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Boundedness properties of fractional integral operators associated to non-doubling measures

José García-Cuerva, A. Eduardo Gatto (2004)

Studia Mathematica

The main purpose of this paper is to investigate the behavior of fractional integral operators associated to a measure on a metric space satisfying just a mild growth condition, namely that the measure of each ball is controlled by a fixed power of its radius. This allows, in particular, non-doubling measures. It turns out that this condition is enough to build up a theory that contains the classical results based upon the Lebesgue measure on Euclidean space and their known extensions for doubling...

Boundedness properties of resolvents and semigroups of operators

J. van Casteren (1997)

Banach Center Publications

Let T: H → H be an operator in the complex Hilbert space H. Suppose that T is square bounded in average in the sense that there exists a constant M(T) with the property that, for all natural numbers n and for all x ∈ H, the inequality 1 / ( n + 1 ) ∑ j = 0 n ∥ T j x ∥ 2 ≤ M ( T ) 2 ∥ x ∥ 2 is satisfied. Also suppose that the adjoint T* of the operator T is square bounded in average with constant M(T*). Then the operator T is power bounded in the sense that s u p ∥ T i n ∥ : n ∈ ℕ is finite. In fact the following inequality is valid for all n ∈ ℕ: ∥Tn∥ ≤ e M(T)M(T*). Suppose...

Bounds for Fractional Powers of Operators in a Hilbert Space and Constants in Moment Inequalities

I. Gil’, Michael (2009)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 47A56, 47A57,47A63We derive bounds for the norms of the fractional powers of operators with compact Hermitian components, and operators having compact inverses in a separable Hilbert space. Moreover, for these operators, as well as for dissipative operators, the constants in the moment inequalities are established.* This research was supported by the Kamea Fund of Israel.

Bounds for the spectral radius of positive operators

Roman Drnovšek (2000)

Commentationes Mathematicae Universitatis Carolinae

Let f be a non-zero positive vector of a Banach lattice L , and let T be a positive linear operator on L with the spectral radius r ( T ) . We find some groups of assumptions on L , T and f under which the inequalities sup { c ≥ 0 : T f ≥ c f } ≤ r ( T ) ≤ inf { c ≥ 0 : T f ≤ c f } hold. An application of our results gives simple upper and lower bounds for the spectral radius of a product of positive operators in terms of positive eigenvectors corresponding to the spectral radii of given operators. We thus extend the matrix result obtained by Johnson and Bru which...

Bounds of Riesz Transforms on L p Spaces for Second Order Elliptic Operators

Zhongwei Shen (2005)

Annales de l’institut Fourier

Let ℒ = -div ( A ( x ) ∇ ) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on ℝ n or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p > 2 , a necessary and sufficient condition is obtained for the boundedness of the Riesz transform ∇ ( ℒ ) - 1 / 2 on the L p space. As an application, for 1 < p < 3 + ϵ , we establish the L p boundedness of Riesz transforms on Lipschitz domains for operators with V M O coefficients. The range of p is sharp. The closely related boundedness of ...

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