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Hardy-Steklov operators and the duality principle in weighted first-order Sobolev spaces on the real axis. (English. Russian original) Zbl 1421.42013

Math. Notes 105, No. 1, 91-103 (2019); translation from Mat. Zametki 105, No. 1, 108-122 (2019).
The authors consider the Hardy-Steklov operator \[ \mathcal Hf(x):=w(x)\int_{\phi(x)}^{\psi(x)}f(y)v(y)\,dy,\ x\in I:=(a,b), \] where \( -\infty\leq c\leq\phi(x)\leq\psi(x)\leq d\leq\infty,\;\phi(a)=\psi(a)=c,\;\phi(b)=\psi(b)=d\), \(\phi,\psi\) are continuous and strictly increasing on \(I\), the weight functions \(w\) and \(v\) belong to \(L^q_{\mathrm{loc}}(a, b)\) and \(L^{p'}_{\mathrm{loc}}(c, d)\), respectively.
Necessary and sufficient conditions for the boundedness of an operator \(\mathcal H\) from \(L^p(c, d)\) to \(L^q(a, b)\) (\(1< p \leq q < \infty\)) are obtained.
This result is the main technical tool in the investigation of the duality principle in weighted Sobolev spaces \(W^1_{p,s}(I)\). This principle consists in proving two-side estimates for \[ J_X(g):=\sup\limits_{0\neq f\in X}\frac{\left|\int_I f(x)g(x) dx\right|}{\|f\|_{W^1_{p,s}(I)}}, \ \mathbf J_X(g):=\sup\limits_{0 \neq f\in X}\frac{\left(\int_I|f(x)g(x)|dx\right)}{\|f\|_{W^1_{p,s}(I)}}, \] where \(X\subset W^1_{p,s}(I)\).
The duality principle allows to reduce the problem of the boundedness of linear operators on Sobolev spaces to the problem of the boundedness of dual operators on associated spaces.

MSC:

42B35 Function spaces arising in harmonic analysis
47G10 Integral operators
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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