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On bilinear Hardy-Steklov operators. (English. Russian original) Zbl 1480.47067

Dokl. Math. 98, No. 3, 634-637 (2018); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 483, No. 6, 596-599 (2018).
The Hardy-Steklov operator on \(L_{p}\)-space on \({\mathbb R}_{+}=(0,\infty)\) is defined by \((Hf)(x)=\int_{a(x)}^{b(x)}f(y)v(y)\,dy\), \(x > 0\), where \(a\) and \(b\) are smooth strictly increasing functions on \({\mathbb R}_{+}\) such that \(0 < a(x) < b(x)\) for all \(x\in {\mathbb R}_{+}\), \(v\) is a measurable nonnegative function on \({\mathbb R}_{+}\). The authors analyse the inequality \(\left( \int_{0}^{\infty} (H_{1}f\cdot H_{2}g)^{q}w^{q}\right)^{1/q}\le C \Vert f \Vert_{p_{1}}\Vert g \Vert_{p_{2}}\), where \(H_{1}\) and \(H_{2}\) are operators of the type mentioned above, \(w\) is a measurable nonnegative function on \({\mathbb R}_{+}\), \(0 < q < \infty\), \(p_{1}\), \(p_{2} > 1\), \(f\in L_{p_{1}}\), \(g\in L_{p_{2}}\) are arbitrary nonnegative functions, and \(C\) is the best possible constant. Under several assumptions, they obtain sharp estimates of this constant.

MSC:

47G10 Integral operators
47A07 Forms (bilinear, sesquilinear, multilinear)
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References:

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