Weighted inequalities for Riemann-Liouville fractional integrals of order one and greater

Martín-Reyes, F. J.; Sawyer, E.

doi:10.1090/S0002-9939-1989-0965246-8

Weighted inequalities for Riemann-Liouville fractional integrals of order one and greater
HTML articles powered by AMS MathViewer

by F. J. Martín-Reyes and E. Sawyer PDF

Proc. Amer. Math. Soc. 106 (1989), 727-733 Request permission

Abstract:

A simple characterization is given for two-weight norm inequalities for generalized Hardy operators ${T_\varphi }f(x) = \smallint _0^x\varphi (\tfrac {t}{x})f(t)dt$, where $\varphi :(0,1) \to (0,\infty )$ is nonincreasing and satisfies $\varphi (ab) \leq D[\varphi (a) + \varphi (b)]$ for $0 < a,b < 1$. Included in particular are the Riemann-Liouville fractional integrals.

References

Kenneth F. Andersen and Benjamin Muckenhoupt, Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 72 (1982), no. 1, 9–26. MR 665888, DOI 10.4064/sm-72-1-9-26
J. Scott Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), no. 4, 405–408. MR 523580, DOI 10.4153/CMB-1978-071-7
Benjamin Muckenhoupt, Hardy’s inequality with weights, Studia Math. 44 (1972), 31–38. MR 311856, DOI 10.4064/sm-44-1-31-38

Two-weighed estimates or Riemann-Liouvile integr

39

Giorgio Talenti, Osservazioni sopra una classe di disuguaglianze, Rend. Sem. Mat. Fis. Milano 39 (1969), 171–185 (Italian, with English summary). MR 280661, DOI 10.1007/BF02924135
Giuseppe Tomaselli, A class of inequalities, Boll. Un. Mat. Ital. (4) 2 (1969), 622–631. MR 0255751