Estimates for the norms of fractional derivatives in terms of integral moduli of continuity and their applications

For functions defined on the entire real axis or a semiaxis, we obtain Kolmogorov-type inequalities that estimate the L _p -norms (1 ≤ p < ∞) of fractional derivatives in terms of the L _p -norms of functions (or the L _p -norms of their truncated derivatives) and their L _p -moduli of continuity and establish their sharpness for p = 1: Applications of the obtained inequalities are given.     

Bernstein-type inequalities for splines defined on the real axis

We obtain exact Bernstein-type inequalities for splines s ? S_m,h?L₂( \mathbbR ) s \in {S_{m,h}}\bigcap {{L_2}\left( \mathbb{R} \right)} , as well as the exact inequalities estimating, for splines s ∈ S _{m, h} , h > 0; the L _p -norms of the Fourier transforms of their kth derivative in terms of the L _p -norms of the Fourier transforms of the splines themselves.     

Integral bounds for simple partial fractions

For p ≥ 2 we obtain bounds for L _p -norms of the Fourier transform of real parts of simple partial fractions. For even p our estimate is sharp. We also prove a new inequality for L _p -norms of simple partial fractions which in some cases is stronger than the corresponding inequality obtained by V. Yu. Protasov.     

Refinement of a Hardy–Littlewood–Pólya-type inequality for powers of self-adjoint operators in a Hilbert space

The well-known Taikov’s refined versions of the Hardy – Littlewood – Pólya inequality for the L ₂ -norms of intermediate derivatives of a function defined on the real axis are generalized to the case of powers of self-adjoint operators in a Hilbert space.     

Estimates for the Second Order Derivatives of Eigenvectors in Thin Anisotropic Plates with Variable Thickness

For the second order derivatives of eigenvectors in a thin anisotropic heterogeneous plate Ω_h, we derive estimates of their weighted L₂-norms with majorants whose dependence on the plate thickness h and on the eigenvalue number is expressed explicitly. These estimates maintain the asymptotic sharpness throughout the entire spectrum, whereas inside its low-frequency band the majorants remain bounded as h → +0. The latter is a rather unexpected fact, because for the first eigenfunction u¹ of a similar boundary-value problem for a scalar second order differential operator with variable coefficients, the norm ‖∇ _x ² u⁰; L₂(Ω_h)‖ is of order h⁻¹ and grows as h tends to zero. Bibliography: 35 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 161–180.     

Hessian-free metric-based mesh adaptation via geometry of interpolation error

The article presents analysis of a new methodology for generating meshes minimizing L ^p -norms of the interpolation error or its gradient, p > 0. The key element of the methodology is the construction of a metric from node-based and edge-based values of a given function. For a mesh with N _h triangles, we demonstrate numerically that L ^∞-norm of the interpolation error is proportional to N _h ⁻¹ and L ^∞-norm of the gradient of the interpolation error is proportional to N _h ^−1/2. The methodology can be applied to adaptive solution of PDEs provided that edge-based a posteriori error estimates are available.     

Special functions associated with a certain fourth-order differential equation

We develop a theory of “special functions” associated with a certain fourth-order differential operator D_m,n\mathcal{D}_{\mu,\nu} on ℝ depending on two parameters μ,ν. For integers μ,ν≥−1 with μ+ν∈2ℕ₀, this operator extends to a self-adjoint operator on L ²(ℝ₊,x ^μ+ν+1 dx) with discrete spectrum. We find a closed formula for the generating functions of the eigenfunctions, from which we derive basic properties of the eigenfunctions such as orthogonality, completeness, L ²-norms, integral representations, and various recurrence relations.     

Comparison of approximation properties of generalized polynomials and splines

We establish that, for p ∈ [2, ∞), q = 1 or p = ∞, q ∈ [ 1, 2], the classes W _p^rof functions of many variables defined by restrictions on the L _p-norms of mixed derivatives of order r = (r ₁, r ₂, ..., r _m) are better approximated in the L _q-metric by periodic generalized splines than by generalized trigonometric polynomials. In these cases, the best approximations of the Sobolev classes of functions of one variable by trigonometric polynomials and by periodic splines coincide. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1011–1020, August, 1998.     

Applications of discrete maximal <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> regularity for finite element operators

In this paper, we present applications of discrete maximal L _p regularity for finite element operators. More precisely, we show error estimates of order h ² for linear and certain semilinear problems in various L _p (Ω)-norms. Discrete maximal regularity allows us to prove error estimates in a very easy and efficient way. Moreover, we also develop interpolation theory for (fractional powers of) finite element operators and extend the results on discrete maximal L _p regularity formerly proved by the author. The author was supported by the DFG-Graduiertenkolleg 853.     

Permutations of the Walsh system that preserve Lebesgue constants

A class of permutations of the Walsh-Paley system that preserve the lebesgue constants and theL ^p -norms of the Dirichlet kernel is distinguished. Thus it is proved, in particular, that Fine’s estimates and calculations of the Lebesgue constants for the Walsh—Paley system hold for the Walsh systems in the enumerations of Walsh and Kaczmarz. A third algorithm for calculating the Lebesgue constants, which is different from those obtained by Fine and which also makes it possible to calculate theL ^p -norms of the Dirichlet kernels, is proposed. It is shown that not all permutations of the Walsh system even within the blocks preserve the Lebesgue constants. The distinguished class of permutations includes theTW-systems of Schipp, which are not, in general, permutations within the blocks. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 36–48, July, 2000.     

Estimates of the solutions to the equations of motion for viscous fluid in a moving cavity with a porous damper

Estimates are obtained for theL ₂-norms of the solutions to the equations of motion for viscous incompressible fluid in a moving ellipsoidal cavity with a porous damper.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 1039–1044, July, 1993.     

Mutual estimates of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-norms and the Bellman function

In this paper, we describe the range of the L^p-norm of a function under fixed L^p-norms with two other different exponents p and under a natural multiplicative restriction of the type of the Muckenhoupt condition. Particular cases of such results are simple inequalities as the interpolation inequality between two L^p-norms as well as such nontrivial inequalities as the Gehring inequality or the reverse H?lder inequality for Mackenhoupt weights. The basic method of our paper is the search for the exact Bellman function of the corresponding extremal problem. Bibliography: 5 Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 81–138.     

A generalization of the Gagliardo inequality

Consider functions u₁, u₂,..., u_n ∈ D(ℝ^k) and assume that we are given a certain set of linear combinations of the form ∑_{i, j} a _ij ^(l) ∂_ju_i. Sufficient conditions in terms of coefficients a _ij ^(l) are indicated under which the norms are controlled in terms of the L¹-norms of these linear combinations. These conditions are mostly transparent if k = 2. The classical Gagliardo inequality corresponds to a single function u₁ = u and the collection of its partial derivatives ∂₁u,..., ∂_ku. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 345, 2007, pp. 120–139.     

Estimates for weak solutions of a nondiagonal parabolic system of two equations

Estimates for the L ^∞-norms of weak solutions are obtained for a model nondiagonal parabolic system of nonlinear differential equations with matrix of coefficients satisfying special structure conditions. A technique based on estimating the certain function of unknowns is employed to this end.     

Estimates for L p -norms of simple partial fractions

We obtain estimates for L _p -norms of simple partial fractions in terms of their L _r -norms on bounded and unbounded segments of the real axis for various p > 1 and r > 1 (S. M. Nikolskii type inequalities). We adduce examples and remarks concerning sharpness of the inequalities and area of their application.     

Two-sided estimates for root vector functions of the Dirac operator

We consider the one-dimensional Dirac operator. We derive a shift formula for its root vector functions and prove anti-a priori and two-sided estimates for various L _p -norms of these functions.     

Asymptotic properties of some goodness-of-fit tests based on the L 1-norm

Some goodness-of-fit tests based on the L ₁-norm are considered. The asymptotic distribution of each statistic under the null hypothesis is the distribution of the L ₁-norm of the standard Wiener process on [0,1]. The distribution function, the density function and a table of some percentage points of the distribution are given. A result for the asymptotic tail probability of the L ₁-norm of a Gaussian process is also obtained. The result is useful for giving the approximate Bahadur efficiency of the test statistics whose asymptotic distributions are represented as the L ₁-norms of Gaussian processes.     

Finite element methods for nonlinear elliptic systems of second order

This paper deals with the finite element displacement method for approximating isolated solutions of general quasilinear elliptic systems. Under minimal assumptions on the structure of the continuous problems it is shown that the discrete analogues also have locally unique solutions which converge with quasi-optimal rates in L₂ and L∞. The essential tools of the proof are a deformation argument and a technique using weighted L₂-norms.     

First non-zero terms for the Taylor expansion at 1 of the Conway potential function

The Conway potential function ∇ _L (t ₁,...,t _l ) of an ordered oriented link L = L ₁ ∪ L ₂ ∪ ... ∪ L _l ⊂ S ³ is considered. In general, this function is not determined by the linking numbers and the Conway potential functions of the components. However, the first two nonzero terms of the Taylor expansion at the point 1 of the function ∇ _L are determined by the linking numbers only. We give the explicit formulas for these terms using summation over trees with l vertices.     

About the Blowup of Quasimodes on Riemannian Manifolds

On any compact Riemannian manifold (M,g) of dimension n, the L ²-normalized eigenfunctions φ _λ satisfy ||f_l||_￥ ￡ Cl^\fracn-12\|\phi_{\lambda}\|_{\infty}\leq C\lambda^{\frac{n-1}{2}} where −Δφ _λ =λ ² φ _λ . The bound is sharp in the class of all (M,g) since it is obtained by zonal spherical harmonics on the standard n-sphere S ⁿ . But of course, it is not sharp for many Riemannian manifolds, e.g., flat tori ℝ ⁿ /Γ. We say that S ⁿ , but not ℝ ⁿ /Γ, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this paper is to determine the (M,g) with maximal eigenfunction growth. In an earlier work, two of us showed that such an (M,g) must have a point x where the set ℒ _x of geodesic loops at x has positive measure in S^*_xMS^{*}_{x}M. We strengthen this result here by showing that such a manifold must have a point where the set ℛ _x of recurrent directions for the geodesic flow through x satisfies |{ℛ} _x |>0. We also show that if there are no such points, L ²-normalized quasimodes have sup-norms that are o(λ ^(n−1)/2), and, in the other extreme, we show that if there is a point blow-down x at which the first return map for the flow is the identity, then there is a sequence of quasimodes with L ^∞-norms that are Ω(λ ^(n−1)/2).