Abstract
Under the standard assumptions on the variable exponent p (x) (log- and decay conditions), we give a characterization of the variable exponent Bessel potential space Bα [Lp (ṡ) (Rn)] in terms of the rate of convergence of the Poisson semigroup Pt. We show that the existence of the Riesz fractional derivative Dα f in the space Lp (ṡ) (Rn) is equivalent to the existence of the limit frac(1, εα) (I - Pε)α f. In the pre-limiting case supx p (x) < frac(n, α) we show that the Bessel potential space is characterized by the condition {norm of matrix} (I - Pε)α f {norm of matrix}p (ṡ) ≦ C εα.
Original language | English |
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Pages (from-to) | 483-497 |
Number of pages | 15 |
Journal | Journal of Mathematical Analysis and Applications |
Volume | 365 |
Issue number | 2 |
DOIs | |
Publication status | Published - May 15 2010 |
Externally published | Yes |
Keywords
- Bessel potential space
- Grünwald-Letnikov approach
- Hypersingular integral
- Riesz fractional derivative
- Riesz potential operator
ASJC Scopus subject areas
- Analysis
- Applied Mathematics