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This is an introduction to Witten’s analytic proof of the Morse inequalities. The text is directed primarily to readers whose main interest is in complex analysis, and the similarities to Hörmander’s -estimates for the -equation is used as motivation. We also use the method to prove -estimates for the -equation with a weight where is a nondegenerate Morse function.
Let be a polynomial of degree at most which does not vanish in the disk , then for and , Boas and Rahman proved
In this paper, we improve the above inequality for by involving some of the coefficients of the polynomial . Analogous result for the class of polynomials having no zero in is also given.
The estimate is shown to hold if and only if is elliptic and canceling. Here is a homogeneous linear differential operator of order on from a vector space to a vector space . The operator is defined to be canceling if . This result implies in particular the classical Gagliardo–Nirenberg–Sobolev inequality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous linear differential...
Let be a non-negative matrix. Denote by the supremum of those that satisfy the inequality
where and and also is an increasing, non-negative sequence of real numbers. If , we use instead of . In this paper we obtain a Hardy type formula for , where is a Hausdorff matrix and . Another purpose of this paper is to establish a lower bound for , where is the Nörlund matrix associated with the sequence and . Our results generalize some works of Bennett, Jameson and present authors....
We study lower estimates for integral fuctionals for which the structure of the integrand is defined by a graph, in particular, by a bipartite graph. Functionals of such kind appear in statistical mechanics and quantum chemistry in the context of Mayer's transformation and Mayer's cluster integrals. Integral functionals generated by graphs play an important role in the theory of graph limits. Specific kind of functionals generated by bipartite graphs are at the center of the famous and much studied...
In this paper we consider some matrix operators on block weighted sequence spaces . The problem is to find the lower bound of some matrix operators such as Hausdorff and Hilbert matrices on . This study is an extension of papers by G. Bennett, G.J.O. Jameson and R. Lashkaripour.
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