Let 0 < T: LP(Y, v) -+ Lq(X, ) be a positive linear operator and let HITIP ,q denote its operator norm. In this paper a method is given to compute 1Tllp, q exactly or to bound 11Tllp q from above. As an application the exact norm 11VIlp,q of the Volterra operator Vf(x) = fo f(t)dt is computed.

Variational inequalities are being used as a mathematical programming tool in modeling a wide class of problems arising in different branches of pure and applied sciences, see [1–26]. Variational inequalities have been extended and generalized in various directions using novel and innovative techniques. A useful and important generalization is called the general variational inclusion involving ...

Let M be a semifinite von Neumann algebra equipped with a semifinite normal faithful trace τ . Let d be an injective positive measurable operator with respect to (M, τ ) such that d is also measurable. Define Lp(d) = {x ∈ L0(M) : dx+ xd ∈ Lp(M)} and ‖x‖Lp(d) = ‖dx+ xd‖p . We show that for 1 6 p0 < p1 6 ∞, 0 < θ < 1 and α0 > 0, α1 > 0 the interpolation equality (Lp0(d 0), Lp1(d α))θ = Lp(d ) hol...

In this paper, we prove the Hyers-Ulam stability in$beta$-homogeneous probabilistic modular spaces via fixed point method for the functional equation[f(x+ky)+f(x-ky)=f(x+y)+f(x-y)+frac{2(k+1)}{k}f(ky)-2(k+1)f(y)]for fixed integers $k$ with $kneq 0,pm1.$

Two uniqueness results for C0 semigroups on weighted Lp spaces over Rn generated by operators of type + r with singular drift are proven. A key ingredient in the proofs is the veri cation of some kind of \weak Kato inequality" which seems to break down exactly for those drift singularities where Lp uniqueness breaks down as well. 3 Lp uniqueness of non{symmetric di usion operators with singular...

We show that a reflexive subspace of the predual of a von Neumann algebra embeds into a noncommutative Lp space for some p > 1. This is a noncommutative version of Rosenthal’s result for commutative Lp spaces. Similarly for 1 ≤ q < 2, an infinite dimensional subspace X of a noncommutative Lq space either contains lq or embeds in Lp for some q < p < 2. The novelty in the noncommutative setting i...

We prove the following noncommutative version of Lewis’s classical result. Every n-dimensional subspace E of Lp(M) (1 < p < ∞) for a von Neumann algebra M satisfies dcb(E,RC n p ) ≤ cp · n ̨̨ ̨ 1 2 − 1 p ̨̨ ̨ for some constant cp depending only on p, where 1 p + 1 p = 1 and RC p = [Rn∩ Cn, Rn +Cn] 1 p . Moreover, there is a projection P : Lp(M) → Lp(M) onto E with ‖P‖cb ≤ cp · n ̨̨ ̨ 1 2 − 1 p ̨̨ ̨ . We fo...

Maximum a-posteriori (MAP) estimation is an important task in many applications of probabilistic graphical models. Although finding an exact solution is generally intractable, approximations based on linear programming (LP) relaxation often provide good approximate solutions. In this paper we present an algorithm for solving the LP relaxation optimization problem. In order to overcome the lack ...