HYPERSINGULAR INTEGRALS IN POWER-WEIGHTED VARIABLE GENERALIZED HÖLDER SPACES OVER METRIC MEASURE SPACES

The weighted variable generalized Hölder spaces \(H^{\,\omega (\cdot )} (\Omega , w)\) $H^{\omega (·)}(\mathrm{\Omega },w)$ defined in terms of the local modulus of continuity are considered. Zygmund-type estimates are obtained for a hypersingular integral operator \(D^\alpha\) $D^{\alpha }$ defined on a bounded open set \(\Omega\) $\mathrm{\Omega }$ of a metric measure space \(\mathcal {X} = (\mathcal {X}, d, \mu )\) $\mathcal{X}=(\mathcal{X},d,\mu )$ , assuming the power weight function \(w(x) = d^{\,\nu }(x, a)\) $w(x)=d^{\nu }(x,a)$ , where \(a, x \in \Omega\) $a,x\in \mathrm{\Omega }$ , \(1< \textrm{Re}\,\nu < N + 1\) $1<\text{Re}\nu <N+1$ , and N is a parameter that characterizes the measure of balls in \(\mathcal {X}\) $\mathcal{X}$ with respect to their radius. Based on these estimates, it is proven that, under specific conditions on the characteristic \(\omega (x, h)\) $\omega (x,h)$ of \(H^{\,\omega (\cdot )}\) $H^{\omega (·)}$ , \(D^\alpha\) $D^{\alpha }$ is a bounded operator from \(H^{\,\omega (\cdot )} (\Omega , w)\) $H^{\omega (·)}(\mathrm{\Omega },w)$ to \(H^{\,\omega _{-\alpha } (\cdot )} (\Omega , w)\) $H^{{\omega }_{-\alpha }(·)}(\mathrm{\Omega },w)$ , where \(\omega _{-\alpha } (x, h):= h^{\,-\textrm{Re}\,\alpha } \, \omega (x, h)\) ${\omega }_{-\alpha }(x,h):=h^{-\text{Re}\alpha }\omega (x,h)$ . This result complements a similar one for the case of \(0 < \textrm{Re}\,\nu \le 1\) $0<\text{Re}\nu \le 1$ that was obtained in a previous study conducted by the authors.