PARABOLIC FRACTIONAL MAXIMAL AND INTEGRAL OPERATORS WITH ROUGH KERNELS IN PARABOLIC GENERALIZED MORREY SPACES

Abstract

Let P be a real nxn matrix, whose all the eigenvalues have positive real part, A(t) = t(P), t > 0, gamma = trP is the homogeneous dimension on R-n and Omega 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-Omega,alpha(P) and I-Omega,alpha(P), 0 < alpha < gamma with rough kernels in the parabolic generalized Morrey space M-p,M-phi,M-P(R-n). We find conditions on the pair (phi(1), phi(2)) for the boundedness of the operators M-Omega,alpha(P) and I-Omega,alpha(P) from the space M-p,M-phi 1,M-P(R-n) to another one M-q,M-phi 2,M-P(R-n), 1 < p < q < infinity, 1/p-1/q = alpha/gamma, and from the space M-1,M-phi 1,M-P(R-n) to the weak space W M-q,M-phi 2,M-P(R-n), 1 <= q < infinity, 1 - 1/q = alpha/gamma. We also find conditions on phi for the validity of the Adams type theorems M-Omega,alpha(P), I-Omega,alpha(P) : M-p,M-phi 1/p,M-P(R-n) -> M-q,M-phi 1/q,M-P(R-n), 1 < p < q < infinity.