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1.
Kati Ain  Eve Oja 《Mathematische Nachrichten》2015,288(14-15):1569-1580
Let and , where is the conjugate index of p. We prove an omnibus theorem, which provides numerous equivalences for a sequence in a Banach space X to be a ‐null sequence. One of them is that is ‐null if and only if is null and relatively ‐compact. This equivalence is known in the “limit” case when , the case of the p‐null sequence and p‐compactness. Our approach is more direct and easier than those applied for the proof of the latter result. We apply it also to characterize the unconditional and weak versions of ‐null sequences.  相似文献   

2.
In this work we show that if is a linear differential operator of order ν with smooth complex coefficients in from a complex vector space E to a complex vector space F, the Sobolev a priori estimate holds locally at any point if and only if is elliptic and the constant coefficient homogeneous operator is canceling in the sense of Van Schaftingen for every which means that Here is the homogeneous part of order ν of and is the principal symbol of . This result implies and unifies the proofs of several estimates for complexes and pseudo‐complexes of operators of order one or higher proved recently by other methods as well as it extends —in the local setup— the characterization of Van Schaftingen to operators with variable coefficients.  相似文献   

3.
Let , where Ω is a bounded domain of , , and . We prove that , where ρ denotes the distance function to the boundary. Then, we show that, up to subsequences, the extremal functions of converge (as ) to the viscosity solutions of a specific Dirichlet problem involving the infinity Laplacian in the punctured domain , for some .  相似文献   

4.
Let satisfy that , for any given , is an Orlicz function and is a Muckenhoupt weight uniformly in . In this article, the authors introduce the weak Musielak–Orlicz Hardy space via the grand maximal function and then obtain its vertical or its non–tangential maximal function characterizations. The authors also establish other real‐variable characterizations of , respectively, in terms of the atom, the molecule, the Lusin area function, the Littlewood–Paley g‐function or ‐function. All these characterizations for weighted weak Hardy spaces (namely, and with and ) are new and part of these characterizations even for weak Hardy spaces (namely, and with ) are also new. As an application, the boundedness of Calderón–Zygmund operators from to in the critical case is presented.  相似文献   

5.
6.
We study the class of univalent analytic functions f in the unit disk of the form satisfying where Ω will be a proper subdomain of which is starlike with respect to . Let be the unique conformal mapping of onto Ω with and and . Let denote the arclength of the image of the circle , . The first result in this paper is an inequality for , which solves the general extremal problem , and contains many other well‐known results of the previous authors as special cases. Other results of this article cover another set of related problems about integral means in the general setting of the class .  相似文献   

7.
We study the well‐posedness of the second order degenerate differential equations with infinite delay: with periodic boundary conditions , where and M are closed linear operators in a Banach space satisfying , . Using operator‐valued Fourier multiplier techniques, we give necessary and sufficient conditions for the well‐posedness of this problem in Lebesgue‐Bochner spaces , periodic Besov spaces and periodic Triebel‐Lizorkin spaces .  相似文献   

8.
In the paper we consider the Bessel differential operator in half‐line , , and its Dirichlet heat kernel . For , by combining analytical and probabilistic methods, we provide sharp two‐sided estimates of the heat kernel for the whole range of the space parameters and every , which complements the recent results given in 1 , where the case was considered.  相似文献   

9.
We prove the existence of large energy positive solutions for a stationary nonlinear Schrödinger equation with peaks on a Clifford type torus. Here where with for all Each is a function and is defined by the generalized notion of spherical coordinates. The solutions are obtained by a or a process.  相似文献   

10.
We consider the equation which is called Holling–Tanner population model where is a bifurcation parameter and are unknown constants. In this paper, we determine the unknown constants from the asymptotic behavior of the bifurcation curve , where .  相似文献   

11.
Using the general formalism of 12 , a study of index theory for non‐Fredholm operators was initiated in 9 . Natural examples arise from (1 + 1)‐dimensional differential operators using the model operator in of the type , where , and the family of self‐adjoint operators in studied here is explicitly given by Here has to be integrable on and tends to zero as and to 1 as (both functions are subject to additional hypotheses). In particular, , , has asymptotes (in the norm resolvent sense) as , respectively. The interesting feature is that violates the relative trace class condition introduced in 9 , Hypothesis 2.1 ]. A new approach adapted to differential operators of this kind is given here using an approximation technique. The approximants do fit the framework of 9 enabling the following results to be obtained. Introducing , , we recall that the resolvent regularized Witten index of , denoted by , is defined by whenever this limit exists. In the concrete example at hand, we prove Here denotes the spectral shift operator for the pair of self‐adjoint operators , and we employ the normalization, , .  相似文献   

12.
Let be two monomial ideals of the polynomial ring . In this paper, we provide two lower bounds for the Stanley depth of . On the one hand, we introduce the notion of lcm number of , denoted by , and prove that the inequality holds. On the other hand, we show that , where denotes the order dimension of the lcm lattice of . We show that I and satisfy Stanley's conjecture, if either the lcm number of I or the order dimension of the lcm lattice of I is small enough. Among other results, we also prove that the Stanley–Reisner ideal of a vertex decomposable simplicial complex satisfies Stanley's conjecture.  相似文献   

13.
The self‐affine measure corresponding to a upper or lower triangle expanding matrix M and the digit set in the space is supported on the generalized spatial Sierpinski gasket, where are the standard basis of unit column vectors in . We consider in this paper the existence of orthogonal exponentials on the Hilbert space , i.e., the spectrality of . Such a property is directly connected with the entries of M and is not completely determined. For this generalized spatial Sierpinski gasket, we present a method to deal with the spectrality or non‐spectrality of . As an application, the spectral property of a class of such self‐affine measures are clarified. The results here generalize the corresponding results in a simple manner.  相似文献   

14.
We consider the chemotaxis system with rotation under no‐flux boundary conditions in the bounded domain , . Here the matrix‐valued function fulfills () for all with some nondecreasing function S0 and is a nonnegative function with for all . Moreover, f satisfies for all with nondecreasing function f0. It is shown that for the nonnegative initial data and with , if at least one of the following assumptions holds:
  • ,
  • , and ,
  • ,
then the corresponding initial‐boundary value problem possesses a unique global classical solution that is uniformly bounded.  相似文献   

15.
We construct , such that maps onto . This shows that a mapping which maps a set of zero measure onto the set of positive measure can be a gradient mapping.  相似文献   

16.
Consider a Schrödinger equation where and are two continuous real functions on , ε is a positive parameter, the nonlinearity f is assumed to be of critical exponential growth in the sense of the Trudinger‐Moser inequality. By truncating the potentials and , we are able to establish some new existence and concentration results for critical Schrödinger equation in by variational methods. As a particular case, we observe that the concentration appears at the maximum set of the nonlinear potential which complements the results in 6 , 23 .  相似文献   

17.
Given a bounded strongly pseudoconvex domain D in with smooth boundary, we characterize ‐Bergman Carleson measures for , , and . As an application, we show that the Bergman space version of the balayage of a Bergman Carleson measure on D belongs to BMO in the Kobayashi metric.  相似文献   

18.
We study existence and multiplicity of solutions of the following nonlocal ‐Kirchhoff equation with critical exponent, via truncation argument on the Sobolev space with variable exponent, where Ω is a bounded smooth domain of , , M, f are continuous functions, , and are real parameter.  相似文献   

19.
The cuspidalization conjecture, which is a consequence of Grothendieck's section conjecture, asserts that for any smooth hyperbolic curve X over a finitely generated field k of characteristic 0 and any non empty Zariski open , every section of lifts to a section of . We consider in this article the problem of lifting Galois sections to the intermediate quotient introduced by Mochizuki 10 . We show that when and is an union of torsion sub‐packets every Galois section actually lifts to . One of the main tools in the proof is the construction of torus torsors and over X and the geometric interpretation .  相似文献   

20.
We give necessary and sufficient conditions of the existence of a left‐invariant metric of strictly negative Ricci curvature on a solvable Lie group the nilradical of whose Lie algebra is a filiform Lie algebra . It turns out that such a metric always exists, except for in the two cases, when is one of the algebras of rank two, or , and is a one‐dimensional extension of , in which cases the conditions are given in terms of certain linear inequalities for the eigenvalues of the extension derivation.  相似文献   

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