An example of a space Lp(x) on which the Hardy-Littlewood maximal operator is not bounded

By L. Pick, M. Růžička. 2001

135 citations recorded by the source corpus.

Abstract We give an example of a function p such that the Hardy-Littlewood maximal operator is not bounded on the generalized Lebesgue space L p(x) .

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b30e6057bd0c3b7be054eeb41caa8f90e8cecd95

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications (2008, background): “Global log-Hölder continuity is the best possible modulus of continuity to imply the boundedness of the maximal operator, see [6,30].”
A capacity approach to the Poincare inequality and Sobolev imbeddings in variable exponent Sobolev spaces (2004, background): “If the exponent is not 0-Hölder continuous, then the maximal operator need not be bounded on Lp(·), [21].”
THE HERZ-TYPE HARDY SPACES WITH VARIABLE EXPONENT AND THEIR APPLICATIONS (2012, background): “There exist some sufficient conditions on p(·) such that the maximal operator M is bounded on Lp(·)(Rn), see [1-4, 11, 12].”
Fredholmness of Singular Integral Operators with Piecewise Continuous Coefficients on Weighted Banach Function Spaces (2003, background): “;∞ is denoted by P. Clearly, P ⊂ Ptfor every t∈ Γ. The class P plays a very important role in questions on the boundedness of maximal functions and singular integrals on (weighted) Nakano spaces (see [10, 30, 46], the references therein, and also Theorem 6.2). Proposition 5.6. A function pbelongs to”
Some remarks on the Hardy-Littlewood maximal function on variable Lp spaces (2005, result): “Pick and Ru uyziy cka [ 14 ] have constructed a function p which does not satisfy (1.1) and p � P(R n ). An analogous example was found in [4] regarding (1.2).”
MAXIMAL FUNCTIONS IN VARIABLE EXPONENT SPACES: LIMITING CASES OF THE EXPONENT (2009, background): “Pick and R•ui£ka [ 36 ] complemented this result by showing that the local log-Holder continuity in (1) is the optimal continuity modulus for this assertion.”
Gradient estimates for the p (x)-Laplacean system (2005, background): “lim R!0 oðRÞ log 1 R 1⁄4 y ð17Þ rules out the possibility to prove that jDujpðxÞ A L for any s > 1, even in the case F 1 0; this fact can be inferred from the counterexamples in [36], [27], [18].”
An atomic decomposition of variable Besov and Triebel-Lizorkin spaces (2009, background): “Moreover, these spaces have been applied to partial differential equations and the calculus of variation, see [1, 3, 4, 5, 6, 7, 8, 10, 11, 12, 16, 17, 18, 19, 21, 22, 29] and references therein.”
Sobolev embeddings for variable exponent riesz potentials on metric spaces (2006, result): “according to the recent works by Diening [3], [4], Pick and R u zi cka [ 27 ] and Cruz-Uribe, Fiorenza and Neugebauer [2].”
On a progress in the theory of lebesgue spaces with variable exponent: maximal and singular operators (2005, unknown): “[63], the following statement was proved.”
Maximal function on Musielak–Orlicz spaces and generalized Lebesgue spaces (2005, background): “It is shown in [28] that this condition is the tightest condition in terms of a uniform local continuity modulus: If ρ(t) is a concave continuity modulus withρ(t)| ln t | → ∞ for t → 0+, then there exist p :R → (1,∞) which isρ-continuous and constant outside some large ball such that M is not contin”
VARIABLE BESOV AND TRIEBEL-LIZORKIN SPACES (2008, background): “omit the details here, one can see [3], [4], [5], [7], [14], [ 15 ].”