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A weighted estimate for an intermediate operator on the cone of nonnegative functions. (Russian, English) Zbl 1043.47035

Sib. Mat. Zh. 43, No. 1, 161-173 (2002); translation in Sib. Math. J. 43, No. 1, 128-139 (2002).
Summary: Consider the following integral operator \[ K_\beta f(x)=\int_{0}^{x} K^\beta (x,t) f(t)\, dt, \quad x>0, \;0\leq\beta\leq 1, \;K\equiv K_1. \] Under some restrictions on a positive continuous function \(K(x,s)\), we obtain necessary and sufficient conditions on weight functions \(u\), \(v\), and \(\rho\) that guarantee the inequality \(\| uK_\beta f\| _q \leq C(\| \rho f\| _p + \| vKf\| _r)\) for \(f\geq 0\), \(1<p,q,r<\infty\), and \(q\geq\max\{p,r\}\).

MSC:

47G10 Integral operators
45P05 Integral operators
47B38 Linear operators on function spaces (general)
26D10 Inequalities involving derivatives and differential and integral operators
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