Approximation by a nonlinear convolution operator in \(s\)-normed function spaces (Q2772748)

Let \(G\) be an abelian, locally convex Hausdorff group with operation \(+:G\times G\to G\), and Haar measure \(\mu\), let \(L^0(G)\) be the space of \(\mu\)-measurable a.e. finite functions on \(G\), and let \(L^1(G)\) denote the subspace of \(L^0(G)\) consisting of \(\mu\)-integrable functions. Let \(\upsilon\in L^0(G)\) be a non-negative function such that \(D(\upsilon)= (\int_G\upsilon (t)^s d\mu(t))^{1/s} <\infty\). If \(\psi:G\times [0,\infty)\to [0,\infty)\) is such that \(\psi(.,u)\) is measurable on \(G\) for all \(u>0\), \(\psi(\overline\omega,.)\) is a continuous and non-decreasing function for \(\overline\omega\in G\) with \(\psi (\overline\omega,0)=0\), \(\psi(\overline\omega,u)>0\) if \(u>0\), \(\psi(\overline \omega,u)\to\infty\) as \(u\to\infty\), then the kernel \(k\) is said to be \((\upsilon, \psi)_0\)-Lipschitz if \(|k(\overline\omega, u)|\leq\upsilon (\overline\omega) \psi(\overline \omega,u)\). NEWLINENEWLINENEWLINEThe results of this paper involve an operator \(T\) defined as a perturbation of a convolution operator by \(T(f)(x)= (\widehat K*f)(x)+ P(f)(x)\), where \((\widehat K*f)(x)= \int_G \widehat K(x-y)f(y) d\mu(y)\); \(\widehat K(x)=K(-x)\); \(P(f)(x)= \int_Gk(t,f(t+x)) d\mu(t)\). In the main theorem of the paper, estimates are derived in terms of \(s\)-homogeneous, monotone norms \(\|\cdot \|^\circ_s\), \(\|\cdot \|^\sim_s\), with constants \(C^\circ_0, h^\circ_0,C^\sim_0, h^\sim_0\). In particular, the results indicate that if \(k\) is \((\upsilon, \psi)_0\)-Lipschitz, and there is a constant \(\gamma\) such that \(\|\psi(t,|F(\cdot) |) \|^\sim_s \leq\gamma \|F(\cdot) \|^\circ_s\), then \(\|T(f) \|^\sim_s <A\|f\|^\circ_s+ B\|f\|^\circ_s +C(h^\circ_0+ h^\sim_0)\), where \(A=C^\circ D(|K|)\), \(B=\gamma C^\circ D(\upsilon)\), \(C=D(|K|)+ \gamma D(\upsilon)\).NEWLINENEWLINEFor the entire collection see [Zbl 0972.00067].