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For wide classes of locally convex spaces, in particular, for the space of continuous real‐valued functions on a Tychonoff space X equipped with the pointwise topology, we characterize the existence of a fundamental bounded resolution (i.e., an increasing family of bounded sets indexed by the irrationals which swallows the bounded sets). These facts together with some results from Grothendieck's theory of ‐spaces have led us to introduce quasi‐ ‐spaces, a class of locally convex spaces containing ‐spaces that preserves subspaces, countable direct sums and countable products. Regular ‐spaces as well as their strong duals are quasi‐ ‐spaces. Hence the space of distributions provides a concrete example of a quasi‐ ‐space not being a ‐space. We show that has a fundamental bounded resolution if and only if is a quasi‐ ‐space if and only if the strong dual of is a quasi‐ ‐space if and only if X is countable. If X is metrizable, then is a quasi‐ ‐space if and only if X is a σ‐compact Polish space. 相似文献
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Ferenc Weisz 《Mathematische Nachrichten》2023,296(4):1687-1705
Let be a measurable function defined on and . In this paper, we generalize the Hardy–Littlewood maximal operator. In the definition, instead of cubes or balls, we take the supremum over all rectangles the side lengths of which are in a cone-like set defined by a given function ψ. Moreover, instead of the integral means, we consider the -means. Let and satisfy the log-Hülder condition and . Then, we prove that the maximal operator is bounded on if and is bounded from to the weak if . We generalize also the theorem about the Lebesgue points. 相似文献
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Giovany M. Figueiredo Marcelo Montenegro Matheus F. Stapenhorst 《Mathematische Nachrichten》2023,296(10):4569-4609
We show the existence of a solution for an equation where the nonlinearity is logarithmically singular at the origin, namely, in with Dirichlet boundary condition. The function f has exponential growth, which can be subcritical or critical with respect to the Trudinger–Moser inequality. We study the energy functional corresponding to the perturbed equation , where is well defined at 0 and approximates . We show that has a critical point in , which converges to a legitimate nontrivial nonnegative solution of the original problem as . We also investigate the problem with replaced by , when the parameter is sufficiently large. 相似文献
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We prove estimates, , for solutions to the tangential Cauchy–Riemann equations on a class of infinite type domains . The domains under consideration are a class of convex ellipsoids, and we show that if is a ‐closed (0,1)‐form with coefficients in , then there exists an explicit solution u satisfying . Moreover, when , we show that there is a gain in regularity to an f‐Hölder space. We also present two applications. The first is a solution to the ‐equation, that is, given a smooth (0,1)‐form ? on with an L1‐boundary value, we can solve the Cauchy–Riemann equation so that where C is independent of and ?. The second application is a discussion of the zero sets of holomorphic functions with zero sets of functions in the Nevanlinna class within our class of domains. 相似文献
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Inspired by some iterative algorithms useful for proving the real analyticity (or the Gevrey regularity) of a solution of a linear partial differential equation with real-analytic coefficients, we consider the following question. Given a smooth function defined on and given an increasing divergent sequence of positive integers such that the derivative of order of f has a growth of the type , when can we deduce that f is a function in the Denjoy–Carleman class ? We provide a positive result and show that a suitable condition on the gaps between the terms of the sequence is needed. 相似文献
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Nikolaos Panagiotis Souris 《Mathematische Nachrichten》2023,296(6):2609-2625
We study the geodesic equation for compact Lie groups G and homogeneous spaces , and we prove that the geodesics are orbits of products of one-parameter subgroups of G, provided that a simple algebraic condition for the Riemannian metric is satisfied. For the group , we relate this type of geodesics to the free motion of a symmetric top. Moreover, by using series of Lie subgroups of G, we construct a wealth of metrics having the aforementioned type of geodesics. 相似文献
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We prove that the kth Gaussian map is surjective on a polarized unnodal Enriques surface with . In particular, as a consequence, when , we obtain the surjectivity of the kth Gauss-Prym map , with , on smooth hyperplane sections . In case , it is sufficient to ask . 相似文献
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The goal of this article is to investigate nontrivial m‐quasi‐Einstein manifolds globally conformal to an n‐dimensional Euclidean space. By considering such manifolds, whose conformal factors and potential functions are invariant under the action of an ‐dimensional translation group, we provide a complete classification when and or . 相似文献
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A. R. Mirotin 《Mathematische Nachrichten》2023,296(9):4108-4124
Necessary and sufficient conditions are given for the boundedness of Hausdorff operators on the generalized Hardy spaces , real Hardy space , , and for compact Abelian group G. Surprisingly, these conditions turned out to be the same for all groups and spaces under consideration. Applications to Dirichlet series are given. The case of the space of continuous functions on G and examples are also considered. 相似文献
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Rytis Juršėnas 《Mathematische Nachrichten》2023,296(8):3411-3448
Let be an isometric boundary pair associated with a closed symmetric linear relation T in a Krein space . Let be the Weyl family corresponding to . We cope with two main topics. First, since need not be (generalized) Nevanlinna, the characterization of the closure and the adjoint of a linear relation , for some , becomes a nontrivial task. Regarding as the (Shmul'yan) transform of induced by Γ, we give conditions for the equality in to hold and we compute the adjoint . As an application, we ask when the resolvent set of the main transform associated with a unitary boundary pair for is nonempty. Based on the criterion for the closeness of , we give a sufficient condition for the answer. From this result it follows, for example, that, if T is a standard linear relation in a Pontryagin space, then the Weyl family corresponding to a boundary relation Γ for is a generalized Nevanlinna family; a similar conclusion is already known if T is an operator. In the second topic, we characterize the transformed boundary pair with its Weyl family . The transformation scheme is either or with suitable linear relations V. Results in this direction include but are not limited to: a 1-1 correspondence between and ; the formula for , for an ordinary boundary triple and a standard unitary operator V (first scheme); construction of a quasi boundary triple from an isometric boundary triple with and (second scheme, Hilbert space case). 相似文献
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For a positive integer N, let be the modular curve over and its Jacobian variety. We prove that the rational cuspidal subgroup of is equal to the rational cuspidal divisor class group of when for any prime p and any squarefree integer M. To achieve this, we show that all modular units on can be written as products of certain functions , which are constructed from generalized Dedekind eta functions. Also, we determine the necessary and sufficient conditions for such products to be modular units on under a mild assumption. 相似文献
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Arpan Dutta 《Mathematische Nachrichten》2023,296(10):4745-4768
Given a valued field and its completion , we study the set of all possible extensions of v to . We show that any such extension is closely connected with the underlying subextension . The connections between these extensions are studied via minimal pairs, key polynomials, pseudo-Cauchy sequences, and implicit constant fields. As a consequence, we obtain strong ramification theoretic properties of . We also give necessary and sufficient conditions for to be dense in . 相似文献