Weighted exponential inequalities (Q2726698)

In the late 1980's, the authors established a necessary and sufficient condition on pairs of weight functions \(u,v\) in order that the two-weight inequality NEWLINE\[NEWLINE \int_{0}^{\infty}\exp\left(\frac{1}{x}\int_{0}^{x}\log f(t) dt\right)u(x) \leq C \int_{0}^{\infty}f(x)v(x) dx NEWLINE\]NEWLINE holds with some positive constant \(C\) independent of positive functions \(f\). The result was announced in [\textit{H. P. Heinig}, ``Nonlinear analysis, function spaces and applications'', Teubner-Texte Math. 119, 42-85 (1990; Zbl 0773.42008)], the full proof can be found in [\textit{H. P. Heinig, R. Kerman} and \textit{M. Krbec}, Preprint No. 79, Math. Inst. Czech. Acad. Sci., Prague, 30 pp. (1992)]. Later, a generalized version, namely NEWLINE\[NEWLINE \left(\int_{0}^{\infty}\left[\exp\left(\frac{1}{x} \int_{0}^{x}\log f(t) dt\right)\right]^qu(x) dx\right)^{1/q} \leq C \left(\int_{0}^{\infty}f^p(x)v(x) dx\right)^{1/p} NEWLINE\]NEWLINE was obtained for various ranges of parameters \(p,q\in(0,\infty]\) by other authors (see, e.g., [\textit{B. Opic} and \textit{P. Gurka}, Proc. Am. Math. Soc. 120, No. 3, 771-779 (1994; Zbl 0806.26012)] or [\textit{L. Pick} and \textit{B. Opic}, J. Math. Anal. Appl. 183, No.~3, 652-662 (1994; Zbl 0815.47041)]). Considering such inequalities as the action of the geometric mean operator \((Gf)(x)=\exp\left(\frac 1x\int_0^x \log f(t) dt\right)\) on weighted Lebesgue spaces, the authors present a number of results in a substantially more general situation. They study, in this context, operators of the form \(\exp(K(\log f))\), \(f>0\), where \(K\) is a kernel integral operator given by \((Kf)(x):=\int_{0}^{\infty}k(x,y)f(y) dy\). Here, the kernel \(k(x,y)\) is assumed to satisfy certain minimal assumptions such as positivity, homogeneity of degree \(-1\) in both of its variables, the identity \(\int_{0}^{\infty}k(1,t) dt=1\), and the convergence of the integral \(\exp\left(-\int_{0}^{\infty}k(1,t)\log t dt\right)\). A sufficient condition for boundedness of such an operator from one weighted Lebesgue space into another is given, which is later shown to be also necessary for quite a few particular instances involving cases when \(K\) is either a general mean operator or the Laplace transform. A higher-dimensional case is touched and discrete versions of the main results are formulated.