Weighted norm inequalities are proved for a rough homogeneous singular integral operator and its corresponding maximal truncated singular operator. Our results are essential improvements as well as extensions of some known results on the weighted boundedness of singular integrals.

We discuss the manner in which one might expect directional maximal functions to control the Fourier extension operator via L weighted inequalities. We prove a general inequality of this type for the extension operator restricted to circles in the plane.

Hardy–Sobolev–type inequalities associated with the operator L := x · ∇ are established, using an improvement to the Sobolev embedding theorem obtained by M. Ledoux. The analysis involves the determination of the operator semigroup {e−tL∗L}t>0.

This paper is devoted to the study of operator equilibrium problems. By using the KKM theorem, we give sufficient conditions for the existence of solutions of these problems. As a consequence, the existence of solutions for operator variational inequalities and operator minimization problems are derived. AMS subject classifications: 49J45

In this paper we introduce operator preinvex functions and establish a Hermite–Hadamard type inequality for such functions. We give an estimate of the right hand side of a Hermite–Hadamard type inequality in which some operator preinvex functions of selfadjoint operators in Hilbert spaces are involved. Also some Hermite–Hadamard type inequalities for the product of two operator preinvex functio...

In this paper, a class of generalized general mixed quasi variational inequalities is introduced and studied. We prove the existence of the solution of the auxiliary problem for the generalized general mixed quasi variational inequalities, suggest a predictor-corrector method for solving the generalized general mixed quasi variational inequalities by using the auxiliary principle technique. If ...

Quantum inequalities are lower bounds for local averages of quantum observables that have positive classical counterparts, such as the energy density or the Wick square. We establish such inequalities in general (possibly interacting) quantum field theories on Minkowski space, using nonperturbative techniques. Our main tool is a rigorous version of the operator product expansion.

The bilinear inequality is derived from the linear one with the help of an operatorvalued version of the Cauchy-Schwarz inequality. All these results, at least in their finite form, are obtained by simple and elegant methods well within the scope of a basic course on Hilbert spaces. (They can alternatively be obtained by tensor product techniques, but in the author’s view, these methods are les...

Let P (z) be a polynomial of degree n ≥ 1. In this paper we define an operator B, as following, B[P (z)] := λ 0 P (z) + λ 1 (nz 2) P ′ (z) 1! + λ 2 (nz 2) 2 P ′′ (z) 2! , where λ 0 , λ 1 and λ 2 are such that all the zeros of u(z) = λ 0 + c(n, 1)λ 1 z + c(n, 2)λ 2 z 2 lie in half plane |z| ≤ |z − n 2 | and obtain a new generalization of some well-known results.