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Karamata regular variation is a basic tool in stochastic process and the boundary blow-up problems for partial differential equations (PDEs). Morrey space is closely related to study of the regularity of solutions to elliptic PDEs. The aim of this paper is trying to bring together these two areas and this paper is intended as an attempt at motivating some further research on these areas. A version of Morrey space associated with Karamata regular variation is introduced. As application, some estimates of operators, especially one-sided operators, on these spaces are considered. [ABSTRACT FROM AUTHOR]

In the setting of the Heisenberg group, based on the rotation method, we obtain the sharp ( p, p) estimate for the Hardy operator. It will be shown that the norm of the Hardy operator on L(ℍ) is still p/( p-1). This goes some way to imply that the L norms of the Hardy operator are the same despite the domains are intervals on ℝ, balls in ℝ, or 'ellipsoids' on the Heisenberg group ℍ. By constructing a special function, we find the best constant in the weak type (1, 1) inequality for the Hardy operator. Using the translation approach, we establish the boundedness for the Hardy operator from H to L. Moreover, we describe the difference between M weights and A weights and obtain the characterizations of such weights using the weighted Hardy inequalities. [ABSTRACT FROM AUTHOR]

We study the properties of the Triebel-Lizorkin spaces associated with the multidimensional Hermite expansions on Rd (d≥1). In addition to the endpoint estimates, we give the continuous characterizations of these spaces via harmonic and thermic extensions from Rd into R+d+1. Based on this result, we obtain the boundedness of Riesz transform associated with the Hermite expansions. [ABSTRACT FROM AUTHOR]