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The boundedness of two classes of integral operators

Xin Wang, Ming-Sheng Liu (2021)

Czechoslovak Mathematical Journal

The aim of this paper is to characterize the L p - L q boundedness of two classes of integral operators from L p ( 𝒰 , d V α ) to L q ( 𝒰 , d V β ) in terms of the parameters a , b , c , p , q and α , β , where 𝒰 is the Siegel upper half-space. The results in the presented paper generalize a corresponding result given in C. Liu, Y. Liu, P. Hu, L. Zhou (2019).

The norms and singular numbers of polynomials of the classical Volterra operator in L₂(0,1)

Yuri Lyubich, Dashdondog Tsedenbayar (2010)

Studia Mathematica

The spectral problem (s²I - ϕ(V)*ϕ(V))f = 0 for an arbitrary complex polynomial ϕ of the classical Volterra operator V in L₂(0,1) is considered. An equivalent boundary value problem for a differential equation of order 2n, n = deg(ϕ), is constructed. In the case ϕ(z) = 1 + az the singular numbers are explicitly described in terms of roots of a transcendental equation, their localization and asymptotic behavior is investigated, and an explicit formula for the ||I + aV||₂ is given. For all a ≠ 0 this...

The power boundedness and resolvent conditions for functions of the classical Volterra operator

Yuri Lyubich (2010)

Studia Mathematica

Let ϕ(z) be an analytic function in a disk |z| < ρ (in particular, a polynomial) such that ϕ(0) = 1, ϕ(z)≢ 1. Let V be the operator of integration in L p ( 0 , 1 ) , 1 ≤ p ≤ ∞. Then ϕ(V) is power bounded if and only if ϕ’(0) < 0 and p = 2. In this case some explicit upper bounds are given for the norms of ϕ(V)ⁿ and subsequent differences between the powers. It is shown that ϕ(V) never satisfies the Ritt condition but the Kreiss condition is satisfied if and only if ϕ’(0) < 0, at least in the polynomial...

Time regularity and functions of the Volterra operator

Zoltán Léka (2014)

Studia Mathematica

Our aim is to prove that for any fixed 1/2 < α < 1 there exists a Hilbert space contraction T such that σ(T) = 1 and | | T n + 1 - T ⁿ | | ≍ ( n ≥ 1 ) . This answers Zemánek’s question on the time regularity property.

Transferring monotonicity in weighted norm inequalities.

Gord Sinnamon (2003)

Collectanea Mathematica

Certain weighted norm inequalities for integral operators with non-negative, monotone kernels are shown to remain valid when the weight is replaced by a monotone majorant or minorant of the original weight. A similar result holds for operators with quasi-concave kernels. To prove these results a careful investigation of the functional properties of the monotone envelopes of a non-negative function is carried-out. Applications are made to function space embeddings of the cones of monotone functions...

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