A note on the spaces of variable integrability and summability of Almeida and Hästö
By H. Kempka, Jan Vyb'iral. 2011
50 citations recorded by the source corpus.
We address an open problem posed recently by Almeida and Hasto in [1]. They defined the spaces lq(·)(Lp(·)) of variable integrability and summability and showed that ‖·|lq(·)(Lp(·))‖ is a norm if q ≥ 1 is constant almost everywhere or if 1/p(x) + 1/q(x) ≤ 1 for almost every x ∈ R. Nevertheless, the natural conjecture (expressed also in [1]) is that the expression is a norm if p(x), q(x) ≥ 1 almost everywhere. We show that ‖ · |lq(·)(Lp(·))‖ is a norm, if 1 ≤ q(x) ≤ p(x) for almost every x ∈ R. Furthermore, we construct an example of p(x) and q(x) with min(p(x), q(x)) ≥ 1 for every x ∈ R such that the triangle inequality does not hold for ‖ · |lq(·)(Lp(·))‖.
Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/52bcbdd9a5cf83609d65f4a879430135e7fa3d1d
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Citations in the corpus
- Spaces of Variable Smoothness and Integrability: Characterizations by Local Means and Ball Means of Differences (2011, methodology): “Here the smoothness in these spaces gets measured by a weight sequence w = (wj )∞j=0.”
- Lorentz spaces with variable exponents (2012, unknown): “Further it is known (see [21]) that it is a norm if either 1 p(·) + 1 q(·) ≤ 1 holds pointwise for all x ∈ Rn or if 1 ≤ q(·) ≤ p(·) ≤ ∞ holds pointwise.”