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Abstract: In this paper, we present an intrinsic characterisation of projective special Kähler manifolds in terms of a symmetric tensor satisfying certain differential and algebraic conditions. We show that this tensor vanishes precisely when the structure is locally isomorphic to a standard projective special Kähler structure on \(\mathrm {SU}(n,1)/\mathrm {S}(\mathrm {U}(n)\mathrm {U}(1))\) . We use this characterisation to classify 4-dimensional projective special Kähler Lie groups. PubDate: 2021-12-01
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Abstract: We make several remarks concerning properties of functions in parabolic De Giorgi classes of order p. There are new perspectives including a novel mechanism of propagating positivity in measure, the reservation of membership under convex composition, and a logarithmic type estimate. Based on them, we are able to give new proofs of known properties. In particular, we prove local boundedness and local Hölder continuity of these functions via Moser’s ideas, thus avoiding De Giorgi’s heavy machinery. We also seize this opportunity to give a transparent proof of a weak Harnack inequality for nonnegative members of some super-class of De Giorgi, without any covering argument. PubDate: 2021-12-01
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Abstract: Let K be affine, that is, \(K=\{z=(x,y)\in {\mathbb {R}}^{n+m}: y_{1}=\cdots =y_{m}=0\}\) . We compute the sharp constant of Hardy inequality related to the distance d(z, K) for polyharmonic operator. Moreover, we show that there exists a constant \(C>0\) such that for each \(u\in C^{\infty }_{0}({\mathbb {R}}^{n+m}\setminus K)\) , there holds $$\begin{aligned} \int _{{\mathbb {R}}^{n+m}} \nabla ^{k} u ^{2}\mathrm{d}x\mathrm{d}y-c_{m,k}\int _{{\mathbb {R}}^{n+m}}\frac{u^{2}}{ y ^{2k}}\mathrm{d}x\mathrm{d}y\ge C\left( \int _{{\mathbb {R}}^{n+m}} y ^{\gamma } u ^{p}\mathrm{d}x\mathrm{d}y\right) ^{\frac{n+m-2k}{n+m}}, \end{aligned}$$ where \(2\le k<\frac{m+n}{2}\) , \(2<p\le \frac{2(n+m)}{n+m-2k}\) , \(\gamma =\frac{(n+m-2k)p}{2}-n-m\) and \(c_{m,k}\) is the sharp Hardy constant. These inequalities generalize the result of Maz’ya (case \(k=1\) ) and Lu and the second author (case \(m=1\) for polyharmonic operators). In order to prove the main result, we establish some Poincaré–Sobolev inequalities on hyperbolic space which is of independent interest. PubDate: 2021-12-01
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Abstract: In this paper, we consider a pseudo-parabolic equation, which was studied extensively in recent years. We generalize and extend the existing results in the following three aspects. First, we consider the vacuum isolating phenomenon with the initial energy \(J(u_0)\) satisfying \(J(u_0)\le 0\) and \(0<J(u_0)<d\) , respectively, where d is a positive constant denoting the potential well depth. By means of potential well method, we find that there are two explicit vacuum regions which are annulus and ball, respectively. Second, we study the asymptotic behaviors of the solutions and the energy functional. Generally speaking, we establish the exponential decay of the solutions and energy functional when the solutions exist globally, and the concrete decay rate is given. As for the blow-up solutions, we prove that the solutions grow exponentially and obtain the behavior of energy functional as the time t tends to the maximal existence time. We get further two necessary and sufficient conditions for the solutions existing globally and blowing up in finite time, respectively, under the assumption that \(J(u_0)<d\) . Finally, we give a new blow-up condition with eigenfunction method; it should be point out that this initial condition is independent of the initial energy. Under this condition, an upper bound estimation of the blow-up time is obtained and we prove that the solutions grow exponentially; the grow speed is given specifically. PubDate: 2021-12-01
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Abstract: We consider a uniform r-bundle E on a complex rational homogeneous space X and show that if E is poly-uniform with respect to all the special families of lines and the rank r is less than or equal to some number that depends only on X, then E is either a direct sum of line bundles or unstable with respect to some numerical class of a line. So we partially answer a problem posted by Muñoz et al. (Eur J Math 6:430–452, 2020). In particular, if X is a generalized Grassmannian \({\mathcal {G}}\) and the rank r is less than or equal to some number that depends only on X, then E splits as a direct sum of line bundles. So we improve the main theorem of Muñoz et al. (J Reine Angew Math (Crelles J) 664:141–162, 2012, Theorem 3.1) when X is a generalized Grassmannian. Moreover, by calculating the relative tangent bundles between two rational homogeneous spaces, we give explicit bounds for the generalized Grauert–Mülich–Barth theorem on rational homogeneous spaces. PubDate: 2021-12-01
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Abstract: We study a free transmission problem in which solution minimizes a functional with different definitions in positive and negative phases. We prove some asymptotic regularity results when the jumps of the diffusion coefficients get smaller along the free boundary. At last, we prove a measure-theoretic result related to the free boundary. PubDate: 2021-12-01
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Abstract: Let \(1 \le p \le \infty \) . A Banach lattice X is said to be p-disjointly homogeneous or \((p-DH)\) (resp. restricted \((p-DH)\) ) if every normalized disjoint sequence in X (resp. every normalized sequence of characteristic functions of disjoint subsets) contains a subsequence equivalent in X to the unit vector basis of \(\ell _p\) . We revisit DH-properties of Orlicz spaces and refine some previous results of this topic, showing that the \((p-DH)\) -property is not stable under duality in the class of Orlicz spaces and the classes of restricted \((p-DH)\) and \((p-DH)\) Orlicz spaces are different. Moreover, we give a characterization of uniform \((p-DH)\) Orlicz spaces and establish also closed connections between this property and the duality of the DH-property. PubDate: 2021-12-01
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Abstract: Given an arbitrary \(C^\infty \) Riemannian manifold \(M^n\) , we consider the problem of introducing and constructing minimal hypersurfaces in \(M\times \mathbb {R}\) which have the same fundamental properties of the standard helicoids and catenoids of Euclidean space \(\mathbb {R}^3=\mathbb {R} ^2\times \mathbb {R}\) . Such hypersurfaces are defined by imposing conditions on their height functions and horizontal sections and then called vertical helicoids and vertical catenoids. We establish that vertical helicoids in \(M\times \mathbb {R}\) have the same fundamental uniqueness properties of the helicoids in \(\mathbb {R}^3.\) We provide several examples of properly embedded vertical helicoids in the case where M is one of the simply connected space forms. Vertical helicoids which are entire graphs of functions on \(\mathrm{Nil}_3\) and \(\mathrm{Sol}_3\) are also presented. We show that vertical helicoids of \(M\times \mathbb {R} \) whose horizontal sections are totally geodesic in M are locally given by a “twisting” of a fixed totally geodesic hypersurface of M. We give a local characterization of hypersurfaces of \(M\times \mathbb {R}\) which have the gradient of their height functions as a principal direction. As a consequence, we prove that vertical catenoids exist in \(M\times \mathbb {R}\) if and only if M admits families of isoparametric hypersurfaces. If so, properly embedded vertical catenoids can be constructed through the solutions of a certain first-order linear differential equation. Finally, we give a complete classification of the hypersurfaces of \(M\times \mathbb {R}\) whose angle function is constant. PubDate: 2021-12-01
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Abstract: In proving his theorem on global maximizers for the sphere adjoint Fourier restriction inequality, D. Foschi studied some quadratic functional and determined its optimal bound. In this note we extend to higher-dimensional spheres his results on this quadratic functional. PubDate: 2021-12-01
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Abstract: We propose a linear independence criterion, and outline an application of it. Down to its simplest case, it aims at solving this problem: given three real numbers, typically as special values of analytic functions, how to prove that the \(\mathbb {Q}\) -vector space spanned by 1 and those three numbers has dimension at least 3, whenever we are unable to achieve full linear independence, by using simultaneous approximations, i.e. those usually arising from Hermite–Padé approximations of type II and their suitable generalizations. It should be recalled that approximations of type I and II are related, at least in principle: when the numerical application consists in specializing actual functional constructions of the two types, they can be obtained, one from the other, as explained in a well-known paper by Mahler (1968) Compos Math 19: 95–166. That relation is reflected in a relation between the asymptotic behavior of the approximations at the infinite place of \(\mathbb {Q}\) . Rather interestingly, the two view-points split away regarding the asymptotic behaviors at finite places (i.e. primes) of \(\mathbb {Q}\) , and this makes the use of type II more convenient for particular purposes. In addition, sometimes we know type II approximations to a given set of functions, for which type I approximations are not known explicitly. Our approach can be regarded as a dual version of the standard linear independence criterion, which essentially goes back to Siegel. PubDate: 2021-12-01
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Abstract: In the context of Euclidean spaces equipped with an arbitrary Radon measure, we prove the equivalence among several different notions of Sobolev space present in the literature and we characterise the minimal weak upper gradient of all Lipschitz functions. PubDate: 2021-12-01
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Abstract: This article studies the global hypoellipticity of a class of overdetermined systems of pseudo-differential operators defined on the torus. The main goal consists in establishing connections between the global hypoellipticity of the system and the global hypoellipticity of its normal form. It is proved that an obstruction of number-theoretical nature appears as a necessary condition to the global hypoellipticity. Conversely, the sufficiency is approached by analyzing three types of hypotheses: a Hörmander condition, logarithmic growth and super-logarithmic growth. PubDate: 2021-12-01
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Abstract: We describe the structure of the projective cover of a module \(M_R\) over a local ring R and its relation with minimal sets of generators of \(M_R\) . The behaviour of local right perfect rings is completely different from the behaviour of local rings that are not right perfect. PubDate: 2021-12-01
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Abstract: In this paper we prove higher order Poincaré inequalities involving radial derivatives namely, $$\begin{aligned} \int _{\mathbb {H}^{N}} \nabla _{r,\mathbb {H}^{N}}^{k} u ^2 \, \mathrm{d}v_{\mathbb {H}^{N}} \ge \bigg (\frac{N-1}{2}\bigg )^{2(k-l)} \int _{\mathbb {H}^{N}} \nabla _{r,\mathbb {H}^{N}}^{l} u ^2 \, \mathrm{d}v_{\mathbb {H}^{N}} \ \ \text { for all } u\in H^k(\mathbb {H}^{N}), \end{aligned}$$ where underlying space is N-dimensional hyperbolic space \(\mathbb {H}^{N}\) , \(0\le l<k\) are integers and the constant \(\big (\frac{N-1}{2}\big )^{2(k-l)}\) is sharp. Furthermore we improve the above inequalities by adding Hardy-type remainder terms and the sharpness of some constants is also discussed. PubDate: 2021-12-01
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Abstract: For \(p \in (1,N)\) and \(\Omega \subseteq {\mathbb {R}}^N\) open, the Beppo-Levi space \({\mathcal {D}}^{1,p}_0(\Omega )\) is the completion of \(C_c^{\infty }(\Omega )\) with respect to the norm \(\left[ \int _{\Omega } \nabla u ^p \ dx \right] ^ \frac{1}{p}.\) Using the p-capacity, we define a norm and then identify the Banach function space \({\mathcal {H}}(\Omega )\) with the set of all g in \(L^1_{loc}(\Omega )\) that admits the following Hardy–Sobolev type inequality: $$\begin{aligned} \int _{\Omega } g u ^p \ dx \le C \int _{\Omega } \nabla u ^p \ dx, \forall \; u \in {\mathcal {D}}^{1,p}_0(\Omega ), \end{aligned}$$ for some \(C>0.\) Further, we characterize the set of all g in \({\mathcal {H}}(\Omega )\) for which the map \(G(u)= \displaystyle \int _{\Omega } g u ^p \ dx\) is compact on \({\mathcal {D}}^{1,p}_0(\Omega )\) . We use a variation of the concentration compactness lemma to give a sufficient condition on \(g\in {\mathcal {H}}(\Omega )\) so that the best constant in the above inequality is attained in \({\mathcal {D}}^{1,p}_0(\Omega )\) . PubDate: 2021-12-01
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Abstract: We prove the existence of an intermediate Banach space between the space where the Gaussian measure lives and its RKHS, thus extending what happens with Wiener measure, where the intermediate space can be chosen as a space of Hölder paths. From this result, it is very simple to deduce a result of exponential tightness for Gaussian probabilities. PubDate: 2021-12-01
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Abstract: We establish weighted \(L^p\) -Fourier extension estimates for \(O(N-k) \times O(k)\) -invariant functions defined on the unit sphere \({\mathbb {S}}^{N-1}\) , allowing for exponents p below the Stein–Tomas critical exponent \(\frac{2(N+1)}{N-1}\) . Moreover, in the more general setting of an arbitrary closed subgroup \(G \subset O(N)\) and G-invariant functions, we study the implications of weighted Fourier extension estimates with regard to boundedness and nonvanishing properties of the corresponding weighted Helmholtz resolvent operator. Finally, we use these properties to derive new existence results for G-invariant solutions to the nonlinear Helmholtz equation $$\begin{aligned} -\Delta u - u = Q(x) u ^{p-2}u, \quad u \in W^{2,p}({\mathbb {R}}^{N}), \end{aligned}$$ where Q is a nonnegative bounded and G-invariant weight function. PubDate: 2021-12-01
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Abstract: An octonionic contact (OC) manifold is always spherical. We construct the OC Yamabe operator on an OC manifold and prove its transformation formula under conformal OC transformations. An OC manifold is scalar positive, negative or vanishing if and only if its OC Yamabe invariant is positive, negative or zero, respectively. On a scalar positive OC manifold, we can construct the Green function of the OC Yamabe operator and apply it to construct a conformally invariant tensor. It becomes an OC metric if the OC positive mass conjecture is true. We also show the connected sum of two scalar positive OC manifolds to be scalar positive if the neck is sufficiently long. On the OC manifold constructed from a convex cocompact subgroup of F4(−20), we construct a Nayatani-type Carnot–Carathéodory metric. As a corollary, such an OC manifold is scalar positive, negative or vanishing if and only if the Poincaré critical exponent of the subgroup is less than, greater than or equal to 10, respectively. PubDate: 2021-12-01
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Abstract: We show that the tangential speed \(v^T_\phi (t)\) of a parabolic semigroup \((\phi _t)\) of holomorphic self-maps in the unit disc is asymptotically bounded from above by \((1/2)\log t\) , proving a conjecture by Bracci. In order to show the proof, we need a result of “asymptotical monotonicity” of the tangential speed for proper pairs of parabolic semigroups with positive hyperbolic step. PubDate: 2021-12-01
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Abstract: Let f be analytic in the unit disk \({\mathbb {D}}=\{z\in {\mathbb {C}}: z <1 \}\) , and \({{\mathcal {S}}}\) be the subclass of normalized univalent functions given by \(f(z)=z+\sum _{n=2}^{\infty }a_n z^n\) for \(z\in {\mathbb {D}}\) . We give sharp bounds for the modulus of the second Hankel determinant \( H_2(2)(f)=a_2a_4-a_3^2\) for the subclass \( {\mathcal F_{O}}(\lambda ,\beta )\) of strongly Ozaki close-to-convex functions, where \(1/2\le \lambda \le 1\) , and \(0<\beta \le 1\) . Sharp bounds are also given for \( H_2(2)(f^{-1}) \) , where \(f^{-1}\) is the inverse function of f. The results settle an invariance property of \( H_2(2)(f) \) and \( H_2(2)(f^{-1}) \) for strongly convex functions. PubDate: 2021-12-01
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