On first order interpolation inequalities with weights on the H-type group

In this paper, sufficient and necessary conditions for a class of first order interpolation inequalities with weights on the H-type group are given. By polar coordinate changes of the H-type group, the necessity is verified. A class of Hardy type inequalities is established via a representation formula for functions, Hardy-Sobolev type inequalities are obtained by interpolation and then the sufficiency is completed through discussion of parameter σ.     

On first order interpolation inequalities with weights on the Heisenberg group

In this paper, sufficient and necessary conditions for the first order interpolation inequalities with weights on the Heisenberg group are given. The necessity is discussed by polar coordinates changes of the Heisenberg group. Establishing a class of Hardy type inequalities via a new representation formula for functions and Hardy-Sobolev type inequalities by interpolation, we derive the sufficiency. Finally, sharp constants for Hardy type inequalities are determined.     

Estimates in the Hardy-Sobolev space of the annulus and stability result

The main purpose of this work is to establish some logarithmic estimates of optimal type in the Hardy-Sobolev space H ^k,∞; k ∈ ?* of an annular domain. These results are considered as a continuation of a previous study in the setting of the unit disk by L.Baratchart and M. Zerner, On the recovery of functions from pointwise boundary values in a Hardy-Sobolev class of the disk, J. Comput. Appl. Math. 46 (1993), 255–269 and by S.Chaabane and I. Feki, Optimal logarithmic estimates in Hardy-Sobolev spaces H ^k,∞, C. R., Math., Acad. Sci. Paris 347 (2009), 1001–1006. As an application, we prove a logarithmic stability result for the inverse problem of identifying a Robin parameter on a part of the boundary of an annular domain starting from its behavior on the complementary boundary part.     

Sharp constants for Moser‐Trudinger inequalities on spheres in complex space ℂn

The main results of this paper concern sharp constants for the Moser‐Trudinger inequalities on spheres in complex space ?ⁿ. We derive Moser‐Trudinger inequalities for smooth functions and holomorphic functions with different sharp constants (see Theorem 1.1). The sharp Moser‐Trudinger inequalities under consideration involve the complex tangential gradients for the functions and thus we have shown here such inequalities in the CR setting. Though there is a close connection in spirit between inequalities proven here on complex spheres and those on the Heisenberg group for functions with compact support in any finite domain proven earlier by the same authors [17], derivation of the sharp constants for Moser‐Trudinger inequalities on complex spheres are more complicated and difficult to obtain than on the Heisenberg group. Variants of Moser‐Onofri‐type inequalities are also given on complex spheres as applications of our sharp inequalities (see Theorems 1.2 and 1.3). One of the key ingredients in deriving the main theorems is a sharp representation formula for functions on the complex spheres in terms of complex tangential gradients (see Theorem 1.4). © 2004 Wiley Periodicals, Inc.     

A simple approach to Hardy inequalities

We describe a simple method of proving Hardy-type inequalities of second and higher order with weights for functions defined in ℝ ⁿ . It is shown that we can obtain such inequalities with sharp constants by applying the divergence theorem to specially chosen vector fields. Another approach to Hardy inequalities based on the application of identities of Rellich-Pokhozhaev type is also proposed. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 563–572, April, 2000.     

On Generalized Csiszár-Kullback Inequalitieys

 Classical Csiszár-Kullback inequalities bound the L ¹-distance of two probability densities in terms of their relative (convex) entropies. Here we generalize such inequalities to not necessarily normalized and possibly non-positive L ¹ functions. Also, we analyse the optimality of the derived Csiszár-Kullback type inequalities and show that they are in many important cases significantly sharper than the classical ones (in terms of the functional dependence of the L ¹ bound on the relative entropy). Moreover our construction of these bounds is rather elementary. (Received 18 February 2000; in revised form 13 June 2000)     

Some continuous functions related to corner polyhedra,II

The group problem on the unit interval is developed, with and without continuous variables. The connection with cutting planes, or valid inequalities, is reviewed. Certain desirable properties of valid inequalities, such as minimality and extremality are developed, and the connection between valid inequalities for P(I, u ₀) and P _- ⁺ (I, u ₀) is developed. A class of functions is shown to give extreme valid inequalities for P _- ⁺ (I, u ₀) and for certain subsetsU ofI. A method is used to generate such functions. These functions give faces of certain corner polyhedra. Other functions which do not immediately give extreme valid inequalities are altered to construct a class of faces for certain corner polyhedra. This class of faces grows exponentially as the size of the group grows.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates and asymptotic behavior of extremal function to Hardy-Sobolev type inequality on the H-type group*

This paper is devoted to discuss the regularity of the weak solution to a class of non-linear equations corresponding to Hardy-Sobolev type inequality on the H-type group. Combining the Serrin's idea and the Moser's iteration, L^p estimates of the weak solution are obtained, which generalize the results of Garofalo and Vassilev in [6, 14]. As an application, asymptotic behavior of the weak solution has been discussed. Finally, doubling property and unique continuation of the weak solution are given. *This material is based upon work funded by Zhejiang Provincial Natural Science Foundation of China under Grant No. Y606144.     

Exact order of Hoffman’s error bounds for elliptic quadratic inequalities derived from vector-valued Chebyshev approximation

In this paper, we introduce the exact order of Hoffman’s error bounds for approximate solutions of elliptic quadratic inequalities. Elliptic quadratic inequalities are closely related to Chebyshev approximation of vector-valued functions (including complex-valued functions). The set of Chebyshev approximations of a vector-valued function defined on a finite set is shown to be Hausdorff strongly unique of order exactly 2 ^s for some nonnegative integer s. As a consequence, the exact order of Hoffman’s error bounds for approximate solutions of elliptic quadratic inequalities is exactly 2 ^-s for some nonnegative integer s. The integer s, called the order of deficiency (which is computable), quantifies how much the Abadie constraint qualification is violated by the elliptic quadratic inequalities. Received: April 15, 1999 / Accepted: February 21, 2000?Published online July 20, 2000     

Nonlinear biharmonic equations with Hardy potential and critical parameter

Some embedding inequalities in Hardy-Sobolev spaces with weighted function α|x| are proved. The procedure is based on decomposition into spherical harmonics, where in addition various new inequalities are obtained. Next, we study the existence of nontrivial solutions of biharmonic equations with Hardy potential and critical parameter.     

Marcinkiewicz–Zygmund Inequalities and Polynomial Approximation from Scattered Data on SO(3)

We consider scattered data approximation problems on SO(3). To this end, we construct a new operator for polynomial approximation on the rotation group. This operator reproduces Wigner-D functions up to a given degree and has uniformly bounded L ^p -operator norm for all 1 ≤ p ≤ ∞. The operator provides a polynomial approximation with the same approximation degree of the best polynomial approximation. Moreover, the operator together with a Markov type inequality for Wigner-D functions enables us to derive scattered data L ^p -Marcinkiewicz–Zygmund inequalities for these functions for all 1 ≤ p ≤ ∞. As a major application of such inequalities, we consider the stability of the weighted least squares approximation problem on SO(3).     

Lagrange multipliers for regular and singular problems in the calculus of variations

A Lagrange multiplier rule is presented for a variational problem of Bolza type under hypotheses that allow certain components of the coefficient matrices involved in the functional being minimized to fail to be integrable near an endpoint of the interval on which the relevant functions are defined. The problem is also addressed when all coefficients are of classL ², but not necessarily bounded. Applications are made to ascertain properties of functions providing equality to certain singular and regular integral inequalities appearing in the literature.     

Subelliptic Cordes Estimates in the Grušin Plane

We apply subelliptic Cordes conditions and Talenti–Pucci type inequalities to prove W ^2,2 and C ^1,α estimates for p-harmonic functions in the Grušin plane for p near 2.J. J. Manfred; partially supported by NSF award DMS-0500983     

Inequalities for upper bounds of functionals on the classes w r H ω and their applications

We show that the well-known results on estimates of upper bounds of functionals on the classes W ^r H ^ω of periodic functions can be regarded as a special case of Kolmogorov-type inequalities for support functions of convex sets. This enables us to prove numerous new statements concerning the approximation of the classes W ^r H ^ω , establish the equivalence of these statements, and obtain new exact inequalities of the Bernstein-Nikol’skii type that estimate the value of the support function of the class H ^ω on the derivatives of trigonometric polynomials or polynomial splines in terms of the L ^ϱ -norms of these polynomials and splines.     

Morse theory for some lower-C 2 functions in finite dimension

Using inf-regularization methods, we prove that Morse inequalities hold for some lower-C ² functions. For this purpose, we first recall some properties of the class of lower-C ² functions and of their Moreau-Yosida approximations. Then, we establish, under some qualification conditions on the critical points, that it is possible to define a Morse index for a lower-C ² functionf. This index is preserved by the Moreau-Yosida approximation process. We prove in particular that the Moreau-Yosida approximations are twice continuolusly differentiable around such a critical point which is shown to be a strict local minimum of the restriction off and of its approximations to some affine space. In a last step, Morse inequalities are written for Moreau-Yosida approximations and with the aid of deformation retractions we prove that these inequalities also hold for some lower-C ² functions.     

On local behavior of holomorphic functions along complex submanifolds of ℂ N

In this paper we establish some general results on local behavior of holomorphic functions along complex submanifolds of ℂ ^N . As a corollary, we present multi-dimensional generalizations of an important result of Coman and Poletsky on Bernstein type inequalities on transcendental curves in ℂ².     

A Stein Conjecture for the Circle

We discuss the manner in which one might expect directional maximal functions to control the Fourier extension operator via L ² weighted inequalities. We prove a general inequality of this type for the extension operator restricted to circles in the plane.     

Ladder estimates for micropolar fluid equations and regularity of global attractor

This paper is devoted to obtain ladder inequalities for 2D micropolar fluid equations on a periodic domain Q=(0, L)². The ladder inequalities are differential inequalities that connect the evolution of L² norms of derivatives of order N with the evolution of the L² norms of derivatives of other (usually lower) order. Moreover, we find (with slight assumption on external fields) long‐time upper bounds on the L² norms of derivatives of every order, which implies that a global attractor is made up from C^∞ functions. Copyright © 2010 John Wiley & Sons, Ltd.     

Poincaré type inequalities on the discrete cube and in the CAR algebra

We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {?1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and Götze. We also get inequalities between the L ^p norms of $ \left\vert \nabla f\right\vert We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {−1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and G?tze. We also get inequalities between the L ^p norms of and moreover L ^p spaces may be replaced by more general ones. Similar results hold true, replacing functions on the cube by matrices in the *-algebra spanned by n fermions and the L ^p norm by the Schatten norm C _p .     

On the Discrete Poincaré–Friedrichs Inequalities for Nonconforming Approximations of the Sobolev Space H 1

We present a direct proof of the discrete Poincaré–Friedrichs inequalities for a class of nonconforming approximations of the Sobolev space H ¹(Ω), indicate optimal values of the constants in these inequalities, and extend the discrete Friedrichs inequality onto domains only bounded in one direction. We consider a polygonal domain Ω in two or three space dimensions and its shape-regular simplicial triangulation. The nonconforming approximations of H ¹(Ω) consist of functions from H ¹ on each element such that the mean values of their traces on interelement boundaries coincide. The key idea is to extend the proof of the discrete Poincaré–Friedrichs inequalities for piecewise constant functions used in the finite volume method. The results have applications in the analysis of nonconforming numerical methods, such as nonconforming finite element or discontinuous Galerkin methods.