A necessary and sufficient condition for Hardy's operator in Lp(.)(0, 1). Issue 5 (29th October 2013) - Mamedov, Farman I.; Mammadova, Firana M.
<x xml:space="preserve">Abstract</x> The equivalent conditions are proved for the Hardy inequality <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg537sm83j" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />xβ(.)−1∫0xf(t)dtLp(.)(0, 1)≤C∥xβ(.)f∥Lp(.)(0, 1), f≥0, </alternatives> to be fulfilled in the norms of a Lebesgue space with variable exponent <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg537sm7rh" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />Lp(.)(0, 1)</alternatives>. It is assumed that the function <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg537sm7qz" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />1β(.)−p′(.)</alternatives> is monotone and positive on (0, 1).