Orlicz-Hardy spaces associated with operators

Let L be a linear operator in L2(Rn) and generate an analytic semigroup {e-tL}t 0 with kernel satisfying an upper bound of Poisson type, whose decay is measured by θ(L) ∈ (0, ∞). Let ω on (0, ∞) be of upper type 1 and of critical lower type p0(ω) ∈ (n/(n + θ(L)), 1] and ρ(t) = t-1/ω-1(t-1) for t ∈ (0, ∞). We introduce the Orlicz-Hardy space Hω, L(Rn) and the BMO-type space BMOρ, L(Rn) and establish the John-Nirenberg inequality for BMOρ, L(Rn) functions and the duality relation between Hω, L(Rn) and BMOρ, L...     

Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent

We prove the boundedness of the maximal operator Mr in the spaces L^p（·）（Г,p） with variable exponent p（t） and power weight p on an arbitrary Carleson curve under the assumption that p（t） satisfies the log-condition on Г. We prove also weighted Sobolev type L^p（·）（Г, p） → L^q（·）（Г, p）-theorem for potential operators on Carleson curves.     

Wavelet characterization of Hörmander symbol classS ρ,δ m and applications

In this paper, we characterize the symbol in Hormander symbol classS _ρ ^m ,δ (m ∈ R, ρ, δ ≥ 0) by its wavelet coefficients. Consequently, we analyse the kerneldistribution property for the symbol in the symbol classS _ρ ^m ,δ (m ∈R, ρ > 0, δ≥ 0) which is more general than known results ; for non-regular symbol operators, we establish sharp L²-continuity which is better than Calderón and Vaillancourt’s result, and establishL ^p (1 ≤p ≤∞) continuity which is new and sharp. Our new idea is to analyse the symbol operators in phase space with relative wavelets, and to establish the kernel distribution property and the operator’s continuity on the basis of the wavelets coefficients in phase space.     

Equivalent Characterizations for Boundedness of Maximal Singular Integrals on ax+b-Groups

Let (S,d,ρ) be the affine group ℝ ⁿ ⋉ℝ⁺ endowed with the left-invariant Riemannian metric d and the right Haar measure ρ, which is of exponential growth at infinity. In this paper, for any linear operator T on (S,d,ρ) associated with a kernel K satisfying certain integral size condition and H?rmander’s condition, the authors prove that the following four statements regarding the corresponding maximal singular integral T ^∗ are equivalent: T ^∗ is bounded from L_c^￥L_{c}^{\infty} to BMO, T ^∗ is bounded on L ^p for all p∈(1,∞), T ^∗ is bounded on L ^p for some p∈(1,∞) and T ^∗ is bounded from L ¹ to L ^1,∞. As applications of these results, for spectral multipliers of a distinguished Laplacian on (S,d,ρ) satisfying certain Mihlin-H?rmander type condition, the authors obtain that their maximal singular integrals are bounded from L_c^￥L_{c}^{\infty} to BMO, from L ¹ to L ^1,∞, and on L ^p for all p∈(1,∞).     

New decay estimates for linear damped wave equations and its application to nonlinear problem

We present new decay estimates of solutions for the mixed problem of the equation v_tt?v_xx+v_t=0, which has the weighted initial data [v₀,v₁]∈(H¹₀(0,∞) ∩L^1,γ(0,∞)) × (L²(0,∞)∩L^1,γ(0,∞)) (for definition of L^1,γ(0,∞), see below) satisfying γ∈[0,1]. Similar decay estimates are also derived to the Cauchy problem in ?^N for u_tt?Δu+u_t=0 with the weighted initial data. Finally, these decay estimates can be applied to the one dimensional critical exponent problem for a semilinear damped wave equation on the half line. Copyright © 2004 John Wiley & Sons, Ltd.     

Non-Diffusive Large Time Behavior for a Degenerate Viscous Hamilton–Jacobi Equation

The convergence to non-diffusive self-similar solutions is investigated for non-negative solutions to the Cauchy problem ? _t u = Δ _p u + |? u| ^q when the initial data converge to zero at infinity. Sufficient conditions on the exponents p > 2 and q > 1 are given that guarantee that the diffusion becomes negligible for large times and the L ^∞-norm of u(t) converges to a positive value as t → ∞.     

Boundary behavior of Gleason''''s problem in hyperbolic harmonic Bergman spaces

Necessary and sufficient conditions are obtained for the boundedness of Berezin transformation on Lebesgue space Lp(B, dVβ) in the real unit ball B in Rn. As an application, we prove that Gleason type problem is solvable in hyperbolic harmonic Bergman spaces. Furthermore we investigate the boundary behavior of the solutions of Gleason type problem.     

Operator-semistable, operator semi-selfdecomposable probability measures and related nested classes on p-adic vector spaces

Let V be a finite dimensional p-adic vector space and let τ be an operator in GL(V). A probability measure μ on V is called τ-decomposable or m ? [(L)\tilde]₀(t)\mu\in {\tilde L}_0(\tau) if μ = τ(μ)* ρ for some probability measure ρ on V. Moreover, when τ is contracting, if ρ is infinitely divisible, so is μ, and if ρ is embeddable, so is μ. These two subclasses of [(L)\tilde]₀(t){\tilde L}_0(\tau) are denoted by L ₀(τ) and L ₀ ^#(τ) respectively. When μ is infinitely divisible τ-decomposable for a contracting τ and has no idempotent factors, then it is τ-semi-selfdecomposable or operator semi-selfdecomposable. In this paper, sequences of decreasing subclasses of the above mentioned three classes, [(L)\tilde]_m(t) é L_m(t) é L^#_m(t), 1 ￡ m ￡ ￥{\tilde L}_m(\tau)\supset L_m(\tau) \supset L^\#_m(\tau), 1\le m\le \infty , are introduced and several properties and characterizations are studied. The results obtained here are p-adic vector space versions of those given for probability measures on Euclidean spaces.     

Cross-Diffusion Limit for a Reaction-Diffusion System with Fast Reversible Reaction

We consider a reaction-diffusion system which models a fast reversible reaction of type C ₁ + C ₂?C ₃ between mobile reactants inside an isolated vessel. Assuming mass action kinetics, we study the limit when the reaction speed tends to infinity in case of unequal diffusion coefficients and prove convergence of a subsequence of solutions to a weak solution of an appropriate limiting pde-system, where the limiting problem turns out to be of cross-diffusion type. The proof combines the L ²-approach to reaction-diffusion systems having at most quadratic reaction terms with a thorough exploitation of the entropy functional for mass action systems. The limiting cross-diffusion system has unique local strong solutions for sufficiently regular initial data, while uniqueness of weak solutions is in general open but is shown to be valid under restrictions on the diffusivities.     

Operators on the Lattice of Pseudovarieties of Finite Semigroups

L (F) of pseudovarieties of finite semigroups that attempts to take full advantage of the underlying lattice structure, Auinger, Hall and the present authors recently introduced fourteen complete congruences on L (F). Such congruences provide a framework from which to study L (F) both locally and globally. For each such congruence ρ and each U∈L (F) the ρ-class of U is an interval [U ^ρ, U _ρ]. This provides a family of operators of the form U⇒Uρ on L (F) that reveal important relationships between elements of L (F). Various aspects of these operators are considered including characterizations of U ^ρ, bases of pseudoidentities for U ^ρ, instances of commutativity (U ^ρ)^σ = U ^σ)^ρ, as well as the semigroups generated by certain pairs of such operators.     

Commutators of Littlewood-Paley operators

Let b ∈ Lloc(Rn) and L denote the Littlewood-Paley operators including the LittlewoodPaley g function,Lusin area integral and gλ* function. In this paper,the authors prove that the Lp boundedness of commutators [b,L] implies that b ∈ BMO(Rn) . The authors therefore get a characterization of the Lp-boundedness of the commutators [b,L]. Notice that the condition of kernel function of L is weaker than the Lipshitz condition and the Littlewood-Paley operators L is only sublinear,so the results obtained in the p...     

Estimates for parametric Marcinkiewicz integrals in BMO and Campanato spaces

In this paper, the authors consider the behaviors of a class of parametric Marcinkiewicz integrals μ _Ω ^ρ , μ _Ω,λ ^*,ρ and μ _Ω,S ^ρ on BMO(ℝ ⁿ ) and Campanato spaces with complex parameter ρ and the kernel Ω in Llog⁺ L(S ⁿ⁻¹). Here μ _Ω,λ ^*,ρ and μ _Ω,S ^ρ are parametric Marcinkiewicz functions corresponding to the Littlewood-Paley g _λ *-function and the Lusin area function S, respectively. Under certain weak regularity condition on Ω, the authors prove that if f belongs to BMO(ℝ ⁿ ) or to a certain Campanato space, then [μ _Ω,λ ^*,ρ (f)]², [μ _Ω,S ^ρ (f)]² and [μ _Ω ^ρ (f)]² are either infinite everywhere or finite almost everywhere, and in the latter case, some kind of boundedness are also established.     

On a resolvent estimate of the Stokes system in a half space arising from a free boundary problem for the Navier–Stokes equations

In this paper we consider a resolvent problem of the Stokes operator with some boundary condition in the half space, which is obtained as a model problem arising in evolution free boundary problems for viscous, incompressible fluid flow. We show standard resolvent estimates in the L_q framework (1 < q < ∞), applying some kernel estimates to concrete solution formulas. The Volevich trick in [21] plays a fundamental role in estimating solutions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonsmooth Neumann-Type Problems Involving the p-Laplacian

This paper deals with the problem ? Δ _p u + α(x)|u| ^p?2 u = β(x)f(|u|) in Ω, subjected to the zero Neumann boundary condition, where p > 1, Ω ? ? ^N is bounded with smooth boundary, α, β ? L ^∞(Ω), essinf_Ωβ > 0, and f:[0,+ ∞) → ? is a not necessarily continuous nonlinearity that oscillates either at the origin or at the infinity. By using nonsmooth variational methods, we establish in both cases the existence of infinitely many distinct non-negative solutions of the Neumann problem. In our framework, α:Ω → ? may be a sign-changing or even a nonpositive potential, which is not permitted usually in earlier works.     

Approximation in Learning Theory

This paper addresses some problems of supervised learning in the setting formulated by Cucker and Smale. Supervised learning, or learning-from-examples, refers to a process that builds on the base of available data of inputs x_i and outputs y_i, i = 1,...,m, a function that best represents the relation between the inputs x ∈ X and the corresponding outputs y ∈ Y. The goal is to find an estimator f_z on the base of given data z := ((x₁,y₁),...,(x_m,y_m)) that approximates well the regression function f_ρ (or its projection) of an unknown Borel probability measure ρ defined on Z = X × Y. We assume that (x_i,y_i), i = 1,...,m, are independent and distributed according to ρ. We discuss the following two problems: I. the projection learning problem (improper function learning problem); II. universal (adaptive) estimators in the proper function learning problem. In the first problem we do not impose any restrictions on a Borel measure ρ except our standard assumption that |y|≤ M a.e. with respect to ρ. In this case we use the data z to estimate (approximate) the L₂(ρ_X) projection (f_ρ)_W of f_ρ onto a function class W of our choice. Here, ρ_X is the marginal probability measure. In [KT1,2] this problem has been studied for W satisfying the decay condition ε_n(W,B) ≤ Dn^-r of the entropy numbers ε_n(W,B) of W in a Banach space B in the case B = C(X) or B = L₂(\rho_X). In this paper we obtain the upper estimates in the case ε_n(W,L₁(ρ_X)) ≤ Dn^-r with an extra assumption that W is convex. In the second problem we assume that an unknown measure ρ satisfies some conditions. Following the standard way from nonparametric statistics we formulate these conditions of the form f_ρ ∈ Θ. Next, we assume that the only a priori information available is that f_ρ belongs to a class Θ (unknown) from a known collection {Θ} of classes. We want to build an estimator that provides approximation of f_ρ close to the optimal for the class Θ. Along with standard penalized least squares estimators we consider a new method of construction of universal estimators. This method is based on a combination of two powerful ideas in building universal estimators. The first one is the use of penalized least squares estimators. This idea works well in the case of general setting with rather abstract methods of approximation. The second one is the idea of thresholding that works very well when we use wavelets expansions as an approximation tool. A new estimator that we call the big jump estimator uses the least squares estimators and chooses a right model by a thresholding criteria instead of the penalization. In this paper we illustrate how ideas and methods of approximation theory can be used in learning theory both in formulating a problem and in solving it.     

On a wave equation with supercritical interior and boundary sources and damping terms

This article addresses nonlinear wave equations with supercritical interior and boundary sources, and subject to interior and boundary damping. The presence of a nonlinear boundary source alone is known to pose a significant difficulty since the linear Neumann problem for the wave equation is not, in general, well‐posed in the finite‐energy space H¹(Ω) × L₂(?Ω) with boundary data in L₂ due to the failure of the uniform Lopatinskii condition. Further challenges stem from the fact that both sources are non‐dissipative and are not locally Lipschitz operators from H¹(Ω) into L₂(Ω), or L₂(?Ω). With some restrictions on the parameters in the model and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution, and establish exponential and algebraic uniform decay rates of the finite energy (depending on the behavior of the dissipation terms). Moreover, we prove a blow up result for weak solutions with nonnegative initial energy.     

Asymptotics for Solutions of Elliptic Equations in Double Divergence Form

We consider weak solutions of an elliptic equation of the form ? _i ? _i (a _ij u) = 0 and their asymptotic properties at an interior point. We assume that the coefficients are bounded, measurable, complex-valued functions that stabilize as x → 0 in that the norm of the matrix (a _ij (x) ? δ _ij ) on the annulus B _2r \ B _r is bounded by a function Ω(r), where Ω²(r) satisfies the Dini condition at r = 0, as well as some technical monotonicity conditions; under these assumptions, solutions need not be continuous. Our main result is an explicit formula for the leading asymptotic term for solutions with at most a mild singularity at x = 0. As a consequence, we obtain upper and lower estimates for the L ^p -norm of solutions, as well as necessary and sufficient conditions for solutions to be bounded or tend to zero in L ^p -mean as r → 0.     

An integrable equation with nonsmooth solitons

We present the bi-Hamiltonian structure and Lax pair of the equation ρ_t = bu_x+(1/2)[(u ² −u_x ² )ρ]_x, where ρ = u − u_xx and b = const, which guarantees its integrability in the Lax pair sense. We study nonsmooth soliton solutions of this equation and show that under the vanishing boundary condition u → 0 at the space and time infinities, the equation has both “W/M-shape” peaked soliton (peakon) and cusped soliton (cuspon) solutions.     

On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities

We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain Ω of R³. We first prove the local existence of solutions (ρ,u) in C([0,T_*]; (ρ^∞ +H³(Ω)) × under the assumption that the data satisfies a natural compatibility condition. Then deriving the smoothing effect of the velocity u in t>0, we conclude that (ρ,u) is a classical solution in (0,T_**)×Ω for some T_** ∈ (0,T_*]. For these results, the initial density needs not be bounded below away from zero and may vanish in an open subset (vacuum) of Ω.     

Regular solutions to the coagulation equations with singular kernels

In this article, we prove the existence of solutions to the coagulation equation with singular kernels. We use weighted L¹‐spaces to deal with the singularities in order to obtain regular solutions. The Smoluchowski kernel is covered by our proof. The weak L¹ compactness methods are applied to suitably chosen approximating equations as a base of our proof. A more restrictive uniqueness result is also mentioned. Copyright © 2014 John Wiley & Sons, Ltd.