Boundedness of the Riesz Potential in Local Morrey-Type Spaces

Abstract

The problem of boundedness of the Riesz potential in local Morreytype spaces is reduced to the problem of boundedness of the Hardy operator in weighted L p -spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for boundedness for all admissible values The research of V. Burenkov was partially supported by the grants of the RFBR (project 09-01-00093a) and 68 V.I. Burenkov et al.of the parameters, which, for a certain range of the parameters wider than known before, coincide with the necessary ones.Keywords Riesz potential · Local and global Morrey-type spaces · Hardy operator on the cone of monotonic functions Mathematics Subject Classifications (2010) Primary 42B20 · 42B25 · 42B35

On weighted iterated Hardy-type inequalities

“…These inequalities play an important role in the theory of Morrey-type spaces and other topics (see [6], [7] and [9]). It is worth to mention that using characterizations of weighted Hardy inequalities we can show that the characterization of the boundedness of bilinear Hardy inequalities, namely of the inequality…”

Equivalence of norms of Riesz potential and fractional maximal function in generalized Morrey spaces

“…It is easy to see that all statements in this section hold true for balls instead of cubes (for direct proof see [4]). …”

Sobolev's theorem and duality for Herz-Morrey spaces of variable exponent

“…Further, Sobolev's inequality was also studied on generalized Morrey spaces (see [31]). This result was extended to local and global Morrey type spaces by Burenkov, Gogatishvili, Guliyev and Mustafayev [8] (see also [7,9,10]). The local Morrey type spaces are also called Herz spaces introduced by Herz [23].…”