Estimate of the modulus of continuity of a cauchy-type integral in a domain and on its boundary

An estimate for the modulus of continuity of a quaternion singular Cauchy integral

“…The existence of these integrals follows from condition (5) by virtue of Theorem 1 in [11]. Applying estimate (2) from [11] to these integrals, we get…”

“…Decomposing the function f into components, we obtain four complex integrals in which i 3 plays the role of imaginary unit. The existence of these integrals follows from condition (5) by virtue of Theorem 1 in [11]. Applying estimate (2) from [11] to these integrals, we get…”

“…The existence of the integral L f α [ ] follows from condition (6) by analogy with the existence of an integral of the form (9) proved in [11]. Let …”

“…In particular, it was established that the broadest class of curves (see in [6,9]) for which it has the same form as on a circle is the class of regular curves (for which the measure of the part of a curve that enters the disk does not exceed a constant multiplied by the radius of the disk). For more general curves (see [6,[9][10][11][12][13]), the majorant worsens and depends on a curve.…”