Reduction method for Wiener-Hopf integral operators with piecewise continuous symbols in \(L^ p\) spaces. (English. Russian original) Zbl 0574.45007
Funct. Anal. Appl. 18, 132-133 (1984); translation from Funkts. Anal. Prilozh. 18, No. 2, 55-56 (1984).
Let W(a) be Wiener-Hopf integral operator with piecewise continuous \(a\in L^{\infty}(R)\). The family of projections \(\{P_{\tau}\}_{\tau >0}\) is defined by \((P_{\tau}\psi)(x)=\psi (x)\) for \(0<x<\tau\) and 0 for \(x>\tau\). We will say that we are applying the reduction method to W(a) in \(L^ p(R_+)\) \((1<p<\infty)\) if for \(\tau \geq \tau_ 0\) the operators \(W_{\tau}(a)=P_{\tau}W(a)P_{\tau}| Im P_{\tau}\) are invertible and \(\sup \{\| W^{-1}_{\tau}(a)| Im P_{\tau}\|:\) \(\tau \geq \tau_ 0\}<\infty\) (we write \(W(a)\in \Pi_ P\{P_{\tau}\})\). Theorem 1. If W(a) is bounded in \(L^ p(R_+)\), \(W(a)\in \Pi_ p\{P_{\tau}\}\), then \(W(a)\in \Pi_ r\{P_{\tau}\}\) for all \(r\in [p,q]\), \(1/p+1/q=1\), and, consequently, W(a) is invertible in \(L^ r(R_+)\), \(r\in [p,q]\). Theorem 2. Let W(a) be bounded in \(L^ s(R_+)\) for all s in some neighborhood of p. If W(a) is invertible in \(L^ r(R_+)\) for all \(r\in [p,q]\), \(1/p+1/q=1\), then \(W(a)\in \Pi_ p\{P_{\tau}\}\).
Reviewer: A.Bukhvalov
MSC:
45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
45P05 | Integral operators |
47Gxx | Integral, integro-differential, and pseudodifferential operators |
47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
Keywords:
piecewise continuous symbol; invertible operator; Wiener-Hopf integral operator; reduction methodReferences:
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