Parabolic fractional maximal and integral operators with rough kernels in parabolic generalized Morrey spaces

Abstract

Abstract. Let P be a real n×n matrix, whose all the eigenvalues have positive real part, A t = t P , t > 0 , γ = trP is the homogeneous dimension on R n and Ω is an A t -homogeneous of degree zero function, integrable to a power s > 1 on the unit sphere generated by the corresponding parabolic metric. We study the parabolic fractional maximal and integral operators M P Ω,α and I P Ω,α , 0 < α < γ with rough kernels in the parabolic generalized Morrey space M p,ϕ,P (R n ) . We find conditions on the pair (ϕ 1 ,ϕ 2 ) for the boundedness of the operators M P Ω,α and I PWe also find conditions on ϕ for the validity of the Adams type theoremsMathematics subject classification (2010): 42B20, 42B25, 42B35.

Adams-Spanne type estimates for parabolic sublinear operators and their commutators with rough kernels on parabolic generalized Morrey spaces

“…Proof. The statement of Theorem 8 follows by Lemma 1 in the same manner as in the proof of Theorem 4.1 in [7].…”

Fractional Integrals and Derivatives: Mapping Properties

“…The operator H α is L p (R n+1 ) → L q (R n+1 ) bounded when 1 < p < (n + 2)/α and 1/q = 1/p − α/n + 2, see [16]. Mapping properties in Lebesgue spaces were obtained in [79,58], and for Morrey type spaces -in [23,24,25,20,26,21].…”