Weakly singular integral operators in weighted \(L^\infty\)-spaces (Q2494148)

Let \(L_u^\infty=\{f:\,fu\in L^\infty(-1, 1)\}\) with \(| | f| | _u=| | fu| | _\infty\), \(C_v=\{f:\, D(v)\to \mathbb C:fv\in C[-1, 1]\}\) with \(| | f| | _{C_v} =| | fv| | _\infty\) and \[ C_v^{\gamma, \delta}:=\{ f\in C_v:| | f| | _{v, \gamma, \delta}=\sup_{n\geq 0} E_n^v(f)(n+1)^{\gamma}\ln^{\delta}(n+2)<\infty\}, \] where \(E_n^v(f)\) are the weighted polynomial best approximation errors of \(f\in C_v\). Integral operators \(K\) defined by \[ (Kf)(x)=\int_{-1}^1 k(x, t)f(t)\, dt, \quad x\in(-1, 1), \] are studied. The kernel \(k(x, t)\) is assumed to satisfy \[ h(x, t)=(t-x)v(x)k(x, t)w(t)\in C([-1, 1]^2), h(t, t)=0 \text{ for } t\in[-1, 1], \] for which \(v\in C[-1, 1]\) is a power weight with \(v^{-1}\in L^1(-1, 1)\), i.e., \[ v(x)=\prod_{i=1}^N| x-x_i| ^{\beta_i} \text{ with } -1\leq x_1<x_2<\cdots <x_N\leq 1\text{ and } 0\leq \beta_i<1 \] and \(w\) is a weight with \[ \text{ess sup}_{ t\in[-1, 1]}| | (t-.)k(., t)w(t)| | _{v, \gamma, \delta}<\infty. \] Then \(K\) is a bounded linear operator on \(L_u^\infty\) to \(C_{uvw}^{\gamma, \delta-1}\) for all \[ u(x)=w^{-1}(x)\prod_{i=1}^N | x-x_i| ^{\alpha_i}, \quad 0\leq \alpha_i<1-\beta_i\text{ for } 1\leq i\leq N. \] The result is then applied to kernels such as \(k(x, t)=\frac{2t^2(t-1)}{\sqrt{| x-t^2| | x-t| }}\) for which \(K\in\mathcal L(L^\infty, C^{1/2, -1})\).