Necessary and sufficient conditions for the boundedness of B-Riesz potential in the B-Morrey spaces

Abstract

We consider the generalized shift operator, associated with the Laplace–Bessel differential operator Δ B = ∑ i = 1 n ∂ 2 ∂ x i 2 + ∑ i = 1 k γ i x i ∂ ∂ x i . The maximal operator M γ ( B-maximal operator) and the Riesz potential I γ α ( B-Riesz potential), associated with the generalized shift operator are investigated. At first, we prove that the B-maximal operator M γ is bounded from the B-Morrey space L p , λ , γ to L p , λ , γ for all 1 < p < ∞ and 0 ⩽ λ < n + | γ | . We prove that the B-Riesz potential I γ α , 0 < α < n + | γ | is bounded from the B-Morrey space L p , λ , γ to L q , λ , γ if and only if α / ( n + | γ | − λ ) = 1 / p − 1 / q , 1 < p < ( n + | γ | − λ ) / α . Also we prove that the B-Riesz potential I γ α is bounded from the B-Morrey space L 1 , λ , γ to the weak B-Morrey space WL q , λ , γ if and only if α / ( n + | γ | − λ ) = 1 − 1 / q .