Multiplicative Inequalities Research Articles

Some Multiplicative Inequalities for Inner Products and of the Carlson Type

We prove a multiplicative inequality for inner products, which enables us to deduce improvements of inequalities of the Carlson type for complex functions and sequences, and also other known inequalities.

Multiplicative inequalities for the L 1 norm: Applications in analysis and number theory

The paper is devoted to multiplicative lower estimates for the L 1 norm and their applications in analysis and number theory. Multiplicative inequalities of the following three types are considered: martingale (for the Haar system), complex trigonometric (for exponential sums), and real trigonometric. A new method for obtaining sharp bounds for the integral norm of trigonometric and power series is proposed; this method uses the number-theoretic and combinatorial characteristics of the spectrum. Applications of the method (both in H 1 and L 1) to an important class of power density spectra, including [n α] with 1 ≤ α < ∞, are developed. A new combinatorial theorem is proved that makes it possible to estimate the arithmetic characteristics of spectra under fairly general assumptions.

A Posteriori Estimates in Local Norms

We derive a posteriori estimates for the difference between exact solutions and approximate solutions to boundary-value problems in terms of local norms. The diffusion problem, linear elasticity and generalizations to other boundary-value elliptic problems are considered. Computable estimates for the deviation from the exact solution are also obtained in terms of linear functionals. Unlike published works of other authors, the construction of such estimates is not connected with any analysis of the adjoint boundary-value problem. On the basis of multiplicative inequalities, local estimates in certain norms subject to the energy norm are derived. Bibliography: 10 titles.

Generalized Multiplicative Inequalities for Ideal Spaces

We study the problem of completely describing the domains that enjoy the generalized multiplicative inequalities of the embedding theorem type. We transfer the assertions for the Sobolev spaces L p 1(Ω) to the function classes that result from the replacement of L p (Ω) with an ideal space of vector-functions. We prove equivalence of the functional and geometric inequalities between the norms of indicators and the capacities of closed subsets of Ω. The most comprehensible results relate to the case of the rearrangement invariant ideal spaces.

“You've Struck a Rock”

In the Montgomery bus boycott and the South African anti-pass campaign, women's autonomous organizations initiated actions that catalyzed the mass movements for racial justice and national liberation. The activism of women and their organizations sprang from their particular positioning within systems of multiple oppressions simultaneously experiencing racial/ethnic, class, and gender oppression. In both the United States and South Africa, the particular structural location and autonomous resistance of women of African descent was an important aspect of the political opportunity structure and served as a catalyst that catapulted their respective movements for racial justice and/or national liberation to higher levels. This study employs a gendered and comparative approach to these two resistance campaigns to understand better the effects of interactive and multiplicative inequalities on movement processes and the gendered nature of political opportunities.

A remark on Kolmogorov‚s theorem

We deduce the pointwise one-dimensional multiplicative inequalities \( |f^{(k)}(x)| \le CMf(x)^{1-k/m}Mf^{(m)}(x)^{k/m} \), where Mh is the Hardy-Littlewood maximal function of h from a classical theorem of Kolmogorov. This implies that in the one-dimensional case the Gagliardo-Nirenberg inequalities in weighted L p spaces with Muckenhoupt‚s weights are consequences of Kolmogorov's theorem.

A converse to the Jensen inequality, its matrix extensions and inequalities for minors and eigenvalues

We obtain a scalar inequality, converse to the Jensen inequality. We also derive an operator converse to the Jensen inequality. As special cases, we obtain inequalities, similar to the Kantorovich one as well as some operator generalizations of them. Using some exterior algebra, we prove a generalization of the Sylvester determinant theorem. We also deduce some determinant analogs of the additive and multiplicative Kantorovich inequalities.

On Convergence and Stability of the Explicit Difference Method for Solution of Nonlinear Schrödinger Equations

The first boundary value problem for a nonlinear Schrodinger equation is investigated. The conditional convergence and stability on the initial data of the explicit three-level difference scheme of DuFort--Frankel type in C and W_2^1$ norms are proved. Grid analogues of energy conservation laws and grid multiplicative inequalities are used.

Best constants in a class of polymultiplicative inequalities for derivatives

Best constants are found in a class of multiplicative inequalities with factors that give an estimate of the -norm of a function (in or on ) in terms of the product of the -norms of fractional powers of the Laplace operator. Special attention is given to the detection of the cases of equality of the corresponding constants on the sphere and in Euclidean space.

Best Constants in Multiplicative Inequalities for Sup-Norms

The paper finds best constants in a class of multiplicative inequalities with one-dimensional spatial variables ∥f(k)∥∞ ⩽ c(k, l)∥f∥(2l−2k−1)/2l∥f(l)∥(2k+1)/2l −½⩽k⩽l−½ where f is a periodic function with zero mean value, and the norms in the right-hand side of the expression are the L2-norms.

Sharp Multidimensional Multiplicative Inequalities for Weighted L_p Spaces with Homogeneous Weights

Let Ω be an arbitrary cone in IR n with the origin as a vertex. A multidimensional multiplicative inequality for weighted Lp Ω -spaces with homogeneous weights is proved. The inequality is sharp and all cases of equality are pointed out. In particular, this inequality may be regarded as a weighted multidimensional extension of previous inequalities of Carlson, Beurling and Levin.

Stability and convergence of Dufort-Frankel-type difference schemes for a nonlinear Schrödinger-type equation

We consider a first boundary problem for the nonlinear Schrodinger equation $$\frac{{\partial u}}{{\partial t}} = ia\frac{{\partial ^2 u}}{{\partial x^2 }} + f(u,u*)u.$$ The convergence of a three-layer explicit difference scheme in theC andW12 norms is proved. The stability of the scheme with respect to the initial data in the same norms is proved. To justify the convergence and stability we use grid analogues of the energy-preservation laws and grid multiplicative inequalities. The relation 2|a|τ/h2≤ν<1 is assumed for the grid stepsizes.

Pointwise multiplicative inequalities and Nirenberg type estimates in weighted Sobolev spaces

Pointwise multiplicative inequalities and Nirenberg type estimates in weighted Sobolev spaces

Multiplicative inequalities in domains with noncompact boundary

Exact embedding theorems of the multiplicative type are established for functions of Sobolev spaces defined in a domain Ω ⊂Rn,n⩾2, whose boundary is not compact. The main condition on the domain is of the isoperimetric type.

Nonlinear multiplicative inequalities and estimates of gradients of solutions of elliptic equations

Nonlinear multiplicative inequalities and estimates of gradients of solutions of elliptic equations

REMARKS ON ESTIMATING THE LEBESGUE FUNCTIONS OF AN ORTHONORMAL SYSTEM

In this paper we clarify the relationship between lower bounds for Lebesgue functions and multiplicative inequalities of Gagliardo-Nirenberg type. We give simple proofs of some theorems on Lebesgue functions. Bibliography: 10 titles.

A weight space invariant with respect to a singular linear operator

For the singular operator $$Su = \int_a^b {\frac{{K(x, s) u (s)}}{{s - x}}} ds$$ invariant weight spacesλαβ,p(u(x)∈λαβ,p if 10,u (x) ρ (x)∈ Hβ0, 20.‖u‖Lp(ρ0) 1H0βis a Holder space. Multiplicative inequalities of the type of Kh. Sh. Mukhtarov are also obtained.

Some inequalities of the type encountered in imbedding theorems

Multiplicative inequalities are established for functions of one variable in subspaces of the space of continuous functions, and these inequalities are applied to linear singular operators.

Inequalities for the norms of derivatives in the spaces Lp,?

We introduce and study spaces of functions on [0, ∞) which generalize the spaces of Stepanov. We derive multiplicative inequalities for the norms of derivatives of functions in these spaces.

Notes on Matrix Theory-XV Multiplicative Inequalities Obtained from Additive Inequalities

Abstract : In this paper, it is shown how multiplicative inequalities for the characteristic values of positive definite matrices can be obtained from additive inequalities. (Author)