Pointwise Inequality Research Articles

Littlewood–Paley characterizations of Triebel–Lizorkin–Morrey spaces via ball averages

Let 1<p≤u<∞ and q∈(1,∞]. In this article, the authors establish equivalent characterizations of the Triebel–Lizorkin–Morrey space Eu,q,pα(Rn) with smoothness order α∈(0,2) in terms of the Lusin-area function and the gλ∗-function generated by the difference between f(x) and its ball average B2−jf(x):=1|B(x,2−j)|∫B(x,2−j)f(y)dy,∀j∈{1,2,…},∀x∈Rn, where, for any j∈{1,2,…} and x∈Rn, B(x,2−j):={y∈Rn:|y−x|<2−j}. As applications, the authors obtain several characterizations of Eu,∞,pα(Rn) via pointwise inequalities involving ball averages.

Approximation by Hölder Functions in Besov and Triebel–Lizorkin Spaces

In this paper, we show that Besov and Triebel–Lizorkin functions can be approximated by a Holder continuous function both in the Lusin sense and in norm. The results are proved in metric measure spaces for Hajlasz–Besov and Hajlasz–Triebel–Lizorkin functions defined by a pointwise inequality. We also prove new inequalities for medians, including a Poincare type inequality, which we use in the proof of the main result.

An extension of the Beckner’s type Poincaré inequality to convolution measures on abstract Wiener spaces

ABSTRACTWe generalize the Beckner’s type Poincaré inequality (Beckner, W. Proc. Amer. Math. Soc. (1989) 105:397–400) to a large class of probability measures on an abstract Wiener space of the form μ⋆ν, where μ is the reference Gaussian measure and ν is a probability measure satisfying a certain integrability condition. As the Beckner inequality interpolates between the Poincaré and logarithmic Sobolev inequalities, we utilize a family of products for functions which interpolates between the usual point-wise multiplication and the Wick product. Our approach is based on the positivity of a quadratic form involving Wick powers and integration with respect to those convolution measures. In addition, we prove that in the finite-dimensional case the class of densities of convolutions measures satisfies a point-wise covariance inequality.

Best constants for the Hardy–Littlewood maximal operator on finite graphs

We study the behavior of averages for functions defined on finite graphs G, in terms of the Hardy–Littlewood maximal operator MG. We explore the relationship between the geometry of a graph and its maximal operator and prove that MG completely determines G (even though embedding properties for the graphs do not imply pointwise inequalities for the maximal operators). Optimal bounds for the p-(quasi)norm of a general graph G in the range 0<p≤1 are given, and it is shown that the complete graph Kn and the star graph Sn are the extremal graphs attaining, respectively, the lower and upper estimates. Finally, we study weak-type estimates and some connections with the dilation and overlapping indices of a graph.

On the Regularity of the Multisublinear Maximal Functions

AbstractThis paper is concerned with the study of the regularity for the multisublinear maximal operator. It is proved that the multisublinear maximal operator is bounded on first-order Sobolev spaces. Moreover, two key point-wise inequalities for the partial derivatives of the multisublinear maximal functions are established. As an application, the quasi-continuity on the multisublinear maximal function is also obtained.

Pointwise Inequalities for Elliptic Boundary Value Problems

We introduce a new approach for obtaining pointwise estimates for solutions of elliptic boundary value problems when the operator satisfies a certain type of weighted integral inequalities. The method is illustrated on several examples, including a scalar secondorder elliptic equation, the 3D Lamé system, and a scalar higher-order elliptic equation. The techniques can be extended to other elliptic boundary value problems provided that the corresponding weighted integral inequalities are satisfied. Bibliography: 36 titles.

A pointwise inequality for the fourth-order Lane–Emden equation

We prove that the following pointwise inequality holds −∆u ≥ √ 2 (p + 1)− cn |x| a 2 u p+1 2 + 2 n− 4 |∇u|2 u in R where cn := 8 n(n−4) , for positive bounded solutions of the fourth order Henon equation that is ∆u = |x|u in R for some a ≥ 0 and p > 1. Motivated by the Moser’s proof of the Harnack’s inequality as well as Moser iteration type arguments in the regularity theory, we develop an iteration argument to prove the above pointwise inequality. As far as we know this is the first time that such an argument is applied towards constructing pointwise inequalities for partial differential equations. An interesting point is that the coefficient 2 n−4 also appears in the fourth order Q-curvature and the Paneitz operator. This in particular implies that the scalar curvature of the conformal metric with conformal factor u 4 n−4 is positive.

A pointwise inequality for fractional laplacians

The fractional laplacian is an operator appearing in several evolution models where diffusion coming from a Lévy process is present but also in the analysis of fluid interphases. We provide an extension of a pointwise inequality that plays a rôle in their study. We begin recalling two scenarios where it has been used. After stating the results, for fractional Laplace–Beltrami and Dirichlet–Neumann operators, we provide a sketch of their proofs, unravelling the underlying principle to such inequalities.

Superconvergence for Neumann boundary control problems governed by semilinear elliptic equations

This paper is concerned with the discretization error analysis of semilinear Neumann boundary control problems in polygonal domains with pointwise inequality constraints on the control. The approximations of the control are piecewise constant functions. The state and adjoint state are discretized by piecewise linear finite elements. In a postprocessing step approximations of locally optimal controls of the continuous optimal control problem are constructed by the projection of the respective discrete adjoint state. Although the quality of the approximations is in general affected by corner singularities a convergence order of $h^2|\ln h|^{3/2}$ is proven for domains with interior angles smaller than $2\pi/3$ using quasi-uniform meshes. For larger interior angles mesh grading techniques are used to get the same order of convergence.

Finite element error estimates on the boundary with application to optimal control

This paper is concerned with the discretization of linear elliptic partial differential equations with Neumann boundary condition in polygonal domains. The focus is on the derivation of error estimates in the L 2 L^2 -norm on the boundary for linear finite elements. Whereas common techniques yield only suboptimal results, a new approach in this context is presented which allows for quasi-optimal ones, i.e., for domains with interior angles smaller than 2 π / 3 2\pi /3 a convergence order two (up to a logarithmic factor) can be achieved using quasi-uniform meshes. In the presence of internal angles greater than 2 π / 3 2\pi /3 which reduce the convergence rates on quasi-uniform meshes, graded meshes are used to maintain the quasi-optimal error bounds. This result is applied to linear-quadratic Neumann boundary control problems with pointwise inequality constraints on the control. The approximations of the control are piecewise constant. The state and the adjoint state are discretized by piecewise linear finite elements. In a postprocessing step approximations of the continuous optimal control are constructed which possess superconvergence properties. Based on the improved error estimates on the boundary and optimal regularity in weighted Sobolev spaces almost second order convergence is proven for the approximations of the continuous optimal control problem. Mesh grading techniques are again used for domains with interior angles greater than 2 π / 3 2\pi /3 . A certain regularity of the active set is assumed.

A class of superconformal surfaces

Superconformal surfaces in Euclidean space are the one for which the ellipse of curvature at any point is a nondegenerate circle. They can be characterized as the surfaces for which a well-known pointwise inequality relating the intrinsic Gauss curvature with the extrinsic normal and mean curvatures, due to Wintgen (C R Acad Sci Paris T Ser A 288:993–995, 1979) and Guadalupe-Rodriguez (Pac J Math 106:95–103, 1983) for any codimension, reaches equality at all points. In this paper, we show that any pedal surface to a $$2$$ -isotropic Euclidean surface is superconformal. Opposed to almost all known examples, superconformal surfaces in this class are not conformally equivalent to minimal surfaces. Moreover, they can be given in an explicit parametric form since $$2$$ -isotropic surfaces admit a Weierstrass-type representation.

Pointwise inequalities of logarithmic type in Hardy-Hölder spaces

We prove some optimal logarithmic estimates in the Hardy space H∞(G) with Holder regularity, where G is the open unit disk or an annular domain of ℂ. These estimates extend the results established by S.Chaabane and I.Feki in the Hardy-Sobolev space Hk,∞ of the unit disk and those of I. Feki in the case of an annular domain. The proofs are based on a variant of Hardy-Landau-Littlewood inequality for Holder functions. As an application of these estimates, we study the stability of both the Cauchy problem for the Laplace operator and the Robin inverse problem.

Variational method for optimal multimaterial composites and optimal design

The paper outlines novel variational technique for finding microstructures of optimal multimaterial composites, bounds of composites properties, and multimaterial optimal designs. The translation method that is used for the exact two-material bounds is complemented by additional pointwise inequalities on stresses in materials within an optimal composites. The method leads to exact multimaterial bounds and provides a hint for optimal structures that may be multi-rank laminates or, for isotropic composites, “wheel assemblages”. The Lagrangian of the formulated nonconvex multiwell variational problem is equal to the energy of the best adapted to the loading microstructures plus the cost of the used materials; the technique improves both the lower and upper bounds for the quasiconvex envelope of that Lagrangian. The problem of 2d elastic composites is described in some details; on particular, the isotropic component of the quasiconvex envelope of three-well Lagrangian for elastic energy is computed. The obtained results are applied for computing of optimal multimaterial elastic designs; an example of such a design is demonstrated. Finally, the optimal “wheel assemblages” are generalized and novel types of exotic microstructures with unusual properties are described.

Higher Order Sobolev-Type Spaces on the Real Line

This paper gives a characterization of Sobolev functions on the real line by means of pointwise inequalities involving finite differences. This is also shown to apply to more general Orlicz-Sobolev, Lorentz-Sobolev, and Lorentz-Karamata-Sobolev spaces.

A pointwise characterization of functions of bounded variation on metric spaces

We give a new characterization of the space of functions of bounded variation in terms of a pointwise inequality connected to the maximal function of a measure. The characterization is new even in Euclidean spaces and it holds also in general metric spaces.

Preference for safety under the Choquet model: in search of a characterization

Victor prefers safety more than Ursula if whenever Ursula prefers a constant to an uncertain act, so does Victor. This paradigm, whose expected utility (EU) version is Arrow and Pratt’s more risk aversion concept, will be studied in the Choquet expected utility (CEU) model. Necessary condition Pointwise inequality between a function of the utility functions and another of the capacities is necessary and sufficient for the preference by Victor of safety over a dichotomous act whenever such is the preference of Ursula. However, increased preference for safety versus dichotomous acts does not imply preference by Victor of safety over a general act whenever such is the preference of Ursula. A counterexample will be provided, via the casino theory of Dubins and Savage. Sufficient condition Separation of the two functions by some convex function is sufficient for Victor to prefer safety more than Ursula, over general acts. Furthermore, a condition on the capacities will be presented for simplicity seeking, the preference by Victor over any act for some dichotomous act that leaves Ursula indifferent. This condition is met in particular if Victor’s capacity is a convex function of Ursula’s capacity. For these cases, the pointwise inequality (necessary) condition is a characterization of greater preference for safety, extending the Arrow–Pratt notion from EU to CEU and rank-dependent utility (RDU). These inequalities preserve the flavor of the “more pessimism than greediness” characterization of monotone risk aversion by Chateauneuf, Cohen and Meilijson in the RDU model and its extension by Grant and Quiggin to CEU. Preferences between safety and dichotomous acts are at the core of the biseparable preferences model of Ghirardato and Marinacci.

A Characteristic Finite Element Method for Constrained Convection-Diffusion-Reaction Optimal Control Problems

In this paper, we develop a priori error estimates for the solution of constrained convection-diffusion-reaction optimal control problems using a characteristic finite element method. The cost functional of the optimal control problems consists of three parts: The first part is about integration of the state over the whole time interval, the second part refers to final-time state, and the third part is a regularization term about the control. We discretize the state and co-state by piecewise linear continuous functions, while the control is approximated by piecewise constant functions. Pointwise inequality function constraints on the control are considered, and optimal a L-norm priori error estimates are obtained. Finally, we give two numerical examples to validate the theoretical analysis.

Optimal control for an elliptic system with convex polygonal control constraints

In recent years, optimal control problems governed by partial differential equations with pointwise constraints on the controls have received a considerable amount of attention. The challenge consists in finding efficient numerical methods in spite of the nonsmoothness introduced by pointwise inequality constraints. Most investigations so far have considered the case of scalar-valued control with simple constraints (e.g. unilateral or bilateral constraints). For vector-valued controls, which appear, for example, in the context of the Navier–Stokes equations, the constraints that were investigated were of a box type. For scalar-valued controls with unilateral or bilateral constraints, the numerical realization of optimal control problems can advantageously be performed with the semismooth Newton method or, equivalently, by the primal–dual active set method. We refer to Hintermuller et al. (2003), Ito & Kunisch (2004), De Los Reyes & Kunisch (2005), Hintermuller & Hinze (2006), Ito & Kunisch (2008) and Ulbrich (2011) and the references cited therein. An alternative approach is based on interior point methods. They were analysed, for example, in Schiela & Weiser (2008) and Weiser et al. (2008). The semismooth Newton method is different from penalty—or barrier—function methods which rely on penalty—or barrier— parameters, respectively. It was proven to be locally superlinearly convergent, and for unilaterally constrained problems to be globally convergent; cf. Hintermuller et al. (2003). For vector-valued controls, there are only a few papers that deal with the case of general convex control constraints. In Wachsmuth (2006a,b) and Wachsmuth (2007), the first-order necessary and the second-order sufficient optimality conditions for general convex constraints are obtained. In Wachsmuth (2006b), a numerical procedure is proposed which, however, does not concretely exploit the structure of

Adaptive characteristic finite element approximation of convection–diffusion optimal control problems

AbstractIn this article, a characteristic finite element approximation of quadratic optimal control problems governed by linear convection–diffusion equations is given. We derive some a posteriori error estimates for both the control and the state approximations, where the control variable is constrained by pointwise inequality. The derived error estimators are then used as an error indicator to guide the mesh refinement. In this sense, they are very important in developing adaptive finite element algorithm for the optimal control problems. Finally, a numerical example is given to validate the efficiency and reliability of the theoretical results. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Estimates for the maximal singular integral in terms of the singular integral: the case of even kernels

Let T be a smooth homogeneous Calder on-Zygmund singular integral operator in R n . In this paper we study the problem of controlling the maximal singular integral T ? f by the singular integral Tf. The most basic form of control one may consider is the estimate of the L 2 (R n ) norm of T ? f by a constant times the L 2 (R n ) norm of Tf. We show that if T is an even higher order Riesz transform, then one has the stronger pointwise inequality T ? f(x) C M(Tf)(x), where C is a constant and M is the Hardy-Littlewood maximal operator. We prove that the L 2 estimate of T ? by T is equivalent, for even smooth homogeneous Calder on-Zygmund operators, to the pointwise inequality between T ? and M(T ). Our main result characterizes the L 2 and pointwise inequalities in terms of an algebraic condition expressed in terms of the kernel ( x) jxjn of T , where is an