Abstract. We consider Hardy-Littlewood maximal operator on the general Lebesgue space L p(x) (R n ) with variable exponent. A sufficient condition on the function p is known for the boundedness of the maximal operator on L p(x) (Ω) with an open bounded Ω . Our main aim is to find an additional condition to p to guarantee the boundedness of the maximal operator on L p(x) (R n ) . From this point of view we put an emphasis on the behavior of functions p near the infinity. We find a sufficient condition on p such that the maximal operator is bounded on L p(x) (R n ) . We also construct a function p for which the maximal operator is unbounded. (2000): 46E30, 26D15.
Mathematics subject classification
Abstract. We show that a variable exponent Bessel potential space coincides with the variable exponent Sobolev space if the Hardy-Littlewood maximal operator is bounded on the underlying variable exponent Lebesgue space. Moreover, we study the Hölder type quasi-continuity of Bessel potentials of the first order.
The relation between a Banach function space X and its subspaces formed by: (i) those functions with absolutely continuous norm; (ii) those with continuous norm; and (iii) the closure of the set of bounded functions are investigated in the case X = L p (x) .
Key wordsHardy averaging operator, optimal source, optimal domain, solid Banach space, variable-exponent Lebesgue space MSC (2000) Primary: 47G10
Dedicated to the memory of Erhard Schmidtf (t) dt be the one-dimensional Hardy averaging operator. It is well-known that A is bounded on L p whenever 1 < p ≤ ∞. We improve this result in the following sense: we introduce a pair of new function spaces, the 'source' space Sp, which is strictly larger than L p , and the 'target' space Tp, which is strictly smaller than L p , and prove that A is bounded from Sp into Tp. Moreover, we show that this result cannot be improved within the environment of solid Banach spaces. We present applications of this result to variable-exponent Lebesgue spaces L p(x) .
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