Two-dimensional mean inequalities in certain Banach function spaces. (English) Zbl 1152.26014

Let \(A= (0,\infty)\times (0,\infty)\) and \(X^p\) (\(p\in\mathbb{R}\), \(p\neq 0\)) be the space of all measurable functions \(f\) on \(A\) for which \(\| f\|_{X^r}=\|\,|f|^p\|^{1/p}_X< \infty\), \(X\) being the underlying Banach function space. The authors obtain necessary and sufficient conditions for the \(L^p\)-\(X^q\) boundedness of the two-dimensional Hardy operator \[ (H_2f)(x,y)= \int^x_0 \int^y_0 f(t,u)\,dt\,du; \] and the two-dimensional geometric mean operator \[ (G_2f)(x,y)= \exp\Biggl({1\over xy} \int^x_0 \int^y_0\log f(t,u)\,dt\,du\Biggr). \] Preliminary results of these types for particular \(X\) or particular \(f\) have been obtained by E. Sawyer [Stud. Math. 82, 1–16 (1985; Zbl 0585.42020)]; A. Wedestig [J. Inequal. Appl. 2005, No. 4, 387–394 (2005; Zbl 1120.26012)]; H. P. Heinig, R. Kerman and M. Krebec [Georgian Math. J. 8, No. 1, 69–86 (2001; Zbl 1009.26014)]; P. Jain and R. Hassija [Appl. Math. Lett. 16, No. 4, 459–464 (2003; Zbl 1041.26009)]; etc. The results are too complicated to be stated here. We mention the two monographs by B. Opić and A. Kufner [Hardy-type inequalities, Pitman Research Notes in Mathematics (1990; Zbl 0698.26007)], as well as by A. Kufner and L.-E. Persson [Weighted inequalities of Hardy type. Singapore: World Scientific (2003; Zbl 1065.26018)] on Hardy type inequalities.


26D10 Inequalities involving derivatives and differential and integral operators
26D15 Inequalities for sums, series and integrals
Full Text: DOI