On a general weighted Hardy type inequality in the variable exponent Lebesgue spaces

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

“…These results are not immediately comparable to ours, even in the one-weight case, since they assume log-Hölder continuity conditions that depend on the weight. See [12] for a discussion of cases where this condition overlaps with our regularity assumptions. Remark 1.11.…”

On the Logarithmic Regularity Conditions for the Variable Exponent Hardy Type Inequality

“…The proof of Theorem 3 also is given in [8]. Here we derive an alternative proof of that theorem using the general results of [2,4]. In the proof of main results we use the following elementary Lemma.…”

“…Two types of conditions arise here: a balance condition on the weights and a regularity condition on the exponents (see below). Necessary and sufficient conditions for the validity of general inequality (1) were found in [1] for the case of ( ) ≤ ( ), in [2] for cases (0) ≥ (0) and (∞) ≥ (∞), in [3] for cases (0) < (0) and (∞) < (∞), and in [4] for mixed cases (0) ≥ (0) and (∞) < (∞) ( (0) < (0) and (∞) ≥ (∞)). Some special cases of (1) are studied in [5][6][7][8][9][10] too.…”

Riemann-Liouville and Higher Dimensional Hardy Operators for NonNegative Decreasing Function inLp(·)Spaces

“…in (⋅) spaces were derived in the papers [4] for powertype weights and in [5][6][7][8][9] for general weights. The Hardy inequality for nonnegative decreasing functions was studied in [10,11].…”