We prove some formulas relating Cauchy-Riemann operators defined on hypercomplex subspaces of an alternative *-algebra to a differential operator associated with the concept slice-regularity and spherical Dirac operator. These results in particular allow introduce definition locally slice-regular function open path for local slice analysis. Since factor corresponding Laplacian operators, proven...

Abstract A depending on a complex parameter k superanalog SL of Calogero operator is constructed; it is related with the root system of the Lie superalgebra gl(n|m). For m = 0 we obtain the usual Calogero operator; for m = 1 we obtain, up to a change of indeterminates and parameter k the operator constructed by Veselov, Chalykh and Feigin [2, 3]. For k = 1, 1 2 the operator SL is the radial par...

purpose: to study the distribution of ocular higher-order aberrations (hoas) and mesopic pupil size in individuals screened for refractive surgery. methods: ocular hoas and mesopic pupil size were studied in 2458 eyes of 1240 patients with myopia, simple myopic astigmatism and compound myopic astigmatism and 215 eyes of 110 patients with hyperopia, simple hyperopic astigmatism and compound hype...

We consider T f = ∫ x1 0 ∫ x2 0 f (t1, t2)dt1dt2 and a corresponding geometric mean operator G f = exp(1/x1x2) ∫ x1 0 ∫ x2 0 log f (t1, t2)dt1dt2. E. T. Sawyer showed that the Hardy-type inequality ‖T f ‖Lq u ≤ C‖ f ‖Lp v could be characterized by three independent conditions on the weights. We give a simple proof of the fact that if the weight v is of product type, then in fact only one condit...

Results analogous to those proved by Rubio de Francia [28] are obtained for a class of maximal functions formed dilations bilinear multiplier operators limited decay. We focus our attention on L2 × → L1 estimates. discuss two applications: the boundedness Bochner-Riesz operator and spherical operator. For latter we improve known results in [1] reducing dimension restriction from n ≥ 8 4.

A mean-value formula for a linear partial differential hyperbolic equation with an operator splitting into first-order factors is obtained. This can be interpreted as extension of the Ásgeirsson principle.