We introduce mixed Morrey spaces and show some basic properties. These properties extend the classical ones. We investigate the boundedness in these spaces of the iterated maximal operator, the fractional integral operator and singular integral operator. Furthermore, as a corollary, we obtain the boundedness of the iterated maximal operator in classical Morrey spaces. We also establish a version of the Fefferman–Stein vector-valued maximal inequality and some weighted inequalities for the iterated maximal operator in mixed Lebesgue spaces.
KeywordsMorrey spaces Mixed norm Hardy–Littlewood maximal operator Fefferman–Stein vector-valued inequality Fractional integral operator Singular integral operator
Mathematics Subject Classification42B25 42B35
The author would like to thank Professor Yoshihiro Sawano for enthusiastic guidance and be also grateful to Professor Hitoshi Tanaka for his kind suggestion on the fractional integral operators. Furthermore, the author thanks the anonymous referee for his/her comments on this paper, which improved readability.
- 7.Duoandikoetxea, J.: Fourier analysis. Translated and revised from the 1995 Spanish original by D. Cruz-Uribe. Graduate Studies in Mathematics, vol. 29. American Mathematical Society, Providence (2001)Google Scholar
- 11.Liu, F., Torres, R., Xue, Q., Yabuta, K.: Multilinear strong maximal operators on mixed Lebesgue spaces (Preprint)Google Scholar
- 20.Tanaka, H.: Personal communicationGoogle Scholar