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Examples of differential and integral operations are given whose dimension is modified as a result of the introduction of new radial variables. Based on the integral measure , , with a weak singularity, we introduce an operator that is interpreted as the Laplace operator in the space of functions of a fractional number of variables. The integration with respect to the measure , , can also be interpreted as the integration over a domain of fractional dimension. The coefficient of hidden spherical symmetry is introduced. A formula is obtained that relates this coefficient to the Hausdorff dimension of a set in and the Euclidean dimension . The existence of hidden spherical symmetries is verified by calculating the dimension of the th generation of the Koch curve for arbitrary positive integer . [ABSTRACT FROM AUTHOR]

Functions satisfying the Laplace equation for the Kipriyanov -operator are said to be -harmonic. The following properties of -harmonic functions are presented and proved: the Green-type integral representation of -functions, the spherical mean theorem, and the maximum principle. As a corollary, the uniqueness of the solution of the interior and exterior Dirichlet problems is proved. [ABSTRACT FROM AUTHOR]

We introduce integral transforms to study problems for the singular Bessel differential operators B−γ with negative parameter −γ < 0. A solution to the singular Bessel equation B−γu+u = 0 is a function u = 핁μ expressed in terms of the Bessel functions of the first kind with positive parameter μ = (γ + 1)/2. Based on the notion of a 핋-pseudoshift, we formulate the Levitan addition theorem. We construct the Bessel 핁-transform and the corresponding class of singular 핁-pseudodifferential operators. We prove a theorem on the order of 핁-pseudodifferential operators in the class of Sobolev–Kipriyanov functions. [ABSTRACT FROM AUTHOR]

Solutions of singular differential equations with the Bessel operator of negative order are studied. In this regard, of great interest are the solutions of the singular differential Bessel equation , which are presented in the paper as linearly independent functions and , . Some properties of the functions expressed in terms of the properties of the Bessel–Levitan -function are considered. The direct and inverse -Bessel transforms are introduced, and the -pseudoshift operator that commutes with the Bessel operator is defined. The fundamental solution of the ordinary singular differential operator is found. A representation of the fundamental solution of the Kipriyanov -operator with a singularity at the point and on the cone in the Euclidean -dimensional half-space is given. [ABSTRACT FROM AUTHOR]

The support theorem for the Radon transform was obtained by L. Helgason. The Radon transform of functions of variables related by spherical symmetry is a special case of a more general transformation, namely, the Radon–Kipriyanov transform K훾. This transformation corresponds to a weight multi-index 훾 = (훾1,... , 훾m) and coincides with the Radon transform if all components of the multi-index 훾 are natural numbers. In general, the K훾-transformation can be interpreted as a transformation of functions of a fraction-dimensional argument. In this paper, we prove a general support theorem. In a special case where 훾 = 0, this theorem coincides with the Helgason theorem. [ABSTRACT FROM AUTHOR]

We prove that any linear partial integral operator acting from the space of continuous functions to the Lebesgue class L p D β n 2 , 1 < p < ∞ , is bounded as an operator from an anisotropic Lp-space to the class of functions with mixed norm. [ABSTRACT FROM AUTHOR]

We obtain conditions for the existence and uniqueness of a solution to the Fredholm equation with multi-dimensional integral operators with partial integrals in anisotropic Lebesgue spaces. [ABSTRACT FROM AUTHOR]

We consider partial integral in the classes of functions with the mixed Lp– and supnorms or in the anisotropic Lebesgue spaces. For second kind Fredholm integral equations with partial integrals we construct solutions in the form of the operator Neumann series by the method of successive approximations. [ABSTRACT FROM AUTHOR]

For the Kipriyanov operator written in the form of the sum of singular differential Bessel operators with, generally speaking, negative parameters, we obtain a representation in spherical coordinates (the Kipriyanov–Beltrami operator). The operator on the sphere and the corresponding spherical functions ( -harmonics) are introduced. The main properties of the operator on the sphere and the differential equation of -harmonics are provided. A solution of the inner singular Dirichlet problem in a ball centered at the origin in is given. The solution is obtained by the Fourier method in the form of Laplace series in -harmonics. The solution is bounded only in the case where all parameters of the Bessel operators occurring in belong to the interval , , . [ABSTRACT FROM AUTHOR]

For the Euler–Poisson–Darboux type equations with the Bessel operators with negative index we consider the Cauchy problem with the boundary conditions. We obtain a representation of the solution in the form of the Poisson formula. [ABSTRACT FROM AUTHOR]

We consider linear integral operators with partial integrals acting in anisotropic Lebesgue spaces Lr(D) of functions defined in a finite rectangle D in the Euclidean space ℝ2. We establish the boundedness of such operators acting from Lr(D) to Lp(D) under certain relations between the multiindices p and r. [ABSTRACT FROM AUTHOR]

We consider linear integral operators with multidimensional partial integrals in anisotropic Lebesgue spaces Lr(D) of functions defined in a finite parallelepiped D in the Euclidean spaces ℝn, where n ≥ 3. We obtain conditions on kernels of partial integrals and connections between multiindices r and p guaranteeing the boundedness of the partial integrals acting from Lr(D) to Lp(D). [ABSTRACT FROM AUTHOR]

The even Radon–Kipriyanov transform (Kγ-transform) is suitable for studying problems with the Bessel singular differential operator . In this work, the odd Radon–Kipriyanov transform and the complete Radon–Kipriyanov transform are introduced to study more general equations containing odd B-derivatives (in particular, gradients of functions). Formulas of the Kγ-transforms of singular differential operators are given. Based on the Bessel transforms introduced by B.M. Levitan and the odd Bessel transform introduced by I.A. Kipriyanov and V.V. Katrakhov, a relationship of the complete Radon–Kipriyanov transform with the Fourier transform and the mixed Fourier–Levitan–Kipriyanov–Katrakhov transform is deduced. An analogue of Helgason's support theorem and an analog of the Paley–Wiener theorem are given. [ABSTRACT FROM AUTHOR]

The definition of a B-derivative is based on the notion of generalized Poisson shift; this derivative coincides, up to a constant, with the singular Bessel differential operator. We introduce the fractional powers of a B-derivative by analogy with fractional Marchaud and Weyl derivatives. We prove statements on the coincidence of these derivatives for the classes of even smooth integrable functions. We obtain analogs of Bernstein's inequality for B-derivatives of integer and fractional order in the space of even Schlömilch j-polynomials with sup-norm and Lpγ-norm (the Lebesgue norm with power weight xγ, γ > 0). The resulting estimates are sharp and define the norms of powers of the Bessel operator in the spaces of even Schlömilch j-polynomials. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the Euler–Poisson–Darboux equation with boundary conditions. The Bessel operators in the equation can have different parameters. We establish a representation of the solution in the form of the Poisson formula with a special shift generated by the product of cylindrical functions of the first kind and different orders. [ABSTRACT FROM AUTHOR]

We study an Ibragimov-Mamontov type equation with singular Bessel operators playing the role of second order derivatives. We obtain a representation of the solution by using a special Bessel transform. [ABSTRACT FROM AUTHOR]

We study invertibility of the Radon Kγ-transform (Kipriyanov-Radon transform) with a natural parameter |γ| and prove an analog of the Plancherel formula. [ABSTRACT FROM AUTHOR]

The article examines the Radon-Kipriyanov transform (RKγ-transform) of radial functions, one-variable functions and weighted spherical means of multivariable functions. It explores the use of fractional differentiation operator with an order generated by the weight index to derive the inversion formulas for RKγ-transform of weighted spherical means, and the influence of the dimension of the Euclidean space on such formulas. Moreover, it analyzes the significant relationship between the Abel transform and the RKγ-transform of particular functions.

This paper considers the Kipriyanov–Radon transform constructed as a special Radon transform adopted for dealing with singular Bessel differential operators of the corresponding indices acting on a part of the variables. The authors obtain inversion formulas generalizing the classical formulas for the Radon transform of axially-symmetric functions and relating to the integro-differentiation of fractional order in a one-dimensional parameter. [ABSTRACT FROM AUTHOR]