SOBOLEV'S THEOREM AND DUALITY FOR HERZ-MORREY SPACES OF VARIABLE EXPONENT

作者：Mizuta Yoshihiro^*; Ohno Takao

来源：Annales Academiae Scientiarum Fennicae-Mathematica, 2014, 39(1): 389-416.

DOI：10.5186/aasfm.2014.3913

• Herz-Morrey spaces of variable exponent
• maximal functions
• Riesz potentials
• Sobolev's inequality
• Trudinger's inequality
• duality

In this paper, we consider the Herz Morrey space H-P{x(0})((.))q,omega(G) of variable exponent consisting of all measurable functions f on a bounded open set G boolean OR R-n satisfying
parallel to f parallel to H-{x0}(p(.),q,omega)(G)= (integral(2dG)(0) (omega(x(0), r)parallel to f parallel to(Lp()(B(x0,r)\B(x0),(r/2))))(q) dr/r)(1/q) < infinity,
and set H-p(.),H-q,H-omega(G) = boolean AND(x0 is an element of G) H-{x0}(p(.),q,omega) (G).
Our first aim in this paper is to give the boundedness of the maximal and Riesz potential operators in H-p(.),H-q,H-w(G) when q = infinity.
In connection with H-{x0}(p(.),q,omega)(G) and H-p(.),H-q,H-omega (G), let us consider the families (H) under bar (p(.),q,omega)({x0})(G) , (H) under bar (p(.),q,omega)(G) and (H) over tilde (p(.),q,omega)(G). Following Fiorenza Rakotoson [18], Di Fratta Fiorenza [17] and Gogatishvili Mustafayev [19], we next discuss the duality properties among these Herz-Morrey spaces.

• 出版日期2014