Refined operator inequalities for relative operator entropies

Analysis Mathematica - We investigate the relative operator entropies in the more general settings of C*-algebras, real C*-algebras and JC-algebras. We show that all the operator inequalities on...     

Equivalence relations among inequalities for some relative operator entropies

Positivity - The relative operator entropy has properties like operator means. In addition, the relative operator entropy has entropy-like properties. In this paper, we prove a Loewner–Heinz...     

On operator inequalities of some relative operator entropies

In our recent paper, we introduced the notions of relative operator (α,β)$(\alpha ,\beta )$-entropy and Tsallis relative operator (α,β)$(\alpha ,\beta )$-entropy as a parameter extensions of relative operator entropy and Tsallis relative operator entropy. In this paper, we give upper and lower bounds of these new notions according to operator (α,β)$(\alpha ,\beta )$-geometric mean introduced in Nikoufar et al. (2013) [14].     

Hardy inequalities for weighted Dirac operator

An inequality of Hardy type is established for quadratic forms involving Dirac operator and a weight r ^−b for functions in \mathbbRⁿ{\mathbb{R}^n}. The exact Hardy constant c _b  = c _b (n) is found and generalized minimizers are given. The constant c _b vanishes on a countable set of b, which extends the known case n = 2, b = 0 which corresponds to the trivial Hardy inequality in \mathbbR²{\mathbb{R}^2}. Analogous inequalities are proved in the case c _b  = 0 under constraints and, with error terms, for a bounded domain.     

On weighted inequalities for singular integrals

In this note we consider singular integrals associated to Calderón-Zygmund kernels. We prove that if the kernel is supported in then the one-sided condition, , is a sufficient condition for the singular integral to be bounded in , , or from into weak- if . This one-sided condition becomes also necessary when we require the uniform boundedness of the singular integrals associated to the dilations of a kernel which is not identically zero in . The two-sided version of this result is also obtained: Muckenhoupts condition is necessary for the uniform boundedness of the singular integrals associated to the dilations of a general Calderón-Zygmund kernel which is not the function zero either in or in .

     

Multidimensional weighted Opial inequalities

In this article we produce Opial-type weighted multidimensional inequalities over balls and arbitrary smooth bounded domains. The inequalities are sharp. The functions under consideration vanish on the boundary.     

On weighted Hermite-Hadamard inequalities

In this paper, we give a weighted form of the Hermite-Hadamard inequalities. Some applications of them are also derived. The results presented here would provide extensions of those given in earlier works. Finally we pose two interesting problems.     

On weighted inequalities for martingale transform operators

For 1 < p < ∞, the almost surely finiteness of is a necessary and sufficient condition in order to have almost surely convergence of the sequences {E(f|?_n)} with f ∈ L^p(v dP). This condition is also equivalent to have weighted inequalities from L^p(v dP) into L^p(u dP) for some weight u for Doob's maximal function, square function and generalized Burkholder martingale transforms. Similarly, E(u|?₁) < ∞ turns out to be necessary and sufficient for the above weighted inequalities to hold for some v.     

On weighted strong type inequalities for the generalized weighted mean operator

The generalized weighted mean operator ${\mathbf{M}^{g}_{w}}$ is given by $$[\mathbf{M}^{g}_{w}f](x) = g^{-1} \left( \frac{1}{W(x)} \int \limits_{0}^{x}w(t)g(f(t))\,{\rm d}t \right),$$ with $$W(x) = \int \limits_{0}^{x} w(s) {\rm d}s, \quad {\rm for} \, x \in (0, + \infty),$$ where w is a positive measurable function on (0, + ∞) and g is a real continuous strictly monotone function with its inverse g ^?1. We give some sufficient conditions on weights u, v on (0, + ∞) for which there exists a positive constant C such that the weighted strong type (p, q) inequality $$\left( \int \limits_{0}^{\infty} u(x) \Bigl( [\mathbf{M}^{g}_{w}f](x) \Bigr)^{q} {\rm d}x \right)^{1 \over q} \leq C \left( \int \limits_{0}^{\infty}v(x)f(x)^{p} {\rm d}x \right)^{1 \over p}$$ holds for every measurable non-negative function f, where the positive reals p,q satisfy certain restrictions.     

On weighted iterated Hardy-type inequalities

In this paper the inequality

$$\begin{aligned} \bigg ( \int _0^{\infty } \bigg ( \int _x^{\infty } \bigg ( \int _t^{\infty } h \bigg )^q w(t)\,dt \bigg )^{r / q} u(x)\,{ ds} \bigg )^{1/r}\le C \,\int _0^{\infty } h v, \quad h \in {\mathfrak {M}}^+(0,\infty ) \end{aligned}$$

is characterized. Here \(0< q ,\, r < \infty \) and \(u,\,v,\,w\) are weight functions on \((0,\infty )\).

     

A characterization of some weighted inequalities for the vector-valued weighted maximal function

We give in this paper a necessary and sufficient condition of weighted weak and strong type norm inequalities for the vector-valued weighted maximal function.     

Harnack inequalities for fuchsian type weighted elliptic equations

Harnack type inequalities for nonnegative (weak) solutions of degenerate elliptic equations, in divergence form, are established. The asymptotic behavior of solutions of Fuchsian type weighted elliptic operators is also investigated.     

Localisation and weighted inequalities for spherical Fourier means

In this work, we establish certain equivalences between the localisation properties with respect to spherical Fourier means of the support of a given Borel measure and the L ²-rate of decay of the Fourier extension operator associated to it. This, in turn, is intimately connected with the property that the X-ray transform of the measure be uniformly bounded. Geometric properties of sets supporting such a measure are studied. The research of this paper has been partially supported by the EU Comission via the network HARP, and by MEC Grant MTM2004-00678. The first author was partially supported by a Leverhulme Study Abroad Fellowship.     

Sharp weighted inequalities for the vector-valued maximal function

We prove in this paper some sharp weighted inequalities for the vector-valued maximal function of Fefferman and Stein defined by

where is the Hardy-Littlewood maximal function. As a consequence we derive the main result establishing that in the range there exists a constant such that

Furthermore the result is sharp since cannot be replaced by . We also show the following endpoint estimate

where is a constant independent of .

     

On some sharp weighted norm inequalities

Given a weight ω, we consider the space which coincides with when ω∈A_p. Sharp weighted norm inequalities on for the Calderón-Zygmund and Littlewood-Paley operators are obtained in terms of the A_p characteristic of ω for any 1<p<∞.     

Nonlinear equations and weighted norm inequalities

We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for the solvability of the Dirichlet problem

on a regular domain in in the ``superlinear case' . The coefficients are arbitrary positive measurable functions (or measures) on . We also consider more general nonlinear differential and integral equations, and study the spaces of coefficients and solutions naturally associated with these problems, as well as the corresponding capacities.

Our characterizations of the existence of positive solutions take into account the interplay between , , and the corresponding Green's kernel. They are not only sufficient, but also necessary, and are established without any a priori regularity assumptions on and ; we also obtain sharp two-sided estimates of solutions up to the boundary. Some of our results are new even if and is a ball or half-space.

The corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so-called -inequality by an elementary ``integration by parts' argument. This also gives a new unified proof for some classical inequalities including the Carleson measure theorem for Poisson integrals and trace inequalities for Riesz potentials and Green potentials.

     

Some weighted Gagliardo-Nirenberg inequalities and applications

We obtain conditions on the measure so that the -norm of a function is controlled by the -norms of the function and its gradient. Applications to eigenvalues of the Schrödinger operator and to other inequalites are also given.

     

On weighted weak type inequalities for modified Hardy operators

We characterize the pairs of weights for which the modified Hardy operator applies into weak- where is a monotone function and .