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.     

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.     

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.     

Lieb-Thirring inequality for L p norms

In this paper, we obtain the Lieb-Thirring inequality for L _p -norms. The proof uses only the standard apparatus of the theory of orthogonal series.     

Estimation of the L p -norms of stress functions for finitely connected plane domains

Let u(x, G) be the classical stress function of a finitely connected plane domain G. The isoperimetric properties of the L ^p -norms of u(x, G) are studied. Payne’s inequality for simply connected domains is generalized to finitely connected domains. It is proved that the L ^p -norms of the functions u(x, G) and u ^?1 (x, G) strictly decrease with respect to the parameter p, and a sharp bound for the rate of decrease of the L ^p -norms of these functions in terms of the corresponding L ^p -norms of the stress function for an annulus is obtained. A new integral inequality for the L ^p -norms of u(x, G), which is an analog of the inequality obtained by F. G. Avkhadiev and the author for the L ^p -norm of conformal radii, is proved.     

Isoperimetric properties of Euclidean boundary moments of a simply connected domain

We consider integral functionals of a simply connected domain which depend on the distance to the domain boundary. We prove an isoperimetric inequality generalizing theorems derived by the Schwarz symmetrization method. For L ^p -norms of the distance function we prove an analog of the Payne inequality for the torsional rigidity of the domain. In compare with the Payne inequality we find new extremal domains different from a disk.     

On the connection between certain inequalities of the Kolmogorov type for periodic and nonperiodic functions

We obtain nonperiodic analogs of the known inequalities that estimateL _p -norms of intermediate derivatives of a periodic function in terms of itsL _∞-norms and higher derivative. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 2, pp. 147–157, February, 1999.     

Sharp quadrature formulas and inequalities between various metrics for rational functions

We obtain the sharp quadrature formulas for integrals of complex rational functions over circles, segments of the real axis, and the real axis itself. Among them there are formulas for calculating the L₂-norms of rational functions. Using the quadrature formulas for rational functions, in particular, for simple partial fractions and polynomials, we derive some sharp inequalities between various metrics (Nikol’ski?-type inequalities).     

Integral properties of the classical warping function of a simply connected domain

Let u(z,G) be the classical warping function of a simply connected domain G. We prove that the L ^p -norms of the warping function with different exponents are related by a sharp isoperimetric inequality, including the functional u(G) = sup _x∈G u(x, G). A particular case of our result is the classical Payne inequality for the torsional rigidity of a domain. On the basis of the warping function, we construct a new physical functional possessing the isoperimetric monotonicity property. For a class of integrals depending on the warping function, we also obtain a priori estimates in terms of the L ^p -norms of the warping function as well as the functional u(G). In the proof, we use the estimation technique on level lines proposed by Payne.     

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.     

The L p -curvature images of convex bodies and L p -projection bodies

Associated with the L _p -curvature image defined by Lutwak, some inequalities for extended mixed p-affine surface areas of convex bodies and the support functions of L _p -projection bodies are established. As a natural extension of a result due to Lutwak, an L _p -type affine isoperimetric inequality, whose special cases are L _p -Busemann-Petty centroid inequality and L _p -affine projection inequality, respectively, is established. Some L _p -mixed volume inequalities involving L _p -projection bodies are also established.     

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.     

Asymptotics for Global Measures of Accuracy of Splines

We obtain central limit theorems for the stochastic parts of L_p-norms of smoothed cubic spline estimators. The proofs are based on the observation that the variance term of the cubic spline is approximately of a form corresponding to a kernel estimator.     

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.     

The Petty Projection Inequality for Lp-Mixed Projection Bodies

Recently, Lutwak, Yang and Zhang posed the notion of Lp-projection body and established the Lp-analog of the Petty projection inequality. In this paper, the notion of Lp-mixed projection body is introduced--the Lp-projection body being a special case. The Petty projection inequality, as well as Lutwak＇s quermassintegrals （Lp-mixed quermassintegrals） extension of the Petty projection inequality, is established for Lp-mixed projection body.     

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.     

BMO- andL p -conditions for power series and Dirichlet series with positive coefficients

It is known that theL _p -norms of the sums of power series can be estimated from below and above by means of their coefficients, provided these coefficients are nonnegative. In the paper we prove analogous estimates for theL _p -norms of the sums of Dirichlet series. Our main result gives exact lower and upper estimates for the BMO-norm of the sums of power series and Dirichlet series, respectively, by means of their coefficients.     

The (p,q)-mixed geominimal surface areas1

Abstract

The (p, q)-mixed geominimal surface areas are introduced. A special case of the new concept is the L_p geominimal surface area introduced by Lutwak. Related inequalities, such as a?ne isoperimetric inequality, monotonous inequality, cyclic inequality, and Brunn-Minkowski inequality, are established. These new inequalities strengthen some well-known inequalities related to the L_p geominimal surface area.     

Error bounds for multistep methods revisited

For linear multistep methods with constant stepsize we consider error bounds in terms of weightedL ₂-norms ofh ^px^(p) rather than ofh ^px^(p+1). The bounds apply to stiff systemsx'=Ax+f(t,x) where the spectrum ofA lies in a sector andf is of moderate size.     

On a problem of littlewood

The theorem on the tending to zero of coefficients of a trigonometric series is proved when theL ¹-norms of partial sums of this series are bounded. It is shown that the analog of Helson's theorem does not hold for orthogonal series with respect to the bounded orthonormal system. Two facts are given that are similar to Weis' theorem on the existence of a trigonometric series which is not a Fourier series and whoseL ¹-norms of partial sums are bounded.