Finding the fundamental solution of the bi-axially symmetric Laplace–Beltrami equation using generalized shift operator

Many physical processes are described by partial differential equations. The relevance of this study is due to the need to solve applied problems of quantum mechanics, the theory of elasticity, and heat capacity. In this paper, an equation is considered that describes the field created by a contour with two axes of symmetry. The purpose of the study is to find a fundamental solution to this equation, which can later be used when solving boundary value problems.

On the boundedness of a B-Riesz potential in the generalized weighted B-Morrey spaces

We consider the generalized shift operator associated with the Laplace–Bessel differential operator

Some Embeddings into the Morrey and Modified Morrey Spaces Associated with the Dunkl Operator

We consider the generalized shift operator, associated with the Dunkl operatorΛα(f)(x)=(d/dx)f(x)+((2α+1)/x)((f(x)-f(-x))/2),α>-1/2. We study some embeddings into the Morrey space (D-Morrey space)Lp,λ,α,0≤λ<2α+2and modified Morrey space (modifiedD-Morrey space)L̃p,λ,αassociated with the Dunkl operator onℝ. As applications we get boundedness of the fractional maximal operatorMβ,0≤β<2α+2, associated with the Dunkl operator (fractionalD-maximal operator) from the spacesLp,λ,αtoL∞(ℝ)forp=(2α+2-λ)/βand from the spacesL̃p,λ,α(ℝ)toL∞(ℝ)for(2α+2-λ)/β≤p≤(2α+2)/β.

Necessary and Sufficient Conditions for the Boundedness of Dunkl-Type Fractional Maximal Operator in the Dunkl-Type Morrey Spaces

We consider the generalized shift operator, associated with the Dunkl operator , . We study the boundedness of the Dunkl-type fractional maximal operator in the Dunkl-type Morrey space , . We obtain necessary and sufficient conditions on the parameters for the boundedness , from the spaces to the spaces , , and from the spaces to the weak spaces , . As an application of this result, we get the boundedness of from the Dunkl-type Besov-Morrey spaces to the spaces , , , , , and .

On Estimating the Approximation of Locally Summable Functions by Gegenbauer Singular Integrals

Using the generalized shift operator (GSO) generated by the Gegenbauer differential operator we introduce the notion of a Lebesgue–Gegenbauer (L-G)-point of a summable function 𝑓 on the interval [1,∞) and prove that almost all points of this interval are (L-G)-points of 𝑓. Furthermore, we give an exact (by order) estimation of the approximation of locally summable functions by singular integrals generated by GSO (Gegenbauer singular integrals).