In this paper we prove that the maximal operator of the Marcinkiewicz means of two-dimensional integrable functions with respect to the Walsh–Kaczmarz system is of weak type (1, 1). Moreover, the Marcinkiewicz means Mnf converge to f almost everywhere, for any integrable function f. © 2006 Elsevier Inc. All rights reserved.

Complete eigenfunctions for an integrable equation linearized around a soliton solution are the key to the development of a direct soliton perturbation theory. In this article, we explicitly construct such eigenfunctions for a large class of integrable equations including the KdV, NLS and mKdV hierarchies. We establish the striking result that the linearization operators of all equations in the...

We derive a set of bilinear functional equations of Hirota type for the partition functions of the sl(2) related integrable statistical models defined on a random lattice. These equations are obtained as deformations of the Hirota equations for the KP integrable hierarchy , which are satisfied by the partition function of the ensemble of planar graphs.

A continuous-time anti-integrable limit is considered in the context of time-dependent Lagrangian systems on the torus. The anti-integrable limit is the singular (or adiabatic) limit in the singularly (adiabatically, resp.) perturbed problems. This paper presents an implicit function theorem version of the results of Bolotin and MacKay (1997 Nonlinearity 1

In this paper we consider the steady water wave problem for waves that possess a merely Lr−integrable vorticity, with r ∈ (1,∞) being arbitrary. We rst establish the equivalence of the three formulations the velocity formulation, the stream function formulation, and the height function formulation in the setting of strong solutions, regardless of the value of r. Based upon this result and using...

This paper considers various aspects of the canonical coordinate transformations in a complex phase space. The main result is given by two theorems which describe two special families of mappings between integrable Hamiltonian systems. The generating function of these transformations is determined by the entropy and a second arbitrary function which we take to be the energy function. For simple...

We prove that any spectral sequence obeying a certain growth law is the quantum spectrum of an equivalence class of classically integrable non-linear oscillators. This implies that exceptions to the BerryTabor rule for the distribution of quantum energy gaps of classically integrable systems, are far more numerous than previously believed. In particular we show that for each finite dimension k,...

We prove that the HRT (Heil, Ramanathan, and Topiwala) conjecture holds for finite Gabor systems generated by square-integrable functions with certain behavior at infinity. These functions include functions ultimately decaying faster than any exponential function, as well as squareintegrable functions ultimately analytic and whose germs are in a Hardy field that is closed under translations. Tw...

There are several generalizations of the space L1(R) of Lebesgue integrable functions taking values in the real numbers R (and defined on the usual Lebesgue measure space (Ω,Σ, μ) on [0, 1] ) to a space of strongly-measurable “integrable” (suitably formulated) functions taking values in a Banach space X. The most common generalization is the space L1(X) of Bochner-Lebesgue integrable functions....

We prove that if a multiple trigonometric series is spherically Abel summable everywhere to an everywhere finite function f(x) which is bounded below by an integrable function, then the series is the Fourier series of f(x) if the coefficients of the multiple trigonometric series satisfy a mild growth condition. As a consequence, we show that if a multiple trigonometric series is spherically con...