An estimate for the local smoothness of singular integrals (Q2386934)

The author studies smoothness of the convolution integral \[ g_\psi(x)=\frac1{2\pi}\int_{-\pi}^\pi g(t)\,\psi(x-t)\,dt, \] where \(\psi\) is a given \(2\pi\)-periodic function having suitable growth properties (denoted \(\psi\in\mathcal D\)), \(g\) is a \(2\pi\)-periodic function which belongs to \(L(-\pi,\pi)\) (the integral on the right-hand side is considered, if \(\psi\) is not integrable, in the sense of its principal value). The estimates of smoothness of \(g_\psi\) are obtained in terms of the function \(\mathcal N_\eta g_\psi\) (recall that \(\mathcal N_\eta f(x)=\sup_{I\ni x} \frac1{| I| \,\eta(| I| )}\int_I| f(t)-f(x)| \,dt\), where \(\eta\) is a nondecreasing function on \([0,\pi]\) such that \(\eta(0)=0\) and \(\eta(t)t^{-1}\) is non-increasing, \(I\) is a segment of the length \(| I| \)). In fact, the estimates are of the type \[ \mathcal N_\sigma g_\psi(x)\leq c M(\mathcal N_\eta g)(x) \] (with \(\eta\) given and \(\sigma\) depending on \(\eta\) or on \(\eta\) and \(\psi\) ) or of the type \[ \mathcal N_\eta g_\psi(x)\leq c M(g)(x) \] (with \(\eta\) depending on \(\psi\)), where \(M\) denotes the Hardy-Littlewood maximal operator. The author studies two cases, when the function \(\psi\) is either odd (\(\psi\in\mathcal D_1\)) or integrable \((\psi\in\mathcal D_2\)) on the interval \((-\pi,\pi)\). The results extend known estimates of the modulus of continuity of the conjugate functions, those of the Weyl integrals and of fractional integrals (cf. [\textit{A. A. Korenovskii}, Properties of functions defined in terms of average oscillations, Odessa University Press (1988)] and [\textit{S. G. Samko, A. A. Kilbas} and \textit{O. I. Marichev}, Integrals and derivatives of fractional order and some of their applications. Nauka, Minsk (1987; Zbl 0617.26004); English translation 1993; Zbl 0818.26003)].