Sorina Barza

Sharp Weighted Multidimensional Integral Inequalities for Monotone Functions

We prove sharp weighted inequalities for general integral operators acting on monotone functions of several variables. We extend previous results in one dimension, and also those in higher dimensio ...

Sharp constants related to the triangle inequality in Lorentz spaces

We study the Lorentz spaces Lp,s(R, μ) in the range 1 < p < s ≤ ∞, for which the standard functional ||f ||p,s = (∫ ∞ 0 (t1/pf∗(t))s dt t )1/s is only a quasi-norm. We find the optimal constant in the triangle inequality for this quasi-norm, which leads us to consider the following decomposition norm: ||f ||(p,s) = inf { ∑

Hardy's inequalities for monotone functions on partly ordered measure spaces

The theory of weighted inequalities for the Hardy operator, acting on monotone functions in R+, was first introduced in [2]. Extensions of these results to higher dimension have been considered only in very specific cases. In particular, in the diagonal case, only for p = 1 (see [5]). The main difficulty in this context is that the level sets of the monotone functions are not totally ordered, contrary to the one-dimensional case where one considers intervals of the form (0, a), a > 0. It is also worth to point out that, with no monotonicity restriction, the boundedness of the Hardy operator is only known in dimension n = 2 (see [15], [12], and also [3] for an extension in the case of product weights). In this work we completely characterize the weighted Hardy’s inequalities for all values of p > 0, namely, the boundedness of the operator:

Duality theorem over the cone of monotone functions and sequences in higher dimensions

Let f be a non-negative function defined on ℝ+n which is monotone in each variable separately. If 1

Carlson type inequalities

A scale of Carlson type inequalities are proved and the best constants are found. Some multidimensional versions of these inequalities are also proved and it is pointed out that also a well-known i ...

Matriceal Lebesgue spaces and Hölder inequality

We introduce a class of spaces of infinite matrices similar to the class of Lebesgue spaces Lp(T), 1≤p≤∞, and we prove matriceal versions of Holder inequality.

Approximation of infinite matrices by matricial Haar polynomials

The main goal of this paper is to extend the approximation theorem of contiuous functions by Haar polynomials (see Theorem A) to infinite matrices (see Theorem C). The extension to the matricial framework will be based on the one hand on the remark that periodic functions which belong toL∞ (T) may be one-to-one identified with Toeplitz matrices fromB(l2) (see Theorem 0) and on the other hand on some notions given in the paper. We mention for instance:ms—a unital commutative subalgebra ofl∞,C(l2) the matricial analogue of the space of all continuous periodic functionsC(T), the matricial Haar polynomials, etc.In Section 1 we present some results concerning the spacems—a concept important for this generalization, the proof of the main theorem being given in the second section.

Best constants between equivalent norms in Lorentz sequence spaces

We find the best constants in inequalities relating the standard norm, the dual norm, and the norm ∥x∥(p,s):=inf⁡{∑k∥x(k)∥p,s}, where the infimum is taken over all finite representations x=∑kx(k) in the classical Lorentz sequence spaces. A crucial point in this analysis is the concept of level sequence, which we introduce and discuss. As an application, we derive the best constant in the triangle inequality for such spaces.

Sharp constants between equivalent norms in weighted Lorentz spaces

For an increasing weight w in Bp (or equivalently in Ap), we find the best constants for the inequalities relating the standard norm in the weighted Lorentz space Λp(w) and the dual norm.

Factorizations of Weighted Hardy Inequalities

We present factorizations of weighted Lebesgue, Cesàro and Copson spaces, for weights satisfying the conditions which assure the boundedness of the Hardy’s integral operator between weighted Lebesgue spaces. Our results enhance, among other, the best known forms of weighted Hardy inequalities.