially on the Ash-Stevens theory of highest weight representations over affinoid algebras.

••

TL;DR: In this paper, a large class of examples of group actions satisfying the Neshveyev-Stormer rigidity phenomenon was provided. But the results were restricted to a class of compact extensions of the group actions.

Abstract: Motivated by Popa’s seminal work Popa (Invent Math 165:409-45, 2006), in this paper, we provide a fairly large class of examples of group actions
$$\Gamma \curvearrowright X$$
satisfying the extended Neshveyev–Stormer rigidity phenomenon Neshveyev and Stormer (J Funct Anal 195(2):239-261, 2002): whenever
$$\Lambda \curvearrowright Y$$
is a free ergodic pmp action and there is a
$$*$$
-isomorphism
$$\Theta :L^\infty (X)\rtimes \Gamma {\rightarrow }L^\infty (Y)\rtimes \Lambda $$
such that
$$\Theta (L(\Gamma ))=L(\Lambda )$$
then the actions
$$\Gamma \curvearrowright X$$
and
$$\Lambda \curvearrowright Y$$
are conjugate (in a way compatible with
$$\Theta $$
). We also obtain a complete description of the intermediate subalgebras of all (possibly non-free) compact extensions of group actions in the same spirit as the recent results of Suzuki (Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems, To appear in Comm Math Phy. ArXiv Preprint:
arXiv:1805.02077
, 2020). This yields new consequences to the study of rigidity for crossed product von Neumann algebras and to the classification of subfactors of finite Jones index.

••

TL;DR: In this paper, the existence of pulsating travelling front solutions for spatially periodic heterogeneous reaction-diffusion equations in arbitrary dimension, in both bistable and more general multistable frameworks, is studied.

Abstract: We devote this paper to the issue of existence of pulsating travelling front solutions for spatially periodic heterogeneous reaction-diffusion equations in arbitrary dimension, in both bistable and more general multistable frameworks. In the multistable case, the notion of a single front is not sufficient to understand the dynamics of solutions, and we instead observe the appearance of a so-called propagating terrace. This roughly refers to a finite family of stacked fronts connecting intermediate stable steady states whose speeds are ordered. Surprisingly, for a given equation, the shape of this terrace (i.e., the involved intermediate states or even the cardinality of the family of fronts) may depend on the direction of propagation.

••

TL;DR: In this article, the authors consider a minimization problem arising from thermal insulation and show that the disk in two dimensions is the worst one, and the same is true for the ball in higher dimension but under different constraints.

Abstract: In this paper we consider a minimization problem which arises from thermal insulation. A compact connected set K, which represents a conductor of constant temperature, say 1, is thermally insulated by surrounding it with a layer of thermal insulator, the open set
$$\Omega {\setminus } K$$
with
$$K\subset \bar{\Omega }$$
. The heat dispersion is then obtained as
$$\begin{aligned} \inf \left\{ \int _{\Omega }|
abla \varphi |^{2}dx +\beta \int _{\partial ^{*}\Omega }\varphi ^{2}d\mathcal H^{n-1} ,\;\varphi \in H^{1}(\mathbb R^{n}), \, \varphi \ge 1\text { in } K\right\} , \end{aligned}$$
for some positive constant
$$\beta $$
. We mostly restrict our analysis to the case of an insulating layer of constant thickness. We let the set K vary, under prescribed geometrical constraints, and we look for the best (or worst) geometry in terms of heat dispersion. We show that under perimeter constraint the disk in two dimensions is the worst one. The same is true for the ball in higher dimension but under different constraints. We finally discuss few open problems.

••

TL;DR: In this paper, a map called Poisson embedding is introduced to identify the points of a Riemannian manifold with distributions on its boundary. But this map is not suitable for the real analytic case of the Calderon problem.

Abstract: We introduce a new approach to the anisotropic Calderon problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large class of Calderon type inverse problems for quasilinear equations in the real analytic case. The approach also leads to a new proof of the result of Lassas et al. (Annales de l’ ENS 34(5):771–787, 2001) solving the Calderon problem on real analytic Riemannian manifolds. The proof uses the Poisson embedding to determine the harmonic functions in the manifold up to a harmonic morphism. The method also involves various Runge approximation results for linear elliptic equations.

••

TL;DR: In this paper, the authors proved a theorem of Tits type for automorphism groups of projective varieties over an algebraically closed field of arbitrary characteristic, which was first conjectured by Keum, Oguiso and Zhang.

Abstract: We prove a theorem of Tits type for automorphism groups of projective varieties over an algebraically closed field of arbitrary characteristic, which was first conjectured by Keum, Oguiso and Zhang for complex projective varieties.

••

TL;DR: In this article, the Hitchin equation for cyclic Higgs bundles is analyzed and a maximum principle for a type of elliptic systems is derived, and a lower and upper bound of the extrinsic curvature of the image of f is obtained.

Abstract: In this paper, we derive a maximum principle for a type of elliptic systems and apply it to analyze the Hitchin equation for cyclic Higgs bundles. We show several domination results on the pullback metric of the (possibly branched) minimal immersion f associated to cyclic Higgs bundles. Also, we obtain a lower and upper bound of the extrinsic curvature of the image of f. As an application, we give a complete picture for maximal $$Sp(4,{\mathbb {R}})$$-representations in the $$2g-3$$ Gothen components and the Hitchin components.

••

TL;DR: In this article, Nevanlinna's theory for a class of holomorphic maps when the source is a disc was extended to disc-based disc foliations. But it was only for disc foliation.

Abstract: We develop Nevanlinna’s theory for a class of holomorphic maps when the source is a disc. Such maps appear in the theory of foliations by Riemann Surfaces.

••

TL;DR: In this paper, mean value properties of harmonic functions on the first Heisenberg group were investigated in connection with the dynamic programming principles of certain stochastic processes, and a game-theoretical interpretation of the subelliptic Laplacian was provided.

Abstract: We study mean value properties of $$\mathbf{p }$$
-harmonic functions on the first Heisenberg group $${\mathbb {H}}$$
, in connection to the dynamic programming principles of certain stochastic processes. We implement the approach of Peres and Sheffield (Duke Math J 145(1):91–120, 2008) to provide a game-theoretical interpretation of the sub-elliptic $$\mathbf{p }$$
-Laplacian; and of Manfredi et al. (Proc Am Math Soc 138(3):881–889, 2010) to characterize its viscosity solutions via asymptotic mean value expansions.

••

TL;DR: In this paper, a conceptual reformulation and inductive formula of the Siegel series is presented, and a new identity between the intersection number of two modular correspondences over the supersingular locus of the Shimura varieties and the sum of the Fourier coefficients of Siegel-Eisenstein series is established.

Abstract: In this paper, we will explain a conceptual reformulation and inductive formula of the Siegel series. Using this, we will explain that both sides of the local intersection multiplicities of Gross and Keating (Invent Math 112(225–245):2051, 1993) and the Siegel series have the same inherent structures, beyond matching values. As an application, we will prove a new identity between the intersection number of two modular correspondences over
$$\mathbb {F}_p$$
and the sum of the Fourier coefficients of the Siegel-Eisenstein series for
$$\mathrm {Sp}_4/\mathbb {Q}$$
of weight 2, which is independent of
$$p \left( > 2\right) $$
. In addition, we will explain a description of the local intersection multiplicities of the special cycles over
$$\mathbb {F}_p$$
on the supersingular locus of the ‘special fiber’ of the Shimura varieties for
$$\mathrm {GSpin}(n,2), n\le 3$$
in terms of the Siegel series directly.

••

TL;DR: In this paper, the authors introduced the twisted Borisov-Libgober classes of Schubert varieties in general homogeneous spaces G/P. While these classes do not depend on any choice, they depend on a set of new variables.

Abstract: We introduce new notions in elliptic Schubert calculus: the (twisted) Borisov–Libgober classes of Schubert varieties in general homogeneous spaces G/P. While these classes do not depend on any choice, they depend on a set of new variables. For the definition of our classes we calculate multiplicities of some divisors in Schubert varieties, which were only known for full flag varieties before. Our approach leads to a simple recursions for the elliptic classes. Comparing this recursion with R-matrix recursions of the so-called elliptic weight functions of Rimanyi–Tarasov–Varchenko we prove that weight functions represent elliptic classes of Schubert varieties.

••

TL;DR: In this article, it was shown that boundedness holds if and only if the signed sum of dyadic dilations of a given function is larger than 4 and the integrability of the function is greater than 4.

Abstract: We obtain a sharp $$L^2\times L^2 \rightarrow L^1$$ boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the $$L^q$$ integrability of this function; precisely we show that boundedness holds if and only if $$q<4$$. We discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. Our second result is an optimal $$L^2\times L^2\rightarrow L^1$$ boundedness criterion for bilinear operators associated with multipliers with $$L^\infty $$ derivatives. This result provides the main tool in the proof of the first theorem and is also manifested in terms of the $$L^q$$ integrability of the multiplier. The optimal range is $$q<4$$ which, in the absence of Plancherel’s identity on $$L^1$$, should be compared to $$q=\infty $$ in the classical $$L^2\rightarrow L^2$$ boundedness for linear multiplier operators.

••

TL;DR: In this paper, a weak Hardy-Littlewood maximal operator for Dirichlet groups on the open right half plane was proposed. But the Hardy-littlewood maximal operators are not applicable to the infinite dimensional torus.

Abstract: Given a frequency
$$\lambda =(\lambda _n)$$
, we study when almost all vertical limits of a
$$\mathcal {H}_1$$
-Dirichlet series
$$\sum a_n e^{-\lambda _ns}$$
are Riesz summable almost everywhere on the imaginary axis. Equivalently, this means to investigate almost everywhere convergence of Fourier series of
$$H_1$$
-functions on so-called
$$\lambda $$
-Dirichlet groups, and as our main technical tool we need to invent a weak-type
$$(1, \infty )$$
Hardy-Littlewood maximal operator for such groups. Applications are given to
$$H_1$$
-functions on the infinite dimensional torus
$$\mathbb {T}^\infty $$
, ordinary Dirichlet series
$$\sum a_n n^{-s}$$
, as well as bounded and holomorphic functions on the open right half plane, which are uniformly almost periodic on every vertical line.

••

TL;DR: In this article, the authors combine semigroup theory with a nonlocal calculus for these hypoelliptic operators to establish new inequalities of Hardy-Littlewood-Sobolev type in the situation when the drift matrix has nonnegative trace.

Abstract: In his seminal 1934 paper on Brownian motion and the theory of gases Kolmogorov introduced a second order evolution equation which displays some challenging features. In the opening of his 1967 hypoellipticity paper Hormander discussed a general class of degenerate Ornstein–Uhlenbeck operators that includes Kolmogorov’s as a special case. In this note we combine semigroup theory with a nonlocal calculus for these hypoelliptic operators to establish new inequalities of Hardy–Littlewood–Sobolev type in the situation when the drift matrix has nonnegative trace. Our work has been influenced by ideas of E. Stein and Varopoulos in the framework of symmetric semigroups. One of our objectives is to show that such ideas can be pushed to successfully handle the present degenerate non-symmetric setting.

••

TL;DR: In this paper, the authors give another proof of a bi-parameter Carleson embedding theorem that avoids the use of bi-tree capacity, based on some rather subtle comparisons of energies of measures on the Bi-tree.

Abstract: Nicola Arcozzi, Pavel Mozolyako, Karl-Mikael Perfekt, and Giulia Sarfatti recently gave the proof of a bi-parameter Carleson embedding theorem. Their proof uses heavily the notion of capacity on the bi-tree. In this note we give another proof of a bi-parameter Carleson embedding theorem that avoids the use of bi-tree capacity. Unlike the proof on a simple tree in a previous paper of the authors (Arcozzi et al. in Bellman function sitting on a tree, arXiv:1809.03397, 2018), which used the Bellman function technique, the proof here is based on some rather subtle comparisons of energies of measures on the bi-tree.

••

TL;DR: In this paper, the Euler's elastica problem was studied for planar elastic curves of clamped endpoints, and a singular perturbation theory for the modified total squared curvature energy was developed.

Abstract: This paper is devoted to a classical variational problem for planar elastic curves of clamped endpoints, so-called Euler’s elastica problem. We investigate a straightening limit that means enlarging the distance of the endpoints, and obtain several new results concerning properties of least energy solutions. In particular we reach a first uniqueness result that assumes no symmetry. As a key ingredient we develop a foundational singular perturbation theory for the modified total squared curvature energy. It turns out that our energy has almost the same variational structure as a phase transition energy of Modica–Mortola type at the level of a first order singular limit.

••

TL;DR: In this paper, a theory of Besov and Triebel-Lizorkin spaces on general noncompact connected Lie groups endowed with a sub-Riemannian structure is developed.

Abstract: In this paper we develop a theory of Besov and Triebel–Lizorkin spaces on general noncompact connected Lie groups endowed with a sub-Riemannian structure. Such spaces are defined by means of hypoelliptic sub-Laplacians with drift, and endowed with a measure whose density with respect to a right Haar measure is a continuous positive character of the group. We prove several equivalent characterizations of their norms, we establish comparison results also involving Sobolev spaces of recent introduction, and investigate their complex interpolation and algebra properties.

••

TL;DR: In this paper, a holomorphic nonlinear singular partial differential equation (SPDE) with totally characteristic type was studied and the precise bound of the order of the formal Gevrey class was given.

Abstract: The paper discusses a holomorphic nonlinear singular partial differential equation
$$(t \partial _t)^mu=F(t,x,\{(t \partial _t)^j \partial _x^{\alpha }u \}_{j+\alpha \le m, j