A note on the Stieltjes transformation

Abstract

SynopsisA necessary and sufficient condition for the boundedness of the generalised Stieltjes transformation on weighted Lebesgue spaces is obtained and applied to extend the Hilbert double series theorem.

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

“…For the case q ≥ 1 this result is due to G. Sinnamon [10], who complemented a previous result of K. F. Andersson [1] concerning e.g. the case 1 ≤ p ≤ q ≤ ∞, λ > 0.…”

“…In this section we give a new proof of an extended form of a result of G. Sinnamon . Proposition Let , , , and suppose that and are non‐negative extended real valued functions defined on (0, ∞).…”

“…For α, β < 1 and β ≤ 1 − α, Theorem 3.1 was proved by G. Sinnamon [10] using the Hardy inequality and the Stieltjes transformation. Our proof is direct and allows us to consider all parameters.…”

“…In Section we present and prove our new gluing theorem for the situation at hand (see Theorem ). In Section we present a new proof of a result of Sinnamon (see [, Theorem 1]), which also covers the case not covered by the Sinnamon technique (see Proposition ). In Section we present the new equivalence theorem for the studied case (see Theorem ).…”

“…can 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 [10] 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.…”