|
In many problems from analysis, the Hardy space, H1.RD/, always appears as a
suitable substitution for L1.RD/. Thanks to the seminal papers of Charles Fefferman
and EliasM. Stein, Ronald R. Coifman and GuidoWeiss, Robert H. Latter and other
mathematicians, the properties of the Hardy spaces Hp.RD/ with p 2 .0; 1, such
as the endpoint spaces as the interpolation spaces, the characterizations in terms
of various maximal functions, the atomic and the molecular decompositions, were
established in the period 1970s to 1980s. Nowadays, the analysis relating to the
Hardy spaces plays an important role in many fields of analysis, such as complex
analysis, partial differential equations, functional analysis and geometrical analysis.
On the other hand, one of the most crucial assumptions in the classical harmonic
analysis relating to the Hardy space is the doubling condition of the underlying
measures. This is because the Vitali covering lemma and the Calder´on–Zygmund
decomposition lemma—two cornerstones of the classical harmonic analysis—
essentially depend on the doubling condition of the underlying measures. For
a long time, mathematicians tried to seek a theory about function spaces and
the boundedness of operators which does not require the doubling condition on
the underlying measures. The motivations for this come from partial differential
equations, complex analysis and harmonic analysis itself. One typical example is
the singular integral operators considered in an open domain RD with the
usual D-dimensional Lebesgue measure, or on a surface with the usual surface area
measure instead of the whole space. If the boundary of is a Lipschitz surface,
then the problem can be reduced to the related problem in spaces of homogeneous
type in the sense of Ronald R. Coifman and Guido Weiss and can be solved by
the standard argument. For the domain with extremely singular boundary (or called
“wild” boundary), the results for singular integral operators with doubling measures
are not suitable anymore. Another famous examples are the so-called Painlev´e
problem and Vitushkin’s conjecture, in which the non-homogeneous T b theorem
plays a key role. To solve the Painlev´e problem, in the 1990s, mathematicians made
a great effort to establish the L2 boundedness for the Cauchy integrals with the
one-dimensional Hausdorff measure satisfying some linear growth condition on R. |
