On the Action of Riesz Transforms on the Class of Bounded Functions
Adam Osekowski
The paper is devoted to the d-dimensional extension of the classical identity of Stein and Weiss concerning the action of the Hilbert transform on characteristic functions. Let ( R j ) j=1 be the collection of Riesz transforms in Rd . For 1 p < , d we determine the least constants c p,d , C p,d such that
-
f (x )| R j f (x )| pdx c p,d || f ||L1(Rd ),
(1 f (x ))| R j f (x )| pdx C p,d || f ||L1(Rd )
for any Borel function f : Rd [0, 1]. The proof rests on probabilistic methods and
the construction of appropriate harmonic functions on [0, 1] R.
1 Introduction
The Hilbert transform H on the line is the operator defined by the principal value
integral
To accomplish this, Laeng proved that
Let E R be a measurable set of finite Lebesgue measure. A classical result of
Stein and Weiss [19] asserts that the distribution function of HE depends only on
the measure of E and is non-sensitive to the inner structure of E . Precisely, for any
t > 0 we have the identity
This fact can be proved using various tools; for different approaches, consult Stein and
Weiss [19,20], Caldern [5] and the two recent papers: [6] by Colzani et al. and [14]
by Laeng. In fact, in the latter paper a stronger statement, concerning the behavior of
HE restricted to E and R \ E , was established. Namely, it was shown that for any
t > 0,
1 < p < ,
and combined it with the fact that any two functions f, g, which have the same p-th
norms for p lying in some interval ( p1, p2), are equidistributed (cf. [6]).
The principal goal of this paper is to provide another proof of the identities (1.5)
and (1.6), with the use of probabilistic methods: the identities will be deduced from
their novel counterparts in martingale theory. In fact, it will allow us to study the more
general question concerning the action of the Hilbert transform on the class of bounded
1
1 p < ,
pt p1
pt p1
functions. Our approach easily leads to appropriate higher-dimensional results, to
formulate which we require some additional background. Suppose that d 1 is a
given integer. The counterpart of the Hilbert transform in Rd is the collection of Riesz
d
transforms (R j ) j=1 (see e.g. Stein [18]). This family of singular integral operators is
given by
x j y j
|x y|d+1
f (y) dy, j = 1, 2, . . . , d.
R j f ( ) = j f( ), for Rd \{0}.
| |
We are ready to formulate the main result.
Theorem 1.1 Let d 1 be a given integer and suppose that f is a Borel function on
Rd taking values in the interval [0, 1]. Then for any j {1, 2, . . . , d} we have
f (x )|R j f (x )| p dx 2
dt || f ||L1(Rd ), 1 p < ,
pt p1
pt p1
(1 f (x ))|R j f (x )| p dx 2
dt || f ||L1(Rd ), 1 < p < . (1.9)
Both inequalities are sharp for each j and d. They are already sharp if f is assumed
to run over the class of characteristic functions of measurable sets.
Here by sharpness we mean that neither of the constants 2 0 epttp+11 dt, 2 0 epttp11 dt
in (1.8) and (1.9) can be replaced by a smaller number. One easily shows (cf. [14])
that these constants can be expressed in terms of Gamma and Riemann zeta functions:
we have
pt p1
pt p1
If we put f = E for a given Borel subset E Rd with |E | < , we obtain the
following d-dimensional analogues of (1.5) and (1.6): for 1 j d,
pt p1
Adding these estimates, we get the inequality
pt p1
sinh( t ) dt |E |, 1 < p < .
We will show that this estimate is also sharp, for any values of d and j .
Let us mention here two interesting related questions. First, can the Riesz transform
R j be replaced in (1.8), (1.9) and (1.10) by its vector version R = (R1, R2, . . . , Rd )
with no change in the constant? (We interpret |R f (x )| as dj=1 |R j f (x )|2 1/2.)
Unfortunately, we have been unable to shed any light on this problem with the
methods developed here. The second question, which we also did not manage to answer,
concerns the d-dimensional version of (1.3) and (1.4). This is closely related to a
longstanding open problem of Stein concerning the weak-type (1, 1) inequality for Riesz
transforms: it is not known whether this estimate holds with a constant independent
of the dimension.
The above results can be applied to the study of the so-called re-expansion operator.
Let Fc and Fs be the cosine and sine Fourier transforms on R+, respectively. That is,
for x > 0 and any Borel function f on R+,
Fc f (x ) =
f (t ) cos t x dt,
Fs f (x ) =
f (t ) sin t x dt.
Both Fc and Fs are unitary and self-adjoint operators on L2(R+). We define the
reexpansion operator on R+ by the identity = Fs Fc. This operator is interesting
from the analytical point of view, as the object of spectral analysis and also appears
naturally in the scattering theory. For more on the subject, consult Birman [3], Gokhberg
and Krupnik [8] and Ilin [12,13].
The question about various norms of has gathered some interest in the literature.
Hollenbeck et al. [11] proved that the re-expansion operator has the same p-th norm
as the Hilbert transform: || ||L p(R+)L p(R+) = | (...truncated)