Refinements of the Converse Hölder and Minkowski Inequalities

We give a refinement of the converse Hölder inequality for functionals using an interpolation result for Jensen’s inequality. Additionally, we obtain similar improvements of the converse of the Beckenbach inequality. We consider the converse Minkowski inequality for functionals and of its continuous form and give refinements of it. Application on integral mixed means is given.

An Interpolation Theorem for Quasimartingales in Noncommutative Symmetric Spaces

Let E be a separable symmetric space on 0 , ∞ and E M the corresponding noncommutative space. In this paper, we introduce a kind of quasimartingale spaces which is like but bigger than E M and obtain the following interpolation result: let E ^ M be the space of all bounded E M -quasimartingales and 1 < p < p E < q E < q < ∞ . Then, there exists a symmetric space F on 0 , ∞ with nontrivial Boyd indices such that E ^ M = L ^ p M , L ^ q M F , K .

A Novel Adaptive Directional Interpolation Algorithm for Digital Video Resolution Enhancement

In this paper, a novel digital video resolution enhancement algorithm based on adaptive directional interpolation is proposed, where the directionality of the edge structure and the nonlocal self-similarity prior within the current frame as well as its adjacent frames are both considered. First, we establish the regularization equation that conforms to the prior model of a video frame and then take the classic bicubic interpolation result as the initial estimation to iteratively solve the restoration equation, in which the edge structures and contours in low resolution (LR) input are reconstructed to estimate and refine the desired high resolution (HR) output. Experimental results show that the proposed algorithm can effectively enhance the clarity of a video frame, with satisfying subjective visual quality and PSNR value.

On the boundary conditions in estimating ∇ω by div ω and curl ω

In this paper, we study under what boundary conditions the inequality$${\rm \Vert }\nabla \omega {\rm \Vert }_{L^2(\Omega )}^2 \les C({\rm \Vert }{\rm curl}\omega {\rm \Vert }_{L^2(\Omega )}^2 + {\rm \Vert }{\rm div}\omega {\rm \Vert }_{L^2(\Omega )}^2 + {\rm \Vert }\omega {\rm \Vert }_{L^2(\Omega )}^2 )$$holds true. It is known that such an estimate holds if either the tangential or normal component ofωvanishes on the boundary ∂Ω. We show that the vanishing tangential component condition is a special case of a more general one. In two dimensions, we give an interpolation result between these two classical boundary conditions.

An inverse problem in elastography involving Lamé systems

This paper deals with some inverse problems for the linear elasticity system with origin in elastography: we try to identify the material coefficients from some extra information on (a part of) the boundary. In our main result, we assume that the total variation of the coefficient matrix is a priori bounded. We reformulate the problem as the minimization of a function in an appropriate constraint set. We prove that this extremal problem possesses at least one solution with the help of some regularity results. Two crucial ingredients are a Meyers-like theorem that holds in the context of linear elasticity and a nonlinear interpolation result by Luc Tartar. We also perform some numerical experiments that provide satisfactory results. To this end, we apply the Augmented Lagrangian algorithm, completed with a limited-memory BFGS subalgorithm. Finally, on the basis of these experiments, we illustrate the influence of the starting guess and the errors in the data on the behavior of the iterates.

EXPLICIT INTERPOLATION BOUNDS BETWEEN HARDY SPACE AND

Every bounded linear operator that maps ${H}^{1} $ to ${L}^{1} $ and ${L}^{2} $ to ${L}^{2} $ is bounded from ${L}^{p} $ to ${L}^{p} $ for each $p\in (1, 2)$, by a famous interpolation result of Fefferman and Stein. We prove ${L}^{p} $-norm bounds that grow like $O(1/ (p- 1))$ as $p\downarrow 1$. This growth rate is optimal, and improves significantly on the previously known exponential bound $O({2}^{1/ (p- 1)} )$. For $p\in (2, \infty )$, we prove explicit ${L}^{p} $ estimates on each bounded linear operator mapping ${L}^{\infty } $ to bounded mean oscillation ($\mathit{BMO}$) and ${L}^{2} $ to ${L}^{2} $. This $\mathit{BMO}$ interpolation result implies the ${H}^{1} $ result above, by duality. In addition, we obtain stronger results by working with dyadic ${H}^{1} $ and dyadic $\mathit{BMO}$. The proofs proceed by complex interpolation, after we develop an optimal dyadic ‘good lambda’ inequality for the dyadic $\sharp $-maximal operator.

Restriction of Fourier transforms to curves: An endpoint estimate with affine arclength measure

Consider the Fourier restriction operators associated to curves in ℝ d , . We prove for various classes of curves the endpoint restricted strong type estimate with respect to affine arclength measure on the curve. An essential ingredient is an interpolation result for multilinear operators with symmetries acting on sequences of vector-valued functions.

Interpolation for predefined types

We give a logic-independent semantics for predefined (data) types within the categorical abstract model theoretic framework of the theory of institutions. We develop a generic interpolation result for this semantics, which can be easily applied to various concrete situations from the theory and practice of specification and programming. Our study of interpolation is motivated by a number of important applications to computing science, especially in the area of structured specifications.