Some scales of equivalent conditions to characterize the Stieltjes inequality: the case

Abstract

We prove that the weighted Stieltjes inequality ()∫0∞()∫0∞h(t)dt(x+t)λqu(x)dx1/q≤C()∫0∞hp(x)v(x)dx1/pcan be characterized by four different scales of conditions also for the case 0<q<p<∞, 1<p. In particular, a new proof of a result of G. Sinnamon is given, which also covers the case 0<q<1. Moreover, for the situation at hand a new gluing theorem and also an equivalence theorem of independent interest are proved and discussed. By applying our new equivalence theorem to weighted inequalities for the Stieltjes transform we obtain the four new scales of conditions for characterization of Stieltjes inequality.

Bilinear weighted Hardy inequality for nonincreasing functions

“…Using the same argument as in (ii) (based on [14]), we can show that B 7 B 6 + A (42) . Next, since the function s → ϕ(s) s α is nonincreasing, we obtain…”

“…which is, in turn, equivalent to A (14) by Proposition 2.3, since u is integrable at the origin. Finally, observe that if u is not integrable at the origin, then necessarily both B 0 = ∞ (see the proof sketch of Proposition 2.3) and A (14) = ∞.…”

“…Finally, observe that if u is not integrable at the origin, then necessarily both B 0 = ∞ (see the proof sketch of Proposition 2.3) and A (14) = ∞. On the other hand, if A (14) < ∞, then u is integrable at the origin.…”

The Calderón operator and the Stieltjes transform on variable Lebesgue spaces with weights

“…For other results on weighted norm inequalities for the Hardy operator, the Calderón operator and the Stieltjes transform, we refer the reader to Sinnammon [41] and Gogatishvili, et al [20,21,22].…”

Calderón Operator on Local Morrey Spaces with Variable Exponents

“…The reader is referred to [20][21][22] for the studies of the Stieltjes transformation and its application on the Hilbert's double series.…”