Estimates for Littlewood-Paley Functions and Extrapolation

We prove certain L ^p -estimates for Littlewood-Paley functions arising from rough kernels. The estimates are useful for extrapolation to prove L ^p -boundedness of the Littlewood-Paley functions under a sharp kernel condition.      

A generalized affine isoperimetric inequality

A purely analytic proof is given for an inequality that has as a direct consequence the two most important affine isoperimetric inequalities of plane convex geometry: The Blaschke-Santaló inequality and the affine isoperimetric inequality of affine differential geometry.     

A note on Markov type constants

We prove that, if a geodesic metric space has Markov type 2 with constant 1, then it is an Alexandrov space of nonnegative curvature. The same technique provides a lower bound of the Markov type 2 constant of a space containing a tripod or a branching point. Received: 28 November 2007     

Nevanlinna inequality for intersection of divisors

A Nevanlinna-type inequality is proved for holomorphic mapf:C ^m →M and for intersection of sections of a line bundle overM, in which the intersection may not be pure dimensional and the map may be degenerate. Partial financial support was provided by the NSF under grant number DMS-8922760.     

Isoperimetric inequality in the Grushin plane

We prove a sharp isoperimetric inequality in the Grushin plane and compute the corresponding isoperimetric sets.     

Extremal functions for Moser-Trudinger type inequality on compact closed 4-manifolds

Given a compact closed four-dimensional smooth Riemannian manifold, we prove existence of extremal functions for Moser-Trudinger type inequality. The method used is blow-up analysis combined with capacity techniques. Acknowledgements and Notes. The second author has been supported by M.U.R.S.T. within the PRIN 2004 Variational methods and nonlinear differential equations.     

Transferring strong boundedness among Laguerre orthogonal systems

Given the family of Laguerre polynomials, it is known that several orthonormal systems of Laguerre functions can be considered. In this paper we prove that an exhaustive knowledge of the boundedness in weighted L ^p of the heat and Poisson semigroups, Riesz transforms and g-functions associated to a particular Laguerre orthonormal system of functions, implies a complete knowledge of the boundedness of the corresponding operators on the other Laguerre orthonormal system of functions. As a byproduct, new weighted L ^p boundedness are obtained. The method also allows us to get new weighted estimates for operators related with Laguerre polynomials. Carlos Segovia passed away on April 3, 2007.     

On certain Mordell-Weil lattices of hyperelliptic type on rational surfaces

We study a class of rational hyperelliptic fibrations that can serve as a natural higher-genus analogue of rational elliptic fibrations from the standpoint of Mordell-Weil lattices. As a corollary, we obtain certain generalizations of the root lattices. We also describe the torsion part. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 62, Algebraic Geometry-10, 1999.     

On a class of slowly changing functions

We study the slowly changing functions and use them in the study of entire functions. An erratum to this article is available at .     

A generalization of an integral of Ramanujan

This paper considers a generalization of an integral introduced by S. Ramanujan in his third notebook. Ramanujan’s integral is itself a version of the dilogarithm,

The inequality of Ky Fan and related results

In this survey paper, we present refinements, extensions, and variants of the inequality

Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form

We give null controllability results for some degenerate parabolic equations in non divergence form on a bounded interval. In particular, the coefficient of the second order term degenerates at the extreme points of the domain. For this reason, we obtain an observability inequality for the adjoint problem. Then we prove Carleman estimates for such a problem. Finally, in a standard way, we deduce null controllability also for semilinear equations.      

Non-Holonomicity of Sequences Defined via Elementary Functions

We present a new method for proving non-holonomicity of sequences, which is based on results about the number of zeros of elementary and of analytic functions. Our approach is applicable to sequences that are defined as the values of an elementary function at positive integral arguments. We generalize several recent results, e.g., non-holonomicity of the logarithmic sequence is extended to rational functions involving log n. Moreover, we show that the sequence that arises from evaluating the Riemann zeta function at an increasing integer sequence with bounded gap lengths is not holonomic. Martin Klazar: ITI is supported as project 1M0021620808 by Ministry of Education of the Czech Republic.     

A generalization of an inequality by N. V. Krylov

In Krylov (Journal of the Juliusz Schauder Center 4 (1994), 355–364), a parabolic Littlewood–Paley inequality and its application to an L ^p -estimate of the gradient of the heat kernel are proved. These estimates are crucial tools in the development of a theory of parabolic stochastic partial differential equations (Krylov, Mathematical Surveys and Monographs vol. 64 (1999), 185–242). We generalize these inequalities so that they can be applied to stochastic integrodifferential equations.      

Topologies on function spaces

Function spaces play an important role in complex analysis, in the theory of differential equations, in functional analysis and in almost every other branch of modern mathematics. In this paper we give and study the notion of clopen convergence. Also, we study the notion of clopen continuity and define new topologies on function spaces. These results generalize basic results of R. Arens, J. Dugundji and A. Di Concilio (see [1], [4], [2] and [3]).     

B-Products of Besov type spaces and some applications

The paper derives a representation of classical Besov zero spaces by means of the so called B-products of the generalized Besov type zero spaces, generated by completely regular polyhedrons. Peetre K-functional is estimated and an interpolation formula of “real method” is proved for pairs of B-products. Some applications of these results are given.     

Skorohod stochastic differential equations of diffusion type

Leta, b beC ²(R ¹)-functions with bounded derivatives of first and second order. We study stochastic differential equations

A Construction of Subsets of the Reals which have a Similarity Decomposition

We construct a subset of the set R of real numbers of cardinality |R| which has a similarity decomposition, and which has an ordertype < that of R. Seymour Ginsburg had posed the question whether there exist sets with another ordertype than that of R which also have a similarity decomposition.      

Littlewood-Paley inequalities in uniformly convex and uniformly smooth Banach spaces

It is proved that the inequality δX(ε)?cεp, p?2, where δX is the modulus of convexity of X, is sufficient and necessary for the inequality