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.     

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.     

Strong Laws for L p -Norms of Empirical and Related Processes

We obtain lim sup and lim inf results for the L _p -norms (1 ≤ p < ∞) of empirical and quantile processes. We prove these results combining theorems for sums of Banach space valued random variables with invariance principles.     

Isoperimetric monotony of the L p -norm of the warping function of a plane simply connected domain

Let G be a simply connected domain and let u(x,G) be its warping function. We prove that L ^p -norms of functions u and u ^?1 are monotone with respect to the parameter p. This monotony also gives isoperimetric inequalities for norms that correspond to different values of the parameter p. The main result of this paper is a generalization of classical isoperimetric inequalities of St.Venant-Pólya and the Payne inequalities.     

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.     

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.     

Hájek-Rényi-type inequality and strong law of large numbers for some dependent sequences

In this paper, we get the Hjek-Rényi-type inequalities for a pairwise NQD sequence, an Lr (r > 1) mixingale and a linear process, which have the concrete coefficients. In addition, we obtain the strong law of large numbers, strong growth rate and the integrability of supremum for the above sequences, which generalize and improve Corollary 2 for Lr (r > 1) mixingale of Hansen.     

Generalized Christoffel functions for Jacobi-exponential weights

Assume that W=e ^?Q where I:=(a,b), ?∞≦a<0<b≦∞, and Q:?I→[0,∞) is continuous and increasing. Let 0<p<∞, a<t _r <t _r?1<?<t ₁<b, p _i >?1/p, i=1,2,…,r, and $U(x)=\prod_{i=1}^{r} {|x-t_{i}|}^{p_{i}}$ . We give the L _p Christoffel functions for the Jacobi-exponential weight WU. In addition, we obtain restricted range inequalities.     

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 .     

Some Inequalities for Derivatives of Trigonometric and Algebraic Polynomials

We investigate inequalities for derivatives of trigonometric and algebraic polynomials in weighted L ^P spaces with weights satisfying the Muckenhoupt A _p condition. The proofs are based on an identity of Balázs and Kilgore [1] for derivatives of trigonometric polynomials. Also an inequality of Brudnyi in terms of rth order moduli of continuity ω_r will be given. We are able to give values to the constants in the inequalities.     

Sections of Functions and Sobolev-Type Inequalities

We study functions of two variables whose sections by the lines parallel to the coordinate axis satisfy the Lipschitz condition of order 0 < α ≤ 1. We prove that if for a function f the Lip α-norms of these sections belong to the Lorentz space L ^p,1(?) (p = 1/α), then f can be modified on a set of measure zero so as to become bounded and uniformly continuous on ?². For α = 1 this gives an extension of Sobolev’s theorem on continuity of functions of the space W ₁ ^2,2 (?²). We show that the exterior L ^p,1-norm cannot be replaced by a weaker Lorentz L ^p,q -norm with q > 1.     

Clarkson and Random Clarkson Inequalities for Lr(X)

We first show how (p,p′) Clarkson inequality for a Banach space X is inherited by Lebesgue-Bochner spaces L_r(X), which extends Clarkson's procedure deriving his inequalities for L_p from their scalar versions. Fairly many previous and new results on Clarkson's inequalities, and also those on Rademacher type and cotype at the same time (by a recent result of the authors), are obtained as immediate consequences. Secondly we show that if the (p, p') Clarkson inequality holds in X, then random Clarkson inequalities hold in L_r(X) for any 1 ≤ r ≤ ∞; the converse is true if r = p'. As corollaries the original Clarkson and random Clarkson inequalities for L_p are both directly derived from the parallelogram law for scalars.     

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.     

On intersections of Sylow 2-subgroups in automorphism groups of finite simple groups of exceptional Lie type

We consider the problem of interpolation of finite sets of numerical data bounded in L _p -norms (1 ≤ p < ∞) by smooth functions that are defined in an n-dimensional Euclidean ball of radius R and vanish on the boundary of the ball. Under some constraints on the location of interpolation nodes, we obtain two-sided estimates with a correct dependence on R for the L _p -norms of the Laplace operators of the best interpolants.     

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.     

Weighted L p Bernstein-type inequalities on a quasismooth curve in the complex plane

We establish L _p , 1≦p<∞, Bernstein-type inequalities for algebraic polynomials considered on a quasismooth (in the sense of Lavrentiev) curve in the complex plane.     

Non-linear Approximations Using Sets of Finite Cardinality or Finite Pseudo-dimension

We investigate optimal non-linear approximations of multivariate periodic functions with mixed smoothness. In particular, we study optimal approximation using sets of finite cardinality (as measured by the classical entropy number), as well as sets of finite pseudo-dimension (as measured by the non-linear widths introduced by Ratsaby and Maiorov). Approximation error is measured in the L_q(T^d)-sense, where T^d is the d-dimensional torus. The functions to be approximated are in the unit ball SB^r_p, θ of the mixed smoothness Besov space or in the unit ball SW^r_p of the mixed smoothness Sobolev space. For 1<p, q<∞, 0<θ⩽∞ and r>0 satisfying some restrictions, we establish asymptotic orders of these quantities, as well as construct asymptotically optimal approximation algorithms. We particularly prove that for either r>1/p and θ⩾p or r>(1/p−1/q)₊ and θ⩾min{q, 2}, the asymptotic orders of these quantities for the Besov class SB^r_p, θ are both n^−r(log n)^{(d−1)(r+1/2−1/θ)}.     

Weighted Polynomial Inequalities with Doubling Weights on a Quasismooth Arc

We establish L _p ,1??p<?? Markov?CBernstein-type inequalities for algebraic polynomials considered on a quasismooth (in the sense of Lavrentiev) arc in the complex plane.     

Estimates of perturbed Oseen semigroups and their applications to the Navier-Stokes system in ℝ n

For perturbed Oseen semigroups in ? ⁿ , we establish their power L _p ? L _q estimates. These estimates are used to prove the existence of small global solutions to perturbed nonlinear Oseen systems and also of estimates of their L _p -norms as t → ∞.