Some Estimates of Integrals with a Composition Operator

Some Estimates of Integrals with a Composition Operator

Global Estimates for Singular Integrals of the Composition of the Maximal Operator and the Green's Operator

Let Ω be a bounded, convex domain and B a ball in R, n ≥ 2. We use σB to denote the ball with the same center as B and with diam σB σ diam B , σ > 0. We do not distinguish the balls from cubes in this paper. We use |E| to denote the n-dimensional Lebesgue measure of the set E ⊆ R. We say that w is a weight if w ∈ Lloc R and w > 0, a.e. Differential forms are extensions of functions in R. For ex...

Operator Stieltjes Integrals with Respect to a Spectral Measure and Solutions of Some Operator Equations

We introduce the notion of Stieltjes integral with respect to the spectral measure corresponding to a normal operator. Sufficient conditions for the existence of this integral are given, and estimates for its norm are established. The results are applied to operator Sylvester and Riccati equations. Assuming that the spectrum of a closed densely defined operator A does not have common points wit...

Operator Integrals with Respect to a Spectral Measure and Solutions to Some Operator Equations

We introduce the concept of Stieltjes integral of an operator-valued function with respect to the spectral measure associated with a normal operator. We give sufficient conditions for the existence of this integral and find bounds on its norm. The results obtained are applied to the Sylvester and Riccati operator equations. Assuming that the entry C is a normal operator, the spectrum of the ent...

Best Constants in Some Operator Smoothness Estimates

We provide a short proof of the inequality (cf. Ben-Artzi and Klainerman [Regularity and decay of evolution equations, preprint] and Kato and Yajima [Rev. I m ~~(1+~~)~ " ~(1-4) " ~e " ~~~/*dt~C~~~~~~-cc with explicit (essentially exact) values for C. Recently Ben-Artzi and Klainerman [l] and Kato and Yajima [S] have focused interest on the estimate I I, 11(1 +x2)) " 2 (1-4)1'4ei'dUJ12dt<C IIU1...