# Construction of Weights for Positive Integral Operators

Department of Mathematics, Brock University, St. Catharines, ON L2S 3A1, Canada

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 $(X,M,\mu )$ be a $\sigma $ -finite measure space and denote by $P\left(X\right)$ the $\mu $ -measurable functions $f:X\to [0,\infty ]$ , $f<\infty $ $\mu $ ae. Suppose $K:X\times X\to [0,\infty )$ is $\mu \times \mu $ -measurable and define the mutually transposed operators T and ${T}^{\prime}$ on $P\left(X\right)$ by $\left(Tf\right)\left(x\right)={\int}_{X}K(x,y)f\left(y\right)\phantom{\rule{0.166667em}{0ex}}d\mu \left(y\right)$ $\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}$ $\left({T}^{\prime}g\right)\left(y\right)={\int}_{X}K(x,y)g\left(x\right)\phantom{\rule{0.166667em}{0ex}}d\mu \left(x\right),f,g\in P\left(X\right),x,y\in X.$ Our interest is in inequalities involving a fixed (weight) function $w\in P\left(X\right)$ and an index $p\in (1,\infty )$ such that: (*): ${\int}_{X}{\left[w\left(x\right)\left(Tf\right)\left(x\right)\right]}^{p}d\mu \left(x\right)\lesssim C{\int}_{X}{\left[w\left(y\right)f\left(y\right)\right]}^{p}d\mu \left(y\right).$ The constant $C>1$ is to be independent of $f\in P\left(X\right).$ 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 ${\varphi}_{1}^{1/p-1}{\varphi}_{2}^{1/p},$ where ${\varphi}_{1},{\varphi}_{2}\in P\left(X\right)$ satisfy $T{\varphi}_{1}\le {C}_{1}{\varphi}_{1}\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}{T}^{\prime}{\varphi}_{2}\le {C}_{2}{\varphi}_{2}.$ Our fundamental result shows that the ${\varphi}_{1}$ and ${\varphi}_{2}$ above are within constant multiples of (**): ${\psi}_{1}+{\sum}_{j=1}^{\infty}{E}^{-j}{T}^{\left(j\right)}{\psi}_{1}\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}{\psi}_{2}+{\sum}_{j=1}^{\infty}{E}^{-j}{{T}^{\prime}}^{\left(j\right)}{\psi}_{2}$ respectively; here ${\psi}_{1},{\psi}_{2}\in P\left(X\right)$ , $E>1$ and ${T}^{\left(j\right)},{T}^{\prime}{}^{\left(j\right)}$ are the jth iterates of T and ${T}^{\prime}$ . 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 ${T}^{\prime}=T$ . This means that only the first series in (**) needs to be studied.
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*Keywords:*weights; positive integral operators; convolution operators

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**MDPI and ACS Style**

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

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