Sharp Weak Type Inequalities for the Dyadic Maximal Operator

We obtain sharp estimates for the localized distribution function of $\mathcal{M}\phi $ , when ? belongs to L ^p,∞ where $\mathcal{M}$ is the dyadic maximal operator. We obtain these estimates given the L ¹ and L ^q norm, q<p and certain weak-L ^p conditions.In this way we refine the known weak (1,1) type inequality for the dyadic maximal operator. As a consequence we prove that the inequality 0.1 is sharp allowing every possible value for the L ¹ and the L ^q norm for a fixed q such that 1<q<p, where ∥?∥ _p,∞ is the usual quasi norm on L ^p,∞.     

Perturbation theory for convolution semigroups

We deal with convolution semigroups (not necessarily symmetric) in Lp(RN) and provide a general perturbation theory of their generators by indefinite singular potentials. Such semigroups arise in the theory of Lévy processes and cover many examples such as Gaussian semigroups, α-stable semigroups, relativistic Schrödinger semigroups, etc. We give new generation theorems and Feynman-Kac formulas. In particular, by using weak compactness methods in L¹, we enlarge the extended Kato class potentials used in the theory of Markov processes. In L² setting, Dirichlet form-perturbation theory is finely related to L¹-theory and the extended Kato class measures is also enlarged. Finally, various perturbation problems for subordinate semigroups are considered.     

Characterization of Scattering Data for the AKNS System

We characterize the scattering data of the AKNS system with vanishing boundary conditions. We prove a 1,1-correspondence between L ¹-potentials without spectral singularities and Marchenko integral kernels which are sums of an L ¹ function (having a reflection coefficient as its Fourier transform) and a finite exponential sum encoding bound states and norming constants. We give characterization results in the focusing and defocusing cases separately.     

Optimal decay rates for the compressible fluid models of Korteweg type

We consider the compressible Navier-Stokes-Korteweg system that models the motions of the compressible isothermal viscous capillary fluids. We prove the optimal L² and Lp, p?2 decay rates for the classical solutions and their spatial derivatives. In particular, the optimal L² decay rate of the second-order spatial derivatives is obtained. The proof is based on the detailed study of the linear decay estimates and nonlinear energy estimates.     

Stability of Riemann solutions with large oscillation for the relativistic Euler equations

We are concerned with entropy solutions of the 2×2 relativistic Euler equations for perfect fluids in special relativity. We establish the uniqueness of Riemann solutions in the class of entropy solutions in L^∞∩BV_loc with arbitrarily large oscillation. Our proof for solutions with large oscillation is based on a detailed analysis of global behavior of shock curves in the phase space and on special features of centered rarefaction waves in the physical plane for this system. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily largeL¹∩L^∞∩BV_loc perturbation of the Riemann initial data, as long as the corresponding solutions are in L^∞ and have local bounded total variation that allows the linear growth in time. We also extend our approach to deal with the uniqueness and stability of Riemann solutions containing vacuum in the class of entropy solutions in L^∞ with arbitrarily large oscillation.     

Bounds for singular fractional integrals and related Fourier integral operators

We prove sharp L^p−L^q endpoint bounds for singular fractional integral operators and related Fourier integral operators, under the nonvanishing rotational curvature hypothesis.     

Sharp estimates for a class of hyperbolic pseudo-differential equations

In this paper we consider the Cauchy problem for a class of hyperbolic pseudodifferential operators. The considered class contains constant coefficient differential equations, also allowing the coefficients to depend on time. We establish sharp L ^p ? L^p, Lipschitz, and other estimates for their solutions. In particular, the ellipticity condition for the roots of the principal symbol is eliminated for certain dimensions. We discuss the situation with no loss of smoothness for solutions. In the space R¹⁺ⁿ with n ≤ 4 (total dimension ≤ 5), we give a complete list of L ^p ? L^p properties. In particular, this contains the very important case R¹⁺³.     

Thermodynamic formalism for null recurrent potentials

We extend Ruelle’s Perron-Frobenius theorem to the case of Hölder continuous functions on a topologically mixing topological Markov shift with a countable number of states. LetP(?) denote the Gurevic pressure of ? and letL _? be the corresponding Ruelle operator. We present a necessary and sufficient condition for the existence of a conservative measure ν and a continuous functionh such thatL _? ^* ν=e ^P(?)ν,L _? h=e ^P(?) h and characterize the case when ∝hdν<∞. In the case whendm=hdν is infinite, we discuss the asymptotic behaviour ofL _? ^k , and show how to interpretdm as an equilibrium measure. We show how the above properties reflect in the behaviour of a suitable dynamical zeta function. These resutls extend the results of [18] where the case ∝hdν<∞ was studied.     

Best constants for Hausdorff operators on n-dimensional product spaces

In this paper, we study some new kinds of Hausdorff operators on n-dimensional product spaces. We obtain their power weights from L ^p to L ^q boundedness and characterize the necessary and sufficient conditions for the operators being bounded on power weight L ^p spaces. Moreover, we get the sharp constants for the case p = q.     

A priori error estimates of finite volume element method for hyperbolic optimal control problems

In this paper,optimize-then-discretize,variational discretization and the finite volume method are applied to solve the distributed optimal control problems governed by a second order hyperbolic equation.A semi-discrete optimal system is obtained.We prove the existence and uniqueness of the solution to the semidiscrete optimal system and obtain the optimal order error estimates in L ∞(J;L 2)-and L ∞(J;H 1)-norm.Numerical experiments are presented to test these theoretical results.     

Inequalities for integral means over symmetric sets

We prove that the integral of n functions over a symmetric set L in Rn, with additional properties, increases when the functions are replaced by their symmetric decreasing rearrangements. The result is known when L is a centrally symmetric convex set, and our result extends it to nonconvex sets. We deduce as consequences, inequalities for the average of a function whose level sets are of the same type as L, over measurable sets in Rn. The average of such a function on E is maximized by the average over the symmetric set E^*.     

On the Convergence of a Weak Greedy Algorithm for the Multivariate Haar Basis

We define a family of weak thresholding greedy algorithms for the multivariate Haar basis for L ₁[0,1] ^d (d≥1). We prove convergence and uniform boundedness of the weak greedy approximants for all f∈L ₁[0,1] ^d .     

Area integral functions for sectorial operators on L spaces

Area integral functions are introduced for sectorial operators on L^p-spaces. We establish the equivalence between the square and area integral functions for sectorial operators on L^p spaces. This follows that the results of Cowling, Doust, McIntosh, Yagi, and Le Merdy on H^infin functional calculus of sectorial operators on L^p-spaces hold true when the square functions are replaced by the area integral functions.     

A limit theorem for order preserving nonexpansive operators inL 1

We prove the following theorem:Let T be an order preserving nonexpansive operator on L ₁ (μ) (or L ₁ ⁺ ) of a σ-finite measure, which also decreases theL _∞-norm, and let S=tI+(1?t)T for 0<t<1. Then for everyf ∈ L_p (1<p<∞),the sequence S ⁿf converges weakly in L_p. (The assumptions do not imply thatT is nonexpansive inL _p for anyp>1, even ifμ is finite.) For the proof we show that ∥S ⁿ⁺¹ f?S ⁿf∥ _p → 0 for everyf ∈L _p, 1<p<∞, and apply toS the following theorem:Let T be order preserving and nonexpansive in L ₁ ⁺ , and assume that T decreases theL _∞-norm. Then forg ∈L _p (1<p<∞) Tⁿg is weakly almost convergent. If forf ∈ L_p we have T ⁿ⁺¹ f?T ⁿ f → 0weakly, then T ⁿf converges weakly in L_p (1<p<∞).     

Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L^∞ norm and weighted L²-norm. The numerical examples are given to illustrate the theoretical results.     

A splitting method for the nonlinear Schrödinger equation

We introduce a splitting method for the semilinear Schrödinger equation and prove its convergence for those nonlinearities which can be handled by the classical well-posedness L²(Rd)-theory. More precisely, we prove that the scheme is of first order in the L²(Rd)-norm for H²(Rd)-initial data.     

Primal-dual interior-point methods for PDE-constrained optimization

This paper provides a detailed analysis of a primal-dual interior-point method for PDE-constrained optimization. Considered are optimal control problems with control constraints in L ^p . It is shown that the developed primal-dual interior-point method converges globally and locally superlinearly. Not only the easier L ^∞-setting is analyzed, but also a more involved L ^q -analysis, q < ∞, is presented. In L ^∞, the set of feasible controls contains interior points and the Fréchet differentiability of the perturbed optimality system can be shown. In the L ^q -setting, which is highly relevant for PDE-constrained optimization, these nice properties are no longer available. Nevertheless, a convergence analysis is developed using refined techniques. In parti- cular, two-norm techniques and a smoothing step are required. The L ^q -analysis with smoothing step yields global linear and local superlinear convergence, whereas the L ^∞-analysis without smoothing step yields only global linear convergence.     

Quasi-steady-state approximation for a reaction-diffusion system with fast intermediate

We consider a prototype reaction-diffusion system which models a network of two consecutive reactions in which chemical components A and B form an intermediate C which decays into two products P and Q. Such a situation often occurs in applications and in the typical case when the intermediate is highly reactive, the species C is eliminated from the system by means of a quasi-steady-state approximation. In this paper, we prove the convergence of the solutions in L², as the decay rate of the intermediate tends to infinity, for all bounded initial data, even in the case of initial boundary layers. The limiting system is indeed the one which results from formal application of the QSSA. The proof combines the recent L²-approach to reaction-diffusion systems having at most quadratic reaction terms, with local L^∞-bounds which are independent of the decay rate of the intermediate. We also prove existence of global classical solutions to the initial system.     

A subclass of Ockham algebras

Here we introduce a subclass of the class of Ockham algebras ( L ; f ) for which L satisfies the property that for every x ∈ L , there exists n ≥ 0 such that fn ( x ) and fn+1 ( x ) are complementary. We characterize the structure of the lattice of congruences on such an algebra ( L ; f ). We show that the lattice of compact congruences on L is a dual Stone lattice, and in particular, that the lattice Con L of congruences on L is boolean if and only if L is finite boolean. We also show that L is congruence coherent if and only if it is boolean. Finally, we give a sufficient and necessary condition to have the subdirectly irreducible chains.     

Existence and bifurcation theorems for nonlinear elliptic eigenvalue problems on unbounded domains

We consider nonlinear elliptic eigenvalue problems on unbounded domains G?$\text{R}$ⁿ. Using an extended Ljusternik-Schnirelman theory we prove the existence of infinitely many eigenfunctions on every sphere in L²(G). Moreover, we establish that the infimum λ1 of the spectrum of the linearized problem L is always a bifurcation point. In addition, there is an infinity of branches emanating at λ1 from the trivial line of solutions if λ1 belongs to the essential spectrum of L.