On the limiting case for boundedness of the B-Riesz potential in B-Morrey spaces

Abstract

We consider the generalized shift operator associated with the Laplace-Bessel differential operator $$ \Delta _B = \sum\limits_{i = 1}^n {\frac{{\partial ^2 }} {{\partial x_j^2 }}} + \sum\limits_{i = 1}^k {\frac{{\gamma _i }} {{x_i }}\frac{\partial } {{\partial x_i }}} $$ , and study the modified B-Riesz potential Ĩ α, β generated by the generalized shift operator acting in the B-Morrey space in the limiting case. We prove that the operator Ĩ α, β, 0 < α < n + |γ|, is bounded from the B-Morrey space L (n+|γ|−λ)/α,λ,γ (ℝ ,+ ) to the B-BMO space BMO γ (ℝ ,+ ).