lp zd estimates for discrete operators of radon type maximal functions and vector valued estimates abstract we prove l p z d bounds for p 1 of discrete maximal functions corresponding to averaging operators and truncated singular integrals of radon type and their applications to pointwise ergodic theory our new approach is based on a unified analysis of both types of operators and also yields an extension to the vector valued form of these results

some new characterizations of sobolev spaces in this paper we present some new characterizations of sobolev spaces here is a typical result let g l p r n 1 p we prove that g w 1 p r n if and only if

composite systems with uncertain couplings of fixed structure scaled riccati equations and the problem of quadratic stability we consider large scale systems consisting of a finite number of separate uncertain subsystems which interact via uncertain couplings of arbitrarily prescribed structure the couplings are viewed as structured perturbations of the block diagonal system representing the collection of the separate nominal subsystems the nominal system we define spectral value sets and stability radii for these time invariant structured perturbations and derive formulas for their computation scaled riccati equations are introduced to obtain explicit formulas for the stability radii with respect to time varying and possibly nonlinear perturbations of the given structure from these we derive necessary and sufficient conditions under which the stability radii with respect to time invariant and time varying perturbations are equal these results are obtained by constructing joint quadratic liapunov functions of optimal robustness with their help we prove necessary and sufficient conditions for quadratic stability and sufficient conditions for the validity of a generalized aizerman conjecture

ell p left mathbb z d right estimates for discrete operators of radon type variational estimates we prove ell p left mathbb z d right bounds for p in 1 infty of r variations r in 2 infty for discrete averaging operators and truncated singular integrals of radon type we shall present a new powerful method which allows us to deal with these operators in a unified way and obtain the range of parameters of p and r which coincide with the ranges of their continuous counterparts

variational estimates for discrete operators modeled on multi dimensional polynomial subsets of primes we prove the extensions of birkhoff s and cotlar s ergodic theorems to multi dimensional polynomial subsets of prime numbers mathbb p k we deduce them from ell p big mathbb z d big boundedness of r variational seminorms for the corresponding discrete operators of radon type where p 1 and r 2

anisotropic fractional sobolev norms abstract bourgain brezis mironescu showed that with suitable scaling the fractional sobolev s seminorm of a function f w 1 p r n converges to the sobolev seminorm of f as s 1 the anisotropic s seminorms of f defined by a norm on r n with unit ball k are shown to converge to the anisotropic sobolev seminorm of f defined by the norm with unit ball z p k the polar l p moment body of k the limiting behavior for s 0 is also determined extending results by maz ya shaposhnikova

integral energy characterization of hajlasz sobolev spaces abstract we provide an integral energy characterization of hajlasz sobolev spaces m 1 p x e with p 1 where x is a metric measure space that satisfies weak poincare inequality and e is a banach space with the radon nikodym property this result extends previously known results which were originally established on r n

estimates for the square variation of partial sums of fourier series and their rearrangements we investigate the square variation operator v2 which majorizes the partial sum maximal operator on general orthonormal systems ons of size n we prove that the l2 norm of the v2 operator is bounded by o ln n on any ons this result is sharp and refines the classical rademacher menshov theorem we show that this can be improved to o ln n for the trigonometric system which is also sharp we show that for any choice of coefficients this truncation of the trigonometric system can be rearranged so that the l2 norm of the associated v2 operator is o lnln n we also show that for p 2 a bounded ons of size n can be rearranged so that the l2 norm of the vp operator is at most op lnln n uniformly for all choices of coefficients this refines bourgain s work on garsia s conjecture which is equivalent to the v case several other results on operators of this form are also obtained the proofs rely on combinatorial and probabilistic methods

discrete analogues in harmonic analysis i l 2 estimates for singular radon transforms this paper studies the discrete analogues of singular radon transforms we prove the 2 boundedness for those operators that are quasi translation invariant the approach used is related to the circle method of hardy and littlewood and requires multi dimensional extensions of weyl sums and gauss sums as well as variants that replace scalar sums by operator sums 1 introduction the purpose of this paper is to study discrete analogues of singular radon transforms their analysis will involve considerations originating in analytic number theory related to the circle method of hardy and littlewood and necessitates an extension of the notion of weyl sums and gauss sums this will require that we pass from the usual setting of scalar valued sums to certain operator analogues of these sums

vinogradov s mean value theorem via efficient congruencing we obtain estimates for vinogradov s integral that for the rst time approach those conjectured to be the best possible several applications of these new bounds are provided in particular the conjectured asymptotic formula in waring s problem holds for sums of s kth powers of natural numbers whenever s 2k 2 2k 3

small gain theorems for large scale systems and construction of iss lyapunov functions we consider interconnections of n nonlinear subsystems in the input to state stability iss framework for each subsystem an iss lyapunov function is given that treats the other subsystems as independent inputs a gain matrix is used to encode the mutual dependencies of the systems in the network under a small gain assumption on the monotone operator induced by the gain matrix a locally lipschitz continuous iss lyapunov function is obtained constructively for the entire network by appropriately scaling the individual lyapunov functions for the subsystems the results are obtained in a general formulation of iss the cases of summation maximization and separation with respect to external gains are obtained as corollaries

gamma convergence sobolev norms and bv functions we prove that the family of functionals i defined by i g rn rn g x g y x y n p dx dy g l r for p 1 and 0 converges in l r as goes to 0 when p 1 hereafter denotes the euclidean norm of r we also introduce a characterization for bv functions which has some advantages in comparison with the classic one based on the notion of essential variation on a e line

obituary to a j pritchard on the 12th of august 2007 tony pritchard professor of mathematics at the university of warwick died suddenly at the age of 70 from a heart attack while on holiday with his family on the island of lanzarote both of the authors have had the privilege of working with him for many years we have lost both a colleague and a friend the following notes are dedicated to his memory

discrete fractional integral operators with binary quadratic forms as phase polynomials abstract we give estimates on discrete fractional integral operators along binary quadratic forms these operators have been studied for 30 years starting with the investigations of arkhipov and oskolkov but efforts have concentrated on cases where the phase polynomial is translation invariant or quasi translation invariant this work presents the first results for operators with neither translation invariant nor quasi translation invariant phase polynomials

systems with structured uncertainty relations between quadratic and robust stability the relation between the notions of robust stability and quadratic stability for uncertain systems with structured uncertainty due to both real and complex parameter variations is discussed examples are presented to demonstrate that for systems containing at least two uncertain blocks the notions of robust stability for complex parameter variations and quadratic stability for real parameter variations are not equivalent a byproduct of these examples is that for this class of systems quadratic stability for real perturbations need not imply quadratic stability for complex perturbations this is in stark contrast with the situation in the case of unstructured uncertainty for which it is known that quadratic stability for either real or complex perturbations is equivalent to robust stability for complex perturbations and thus equivalent to a small gain condition on the transfer matrix that the perturbation experiences

criteria of asymptotic stability of differential and difference inclusions encountered in control theory abstract for a class of differential inclusions to which many of the practically important control systems can be reduced necessary and sufficient conditions for asymptotic stability of the zero solution are established by the method of lyapunov functions similar results are obtained for the corresponding class of difference inclusions

discrete radon transforms and applications to ergodic theory we prove lp boundedness of certain non translation invariant discrete maximal radon transforms and discrete singular radon transforms we also prove maximal pointwise and lp ergodic theorems for certain families of non commuting operators

interconnected systems with uncertain couplings explicit formulae for mu values spectral value sets and stability radii in this paper we study the variation of the spectrum of block diagonal systems under perturbations of compatible block structure with fixed zero blocks at arbitrarily prescribed locations gershgorin type perturbations we derive explicit and computable formulae for the associated mu values the results are then applied to characterize spectral value sets and stability radii for such perturbed systems by specializing our results to the scalar diagonal case the classical eigenvalue inclusion theorems of gershgorin brauer and brualdi are obtained as corollaries moreover it follows that the inclusion regions of brauer and brualdi are optimal for the corresponding perturbation structures

discrete analogues in harmonic analysis spherical averages in this paper we prove an analogue in the discrete setting of d o fthe spherical maximal theorem for d the methods used are two fold the application of certain sampling techniques and ideas arising in the study of the number of representations of an integer as a sum of d squares in particular the circle method the results we obtained are by necessity limited to d 5 and moreover the range of p for the l p estimates differs from its analogue in d

stability radius for structured perturbations and the algebraic riccati equation abstract in this paper we generalize the notion of stability radius introduced in 1 to allow for structured perturbations we then relate the stability radius to the existence of hermitian solutions of an algebraic riccati equation and give some applications of this result

discrete maximal functions in higher dimensions and applications to ergodic theory we establish a higher dimensional counterpart of bourgain s pointwise ergodic theorem along an arbitrary integer valued polynomial mapping we achieve this by proving variational estimates v r on l p spaces for all 1

qualitative theory of compartmental systems dynamic models of many processes in the biological and physical sciences which depend on local mass balance conditions give rise to systems of ordinary differential equations many nonlinear that are called compartmental systems in this paper the authors define compartmental systems specify their relations to other nonnegative systems and discuss examples of applications the authors review the qualitative results on linear and nonlinear compartmental systems including their relation to cooperative systems they review the results for linear compartmental systems and then integrate and expand the results on nonlinear compartmental systems providing a framework for unifying them under a few general theorems in the course of that they complete the solution of a problem posed by bellman and show that closed nonlinear autonomous n compartment systems can show the full gamut of possible behaviors of systems of odes finally to provide additional structure to this study the authors show how to partiti

new characterizations of sobolev metric spaces abstract we provide new characterizations of sobolev ad bv spaces in doubling and poincare metric spaces in the spirit of the bourgain brezis mironescu and nguyen limit formulas holding in domains of r n