Consider p : Ω → [1,+∞[, a measurable bounded function on a bounded set Ω with decreasing rearrangement p∗ : [0, |Ω| ] → [1,+∞[. We construct a rearrangement invariant space with variable exponent p∗ denoted by Lp∗(·)∗∗ (Ω). According to the growth of p∗, we compare this space to the Lebesgue spaces or grand Lebesgue spaces. In particular, if p∗(·) satisfies the log-H¨older continuity at zero, then it is contained in the grand Lebesgue space Lp∗(0))(Ω). This inclusion fails to be true if we impose a slower growth as |p∗(t) − p∗(0)| ≥ ALn |Ln t| at zero. Some other results are discussed.
Variable exponents and grand Lebesgue spaces: Some optimal results / Fiorenza, Alberto; Rakotoson, Jean Michel; Sbordone, Carlo. - In: COMMUNICATIONS IN CONTEMPORARY MATHEMATICS. - ISSN 0219-1997. - 17:6(2015), pp. 1-14. [10.1142/S0219199715500236]
Variable exponents and grand Lebesgue spaces: Some optimal results
FIORENZA, ALBERTO;SBORDONE, CARLO
2015
Abstract
Consider p : Ω → [1,+∞[, a measurable bounded function on a bounded set Ω with decreasing rearrangement p∗ : [0, |Ω| ] → [1,+∞[. We construct a rearrangement invariant space with variable exponent p∗ denoted by Lp∗(·)∗∗ (Ω). According to the growth of p∗, we compare this space to the Lebesgue spaces or grand Lebesgue spaces. In particular, if p∗(·) satisfies the log-H¨older continuity at zero, then it is contained in the grand Lebesgue space Lp∗(0))(Ω). This inclusion fails to be true if we impose a slower growth as |p∗(t) − p∗(0)| ≥ ALn |Ln t| at zero. Some other results are discussed.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.