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Open AccessArticle

# Construction of Weights for Positive Integral Operators

by Ron Kerman
Department of Mathematics, Brock University, St. Catharines, ON L2S 3A1, Canada
Symmetry 2020, 12(6), 1004; https://doi.org/10.3390/sym12061004
Received: 15 March 2020 / Revised: 13 April 2020 / Accepted: 16 April 2020 / Published: 12 June 2020
(This article belongs to the Special Issue Integral Transformations, Operational Calculus and Their Applications)
Let $( X , M , μ )$ be a $σ$ -finite measure space and denote by $P ( X )$ the $μ$ -measurable functions $f : X → [ 0 , ∞ ]$ , $f < ∞$ $μ$ ae. Suppose $K : X × X → [ 0 , ∞ )$ is $μ × μ$ -measurable and define the mutually transposed operators T and $T ′$ on $P ( X )$ by $( T f ) ( x ) = ∫ X K ( x , y ) f ( y ) d μ ( y )$ $and$ $( T ′ g ) ( y ) = ∫ X K ( x , y ) g ( x ) d μ ( x ) , f , g ∈ P ( X ) , x , y ∈ X .$ Our interest is in inequalities involving a fixed (weight) function $w ∈ P ( X )$ and an index $p ∈ ( 1 , ∞ )$ such that: (*): $∫ X [ w ( x ) ( T f ) ( x ) ] p d μ ( x ) ≲ C ∫ X [ w ( y ) f ( y ) ] p d μ ( y ) .$ The constant $C > 1$ is to be independent of $f ∈ P ( X ) .$ We wish to construct all w for which (*) holds. Considerations concerning Schur’s Lemma ensure that every such w is within constant multiples of expressions of the form $ϕ 1 1 / p − 1 ϕ 2 1 / p ,$ where $ϕ 1 , ϕ 2 ∈ P ( X )$ satisfy $T ϕ 1 ≤ C 1 ϕ 1 and T ′ ϕ 2 ≤ C 2 ϕ 2 .$ Our fundamental result shows that the $ϕ 1$ and $ϕ 2$ above are within constant multiples of (**): $ψ 1 + ∑ j = 1 ∞ E − j T ( j ) ψ 1 and ψ 2 + ∑ j = 1 ∞ E − j T ′ ( j ) ψ 2$ respectively; here $ψ 1 , ψ 2 ∈ P ( X )$ , $E > 1$ and $T ( j ) , T ′ ( j )$ are the jth iterates of T and $T ′$ . This result is explored in the context of Poisson, Bessel and Gauss–Weierstrass means and of Hardy averaging operators. All but the Hardy averaging operators are defined through symmetric kernels $K ( x , y ) = K ( y , x )$ , so that $T ′ = T$ . This means that only the first series in (**) needs to be studied. View Full-Text
MDPI and ACS Style

Kerman, R. Construction of Weights for Positive Integral Operators. Symmetry 2020, 12, 1004.