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Open AccessArticle

Construction of Weights for Positive Integral Operators

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
Keywords: weights; positive integral operators; convolution operators weights; positive integral operators; convolution operators
MDPI and ACS Style

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

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