A note on anisotropic potentials associated with the Laplace-Bessel differential operator

Abstract

In this note the anisotropic maximal operator and anisotropic Riesz potentials generated by the generalized shift operator are investigated in the anisotropic B -Morrey space Lp,λ ,γ (R n k,+) . We prove that the anisotropic B -maximal operator Mγ is bounded on the anisotropic B -Morrey space Lp,λ ,γ (R n k,+) . Also the anisotropic B -Riesz potential R α γ is bounded from the anisotropic B -Morrey spaces Lp,λ ,γ (R n k,+) to Lq,λ ,γ (R n k,+) if and only if 1/p− 1/q = α/(|a| + (a, γ ) − λ) and 1 < p < (|a| + (a, γ )−λ)/α , and its modified version Rα γ is bounded from the anisotropic B -Morrey space to the anisotropic B -BMO space. Furthermore, we obtain some imbedding relations between the space Lp,λ ,γ (R n k,+) and the anisotropic B -Stummel-Kato class Sp,θ,γ (R n k,+) . Mathematics subject classification (2000): 42B20, 42B25, 42B35.