The averaging integral operator between weighted Lebesgue spaces and reverse Hölder inequalities

0342832 - MÚ 2011 RIV GB eng J - Journal Article
Opic, Bohumír
The averaging integral operator between weighted Lebesgue spaces and reverse Hölder inequalities.
Complex Variables and Elliptic Equations. An International Journal. Roč. 55, 8-10 (2010), s. 965-972. ISSN 1747-6933. E-ISSN 1747-6941
R&D Projects: GA ČR GA201/08/0383
Institutional research plan: CEZ:AV0Z10190503
Keywords : averaging integral operator * weighted Lebesque spaces * weights * Hardy-type inequalities * reverse Höldet inequalities
Subject RIV: BA - General Mathematics
Impact factor: 0.409, year: 2010
http://www.tandfonline.com/doi/full/10.1080/17476930903276027

Let 1 < p ≤ q < +∞ and v, w be weights on (0, +∞) such that v(x)xρ is equivalent to a non-decreasing function on (0, +∞) for some ρ ≥ 0, and ... First, we prove that the operator ... if and only if the operator ... Second, we show that the boundedness of the averaging operator A on the space Lp((0, +∞); v) implies that, for all r > 0, the weight v1-p' satisfies the reverse Hlder inequality over the interval (0, r) with respect to the measure dt, while the weight v satisfies the reverse Hlder inequality over the interval (r, +∞) with respect to the measure t-p dt. As a corollary, we obtain that the boundedness of the averaging operator A on the space Lp((0, +∞); v) is equivalent to the boundedness of the averaging operator A on the space Lp((0, +∞); v1+δ) for some δ > 0.
Permanent Link: http://hdl.handle.net/11104/0185455