Weighted norm inequalities

for integral operators

Authors:
Igor E. Verbitsky and Richard L. Wheeden

Journal:
Trans. Amer. Math. Soc. **350** (1998), 3371-3391

MSC (1991):
Primary 42B20, 42B25

DOI:
https://doi.org/10.1090/S0002-9947-98-02017-0

MathSciNet review:
1443202

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a large class of positive integral operators acting on functions which are defined on a space of homogeneous type with a group structure. We show that any such operator has a discrete (dyadic) version which is always essentially equivalent in norm to the original operator. As an application, we study conditions of ``testing type,'' like those initially introduced by E. Sawyer in relation to the Hardy-Littlewood maximal function, which determine when a positive integral operator satisfies two-weight weak-type or strong-type estimates. We show that in such a space it is possible to characterize these estimates by testing them only over ``cubes''. We also study some pointwise conditions which are sufficient for strong-type estimates and have applications to solvability of certain nonlinear equations.

**[C]**M. Christ,*A T(b) theorem with remarks on analytic capacity and the Cauchy integral*, Colloq. Math.**60, 61**(1990), 601-628. MR**92k:42020****[CW]**R. Coifman and G. Weiss,*Extensions of Hardy spaces and their use in analysis*, Bull. Amer. Math. Soc.**83**(1977), 569-645. MR**56:6264****[KV]**N. J. Kalton and I. E. Verbitsky,*Nonlinear equations and weighted norm inequalities*, Trans. Amer. Math. Soc. (to appear).**[MV]**V. G. Maz'ya and I. E. Verbitsky,*Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers*, Arkiv för Matem.**33**(1995), 81- 115. MR**96i:26021****[S1]**E. T. Sawyer,*A characterization of a two-weight norm inequality for maximal operators*, Studia Math.**75**(1982), 1-11. MR**84i:42032****[S2]**E. T. Sawyer,*A two weight weak type inequality for fractional integrals*, Trans. Amer. Math. Soc.**281**(1984), 339-345. MR**85j:26010****[S3]**E. T. Sawyer,*A characterization of two weight norm inequalities for fractional and Poisson integrals*, Trans. Amer. Math.**308**(1988), 533-545. MR**89d:26009****[SW]**E. T. Sawyer and R. L. Wheeden,*Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces*, Amer. J. Math.**114**(1992), 813-874. MR**94i:42024****[SWZ]**E. T. Sawyer, R. L. Wheeden, and S. Zhao,*Weighted norm inequalities for operators of potential type and fractional maximal functions*, Potential Analysis**5**(1996), 523-580. CMP**97:09****[VW]**I. E. Verbitsky and R. L. Wheeden,*Weighted trace inequalities for fractional integrals and applications to semilinear equations*, J. Funct. Analysis**129**(1995), 221-241. MR**95m:42025****[WZ]**R. L. Wheeden and S. Zhao,*Weak-type estimates for operators of potential type*, Studia Math.**119**(1996), 149-160. MR**97d:42013**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
42B20,
42B25

Retrieve articles in all journals with MSC (1991): 42B20, 42B25

Additional Information

**Igor E. Verbitsky**

Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Email:
igor@math.missouri.edu

**Richard L. Wheeden**

Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

Email:
wheeden@math.rutgers.edu

DOI:
https://doi.org/10.1090/S0002-9947-98-02017-0

Received by editor(s):
March 28, 1996

Received by editor(s) in revised form:
October 1, 1996

Additional Notes:
The first author was partially supported by NSF Grant DMS94-01493 and the second by NSF Grant DMS95-00799.

Article copyright:
© Copyright 1998
American Mathematical Society