\(L_ p\to L_ q\)-estimates for some potential-type operators with oscillating kernels. (English) Zbl 1035.47017
Summary: We obtain \(L_p\to L_p\)-estimates for the potential operator \(K_a^\alpha\) in \(\mathbb{R}^n\) with a radial kernel of the form \(a(| t|) e^{i| t|}/ | t|^{n-\alpha}\), \(0<\alpha <n\), where \(a(\infty)\neq 0\) and the characteristic \(a(r)\) is sufficiently smooth in some neighborhood of infinity and locally satisfies some general assumptions. In particular, \(a(r)\) may have power singularities. We construct some convex sets on the \((1/p,1/q)\)-plane for which the operator \(K_a^\alpha\) is bounded from \(L_p\) into \(L_q\), as well as the domains where \(K_a^\alpha\) is not bounded.
MSC:
47B38 | Linear operators on function spaces (general) |
46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
26A33 | Fractional derivatives and integrals |
31B15 | Potentials and capacities, extremal length and related notions in higher dimensions |
47G10 | Integral operators |