In
mathematics,
singular integral operators In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator
: T(f)(x) = \int K(x,y)f(y) \, dy,
w ...
of convolution type are the
singular integral operator In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator
: T(f)(x) = \int K(x,y)f(y) \, dy,
wh ...
s that arise on R
''n'' and T
''n'' through convolution by distributions; equivalently they are the singular integral operators that commute with translations. The classical examples in
harmonic analysis are the
harmonic conjugation operator on the circle, the
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the functi ...
on the circle and the real line, the
Beurling transform
In mathematics, singular integral operators of convolution type are the singular integral operators that arise on R''n'' and T''n'' through convolution by distributions; equivalently they are the singular integral operators that commute with transl ...
in the complex plane and the
Riesz transform In the mathematical theory of harmonic analysis, the Riesz transforms are a family of generalizations of the Hilbert transform to Euclidean spaces of dimension ''d'' > 1. They are a type of singular integral operator, meaning that they a ...
s in Euclidean space. The continuity of these operators on ''L''
2 is evident because the
Fourier transform converts them into
multiplication operators. Continuity on ''L
p'' spaces was first established by
Marcel Riesz
Marcel Riesz ( hu, Riesz Marcell ; 16 November 1886 – 4 September 1969) was a Hungarian mathematician, known for work on summation methods, potential theory, and other parts of analysis, as well as number theory, partial differential equations ...
. The classical techniques include the use of
Poisson integrals,
interpolation theory {{about, Interpolation Theory in biology, other types of interpolation, interpolation (disambiguation)
The Interpolation Theory, also known as the Intercalation Theory or the Antithetic Theory, is a theory that attempts to explain the origin of the ...
and the
Hardy–Littlewood maximal function In mathematics, the Hardy–Littlewood maximal operator ''M'' is a significant non-linear operator used in real analysis and harmonic analysis.
Definition
The operator takes a locally integrable function ''f'' : R''d'' → C and returns another ...
. For more general operators, fundamental new techniques, introduced by
Alberto Calderón
Alberto Pedro Calderón (September 14, 1920 – April 16, 1998) was an Argentinian mathematician. His name is associated with the University of Buenos Aires, but first and foremost with the University of Chicago, where Calderón and his mentor, t ...
and
Antoni Zygmund
Antoni Zygmund (December 25, 1900 – May 30, 1992) was a Polish mathematician. He worked mostly in the area of mathematical analysis, including especially harmonic analysis, and he is considered one of the greatest analysts of the 20th century. ...
in 1952, were developed by a number of authors to give general criteria for continuity on ''L
p'' spaces. This article explains the theory for the classical operators and sketches the subsequent general theory.
''L''2 theory
Hilbert transform on the circle
The theory for ''L''
2 functions is particularly simple on the circle. If ''f'' ∈ ''L''
2(T), then it has a Fourier series expansion
Hardy space
In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . I ...
''H''
2(T) consists of the functions for which the negative coefficients vanish, ''a
n'' = 0 for ''n'' < 0. These are precisely the square-integrable functions that arise as boundary values of holomorphic functions in the open unit disk. Indeed, ''f'' is the boundary value of the function
in the sense that the functions
defined by the restriction of ''F'' to the concentric circles , ''z'', = ''r'', satisfy
The orthogonal projection ''P'' of ''L''
2(T) onto H
2(T) is called the Szegő projection. It is a bounded operator on ''L''
2(T) with
operator norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Introd ...
1. By Cauchy's theorem
Thus
When ''r'' = 1, the integrand on the right-hand side has a singularity at θ = 0. The truncated Hilbert transform is defined by
where δ = , 1 – ''e''
''iε'', . Since it is defined as convolution with a bounded function, it is a bounded operator on ''L''
2(T). Now
If ''f'' is a polynomial in ''z'' then
By Cauchy's theorem the right-hand side tends to 0 uniformly as ''ε'', and hence ''δ'', tends to 0. So
uniformly for polynomials. On the other hand, if ''u''(''z'') = ''z'' it is immediate that
Thus if ''f'' is a polynomial in ''z''
−1 without constant term
:
uniformly.
Define the Hilbert transform on the circle by
Thus if ''f'' is a trigonometric polynomial
:
uniformly.
It follows that if ''f'' is any ''L''
2 function
:
in the ''L''
2 norm.
This is an immediate consequence of the result for trigonometric polynomials once it is established that the operators ''H''
''ε'' are uniformly bounded in
operator norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Introd ...
. But on
€“''Ï€'',''Ï€''
The first term is bounded on the whole of
€“Ï€,Ï€ so it suffices to show that the convolution operators ''S''
''ε'' defined by
are uniformly bounded. With respect to the orthonormal basis ''e''
''inθ'' convolution operators are diagonal and their operator norms are given by taking the supremum of the moduli of the Fourier coefficients. Direct computation shows that these all have the form
with 0 < ''a'' < ''b''. These integrals are well known to be uniformly bounded.
It also follows that, for a continuous function ''f'' on the circle, ''H''
''ε''''f'' converges uniformly to ''Hf'', so in particular pointwise. The pointwise limit is a
Cauchy principal value
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.
Formulation
Depending on the type of singularity in the integrand ...
, written
If ''f'' is just in ''L''
2 then ''H''
''ε''''f'' converges to ''Hf'' pointwise almost everywhere. In fact define the
Poisson operators on ''L''
2 functions by
for ''r'' < 1. Since these operators are diagonal, it is easy to see that ''T
rf'' tends to ''f'' in L
2 as ''r'' increases to 1. Moreover, as Lebesgue proved, ''T
rf'' also tends pointwise to ''f'' at each
Lebesgue point of ''f''. On the other hand, it is also known that ''T
rHf'' − ''H''
1 − ''r'' ''f'' tends to zero at each Lebesgue point of ''f''. Hence ''H''
1 – ''r'' ''f'' tends pointwise to ''f'' on the common Lebesgue points of ''f'' and ''Hf'' and therefore almost everywhere.
Results of this kind on pointwise convergence are proved more generally below for ''L
p'' functions using the Poisson operators and the Hardy–Littlewood maximal function of ''f''.
The Hilbert transform has a natural compatibility with orientation-preserving diffeomorphisms of the circle. Thus if ''H'' is a diffeomorphism of the circle with
then the operators
are uniformly bounded and tend in the
strong operator topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space ''H'' induced by the seminorms of the form T\mapsto\, Tx\, , as ...
to ''H''. Moreover, if ''Vf''(''z'') = ''f''(''H''(''z'')), then ''VHV''
−1 − ''H'' is an operator with smooth kernel, so a
Hilbert–Schmidt operator
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm
\, A\, ^2_ \ \stackrel\ \sum_ \, Ae_i\, ...
.
In fact if ''G'' is the inverse of ''H'' with corresponding function ''g''(''θ''), then
Since the kernel on the right hand side is smooth on T × T, it follows that the operators on the right hand side are uniformly bounded and hence so too are the operators ''H''
''ε''''h''. To see that they tend strongly to ''H'', it suffices to check this on trigonometric polynomials. In that case
In the first integral the integrand is a trigonometric polynomial in ''z'' and ζ and so the integral is a trigonometric polynomial in ''ζ''. It tends in ''L''
2 to the trigonometric polynomial
The integral in the second term can be calculated by the
principle of the argument. It tends in ''L''
2 to the constant function 1, so that
where the limit is in ''L''
2. On the other hand, the right hand side is independent of the diffeomorphism. Since for the identity diffeomorphism, the left hand side equals ''Hf'', it too equals ''Hf'' (this can also be checked directly if ''f'' is a trigonometric polynomial). Finally, letting ε → 0,
The direct method of evaluating Fourier coefficients to prove the uniform boundedness of the operator ''H''
''ε'' does not generalize directly to ''L
p'' spaces with 1 < ''p'' < ∞. Instead a direct comparison of ''H''
''ε''''f'' with the
Poisson integral of the Hilbert transform is used classically to prove this. If ''f'' has Fourier series
its Poisson integral is defined by
where the
Poisson kernel
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the deriva ...
''K''
''r'' is given by
In ''f'' is in L
''p''(T) then the operators ''P''
''r'' satisfy
In fact the ''K''
''r'' are positive so
Thus the operators ''P
r'' have operator norm bounded by 1 on ''L
p''. The convergence statement above follows by continuity from the result for trigonometric polynomials, where it is an immediate consequence of the formula for the Fourier coefficients of ''K''
''r''.
The uniform boundedness of the operator norm of ''H''
''ε'' follows because ''HP
r'' − ''H''
1−''r'' is given as convolution by the function ''ψ''
''r'', where
for 1 − ''r'' ≤ , θ, ≤ ''π'', and, for , θ, < 1 − ''r'',
These estimates show that the ''L''
1 norms ∫ , ψ
''r'', are uniformly bounded. Since ''H'' is a bounded operator, it follows that the operators ''H''
ε are uniformly bounded in operator norm on ''L''
2(T). The same argument can be used on ''L
p''(T) once it is known that the Hilbert transform ''H'' is bounded in operator norm on ''L
p''(T).
Hilbert transform on the real line
As in the case of the circle, the theory for ''L''
2 functions is particularly easy to develop. In fact, as observed by Rosenblum and Devinatz, the two Hilbert transforms can be related using the Cayley transform.
The Hilbert transform ''H''
R on L
2(R) is defined by
where the
Fourier transform is given by
Define the Hardy space H
2(R) to be the closed subspace of ''L''
2(R) consisting of functions for which the Fourier transform vanishes on the negative part of the real axis. Its orthogonal complement is given by functions for which the Fourier transform vanishes on the positive part of the real axis. It is the complex conjugate of H
2(R). If ''P''
R is the orthogonal projection onto H
2(R), then
The Cayley transform
carries the extended real line onto the circle, sending the point at ∞ to 1, and the upper halfplane onto the unit disk.
Define the unitary operator from ''L''
2(T) onto ''L''
2(R) by
This operator carries the Hardy space of the circle H
2(T) onto H
2(R). In fact for , ''w'', < 1, the linear span of the functions
is dense in H
2(T). Moreover,
where
On the other hand, for ''z'' ∈ H, the linear span of the functions
is dense in ''L''
2((0,∞)). By the
Fourier inversion formula, they are the Fourier transforms of
so the linear span of these functions is dense in H
2(R). Since ''U'' carries the ''f''
w's onto multiples of the ''h''
z's, it follows that ''U'' carries H
2(T) onto H
2(R). Thus
In , part of the L
2 theory on the real line and the upper halfplane is developed by transferring the results from the circle and the unit disk. The natural replacements for concentric circles in the disk are lines parallel to the real axis in H. Under the Cayley transform, these correspond to circles in the disk that are tangent to the unit circle at the point one. The behaviour of functions in H
2(T) on these circles is part of the theory of
Carleson measures. However, the theory of singular integrals can be developed more easily by working directly on R.
H
2(R) consists exactly of L
2 functions ''f'' that arise of boundary values of holomorphic functions on H in the following sense: ''f'' is in H
2 provided that there is a holomorphic function ''F''(''z'') on H such that the functions ''f
y''(''x'') = ''f''(''x'' + ''iy'') for ''y'' > 0 are in L
2 and ''f
y'' tends to ''f'' in L
2 as ''y'' → 0. In this case ''F'' is necessarily unique and given by
Cauchy's integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary ...
:
In fact, identifying H
2 with ''L''
2(0,∞) via the Fourier transform, for ''y'' > 0 multiplication by ''e''
−''yt'' on ''L''
2(0,∞) induces a contraction semigroup ''V''
''y'' on H
2. Hence for ''f'' in L
2
If ''f'' is in H
2, ''F''(''z'') is holomorphic for Im ''z'' > 0, since the family of L
2 functions ''g
z'' depends holomorphically on ''z''. Moreover, ''f''
''y'' = ''V
yf'' tends to ''f'' in ''H''
2 since this is true for the Fourier transforms. Conversely if such an ''F'' exists, by Cauchy's integral theorem and the above identity applied to ''f''
''y''
for ''t'' > 0. Letting ''t'' tend to ''0'', it follows that ''Pf
y'' = ''f
y'', so that ''f
y'' lies in H
2. But then so too does the limit ''f''. Since
uniqueness of ''F'' follows from
For ''f'' in L
2, the truncated Hilbert transforms are defined by
The operators ''H''
''ε'',''R'' are convolutions by bounded functions of compact support, so their operator norms are given by the uniform norm of their Fourier transforms. As before the absolute values have the form
with 0 < ''a'' < ''b'', so the operators ''H''
''ε'',''R'' are uniformly bounded in operator norm. Since ''H''
''ε'',''R''''f'' tends to ''H''
ε''f'' in ''L''
2 for ''f'' with compact support, and hence for arbitrary ''f'', the operators ''H''
''ε'' are also uniformly bounded in operator norm.
To prove that ''H''
ε ''f'' tends to ''Hf'' as ''ε'' tends to zero, it suffices to check this on a dense set of functions. On the other hand,
so it suffices to prove that ''H''
ε''f'' tends to ''if'' for a dense set of functions in H
2(R), for example the Fourier transforms of smooth functions ''g'' with compact support in (0,∞). But the Fourier transform ''f'' extends to an entire function ''F'' on C, which is bounded on Im(''z'') ≥ 0. The same is true of the derivatives of ''g''. Up to a scalar these correspond to multiplying ''F''(''z'') by powers of ''z''. Thus ''F'' satisfies a
Payley-Wiener estimate for Im(''z'') ≥ 0:
for any ''m'', ''N'' ≥ 0. In particular, the integral defining ''H''
''ε''''f''(''x'') can be computed by taking a standard semicircle contour centered on ''x''. It consists of a large semicircle with radius ''R'' and a small circle radius ε with the two portions of the real axis between them. By Cauchy's theorem, the integral round the contour is zero. The integral round the large contour tends to zero by the Paley-Wiener estimate. The integral on the real axis is the limit sought. It is therefore given as minus the limit on the small semicircular contour. But this is the limit of
Where Γ is the small semicircular contour, oriented anticlockwise. By the usual techniques of contour integration, this limit equals ''if''(''x''). In this case, it is easy to check that the convergence is dominated in ''L''
2 since
so that convergence is dominated by
which is in ''L''
2 by the Paley-Wiener estimate.
It follows that for ''f'' on ''L''
2(R)
This can also be deduced directly because, after passing to Fourier transforms, ''H''
''ε'' and ''H'' become multiplication operators by uniformly bounded functions. The multipliers for ''H''
''ε'' tend pointwise almost everywhere to the multiplier for ''H'', so the statement above follows from the
dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary t ...
applied to the Fourier transforms.
As for the Hilbert transform on the circle, ''H''
ε''f'' tends to ''Hf'' pointwise almost everywhere if ''f'' is an L
2 function. In fact, define the
Poisson operators on L
2 functions by
where the Poisson kernel is given by
for ''y'' > 0. Its Fourier transform is
from which it is easy to see that ''T
yf'' tends to ''f'' in ''L''
2 as ''y'' increases to 0. Moreover, as Lebesgue proved, ''T
yf'' also tends pointwise to ''f'' at each
Lebesgue point of ''f''. On the other hand, it is also known that ''T
yHf'' – ''H
yf'' tends to zero at each Lebesgue point of ''f''. Hence ''H''
''ε''''f'' tends pointwise to ''f'' on the common Lebesgue points of ''f'' and ''Hf'' and therefore almost everywhere. The absolute values of the functions ''T
yf'' − ''f'' and ''T
yHf'' – ''H
yf'' can be bounded pointwise by multiples of the maximal function of ''f''.
As for the Hilbert transform on the circle, the uniform boundedness of the operator norms of ''H''
''ε'' follows from that of the ''T''
''ε'' if ''H'' is known to be bounded, since ''HT''
''ε'' − ''H''
''ε'' is the convolution operator by the function
The ''L''
1 norms of these functions are uniformly bounded.
Riesz transforms in the complex plane
The complex Riesz transforms ''R'' and ''R''* in the complex plane are the unitary operators on ''L''
2(C) defined as multiplication by ''z''/, ''z'', and its conjugate on the Fourier transform of an ''L''
2 function ''f'':
Identifying C with R
2, ''R'' and ''R''* are given by
where ''R''
1 and ''R''
2 are the Riesz transforms on R
2 defined below.
On ''L''
2(C), the operator ''R'' and its integer powers are unitary. They can also be expressed as singular integral operators:
where
Defining the truncated higher Riesz transforms as
these operators can be shown to be uniformly bounded in operator norm. For odd powers this can be deduced by the method of rotation of Calderón and Zygmund, described below. If the operators are known to be bounded in operator norm it can also be deduced using the Poisson operators.
The Poisson operators ''T''
''s'' on R
2 are defined for ''s'' > 0 by
They are given by convolution with the functions
''P''
''s'' is the Fourier transform of the function ''e''
− ''s'', ''x'', , so under the Fourier transform they correspond to multiplication by these functions and form a contraction semigroup on L
2(R
2). Since ''P''
''y'' is positive and integrable with integral 1, the operators ''T''
''s'' also define a contraction semigroup on each L
''p'' space with 1 < ''p'' < ∞.
The higher Riesz transforms of the Poisson kernel can be computed:
for ''k'' ≥ 1 and the complex conjugate for − ''k''. Indeed, the right hand side is a harmonic function ''F''(''x'',''y'',''s'') of three variable and for such functions
As before the operators
are given by convolution with integrable functions and have uniformly bounded operator norms. Since the Riesz transforms are unitary on L
2(C), the uniform boundedness of the truncated Riesz transforms implies that they converge in the strong operator topology to the corresponding Riesz transforms.
The uniform boundedness of the difference between the transform and the truncated transform can also be seen for odd ''k'' using the Calderón-Zygmund method of rotation. The group T acts by rotation on functions on C via
This defines a unitary representation on L
2(C) and the unitary operators ''R''
θ commute with the Fourier transform. If ''A'' is a bounded operator on L
2(R) then it defines a bounded operator ''A''
(1) on
L
2(C) simply by making ''A'' act on the first coordinate. With the identification L
2(R
2) = L
2(R) ⊗ L
2(R), ''A''
(1) = ''A'' ⊗ ''I''. If φ is a continuous function on the circle then a new operator can be defined by
This definition is understood in the sense that
for any ''f'', ''g'' in L
2(C). It follows that
Taking ''A'' to be the Hilbert transform ''H'' on ''L''
2(R) or its truncation ''H''
ε, it follows that
Taking adjoints gives a similar formula for ''R*'' and its truncation. This gives a second way to verify estimates of the norms of ''R'', ''R''* and their truncations. It has the advantage of being applicable also for ''L
p'' spaces.
The Poisson operators can also be used to show that the truncated higher Riesz transforms of a function tend to the higher Riesz transform at the common Lebesgue points of the function and its transform. Indeed, (''R
kT''
''ε'' − ''R''
(''k'')''ε'')''f'' → 0 at each Lebesgue point of ''f''; while (''R
k'' − ''R
kT''
''ε'')''f'' → 0 at each Lebesgue point of ''R
kf''.
Beurling transform in the complex plane
Since
the Beurling transform ''T'' on ''L''
2 is the unitary operator equal to ''R''
2. This relation has been used classically in and to establish the continuity properties of ''T'' on ''L
p'' spaces. The results on the Riesz transform and its powers show that ''T'' is the limit in the strong operator topology of the truncated operators
Accordingly, ''Tf'' can be written as a Cauchy principal value integral:
From the description of ''T'' and ''T''* on Fourier transforms, it follows that if ''f'' is smooth of compact support
Like the Hilbert transform in one dimension, the Beurling transform has a compatibility with conformal changes of coordinate. Let Ω be a bounded region in C with smooth boundary ∂Ω and let φ be a univalent holomorphic map of the
unit disk ''D'' onto Ω extending to a smooth diffeomorphism of the circle onto ∂Ω. If χ
Ω is the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
of Ω, the operator can ''χ''
Ω''Tχ''
Ω defines an operator ''T''(Ω) on ''L''
2(Ω). Through the conformal map ''φ'', it induces an operator, also denoted ''T''(Ω), on L
2(''D'') which can be compared with ''T''(''D''). The same is true of the truncations ''T''
''ε''(Ω) and ''T''
''ε''(''D'').
Let ''U''
''ε'' be the disk , ''z'' − ''w'', < ε and ''V''
ε the region , φ(''z'') − φ(''w''), < ''ε''. On ''L''
2(''D'')
and the operator norms of these truncated operators are uniformly bounded. On the other hand, if
then the difference between this operator and ''T''
''ε''(Ω) is a truncated operator with smooth kernel ''K''(''w'',''z''):
So the operators ''T′''
ε(''D'') must also have uniformly bounded operator norms. To see that their difference tends to 0 in the strong operator topology, it is enough to check this for ''f'' smooth of compact support in ''D''. By Green's theorem
All four terms on the right hand side tend to 0. Hence the difference ''T''(Ω) − ''T''(''D'') is the
Hilbert–Schmidt operator
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm
\, A\, ^2_ \ \stackrel\ \sum_ \, Ae_i\, ...
with kernel ''K''.
For pointwise convergence there is simple argument due to showing that the truncated integrals converge to ''Tf'' precisely at its Lebesgue points, that is almost everywhere. In fact ''T'' has the following symmetry property for ''f'', ''g'' ∈ ''L''
2(C)
On the other hand, if ''χ'' is the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
of the disk ''D''(''z'',ε) with centre ''z'' and radius ''ε'', then
Hence
By the
Lebesgue differentiation theorem
In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages taken about the point. The theorem is named for ...
, the right-hand side converges to ''Tf'' at the Lebesgue points of ''Tf''.
Riesz transforms in higher dimensions
For ''f'' in the Schwartz space of R
''n'', the ''j''th Riesz transform is defined by
where
Under the Fourier transform:
Thus ''R
j'' corresponds to the operator ∂
''j''Δ
−1/2, where Δ = −∂
12 − ⋯ −∂
''n''2 denotes the Laplacian on R
''n''. By definition ''R
j'' is a bounded and skew-adjoint operator for the ''L''
2 norm and
The corresponding truncated operators
are uniformly bounded in the operator norm. This can either be proved directly or can be established by the Calderón−Zygmund method of rotations for the group SO(''n''). This expresses the operators ''R
j'' and their truncations in terms of the Hilbert transforms in one dimension and its truncations. In fact if ''G'' = SO(''n'') with normalised Haar measure and ''H''
(1) is the Hilbert transform in the first coordinate, then
where ''φ''(''g'') is the (1,''j'') matrix coefficient of ''g''.
In particular for ''f'' ∈ ''L''
2, ''R''
''j'',ε''f'' → ''R
jf'' in ''L''
2. Moreover, ''R''
''j'',ε''f'' tends to ''R
j'' almost everywhere. This can be proved exactly as for the Hilbert transform by using the Poisson operators defined on ''L''
2(R
''n'') when R
''n'' is regarded as the boundary of a halfspace in R
''n''+1. Alternatively it can be proved directly from the result for the Hilbert transform on R using the expression of ''R
j'' as an integral over ''G''.
The Poisson operators ''T
y'' on R
''n'' are defined for ''y'' > 0 by
They are given by convolution with the functions
''P''
''y'' is the Fourier transform of the function ''e''
−''y'', ''x'', , so under the Fourier transform they correspond to multiplication by these functions and form a contraction semigroup on L
2(R
''n''). Since ''P
y'' is positive and integrable with integral 1, the operators ''T''
''y'' also define a contraction semigroup on each ''L
p'' space with 1 < ''p'' < ∞.
The Riesz transforms of the Poisson kernel can be computed
The operator ''R
jT''
''ε'' is given by convolution with this function. It can be checked directly that the operators ''R
jT''
''ε'' − ''R''
''j'',''ε'' are given by convolution with functions uniformly bounded in ''L''
1 norm. The operator norm of the difference is therefore uniformly bounded. We have (''R
jT''
''ε'' − ''R''
''j'',''ε'')''f'' → 0 at each Lebesgue point of ''f''; while (''R
j'' − ''R
jT''
''ε'')''f'' → 0 at each Lebesgue point of ''R
jf''. So ''R''
''j'',''ε''''f'' → ''R
jf'' on the common Lebesgue points of ''f'' and ''R
jf''.
''Lp'' theory
Elementary proofs of M. Riesz theorem
The theorem of
Marcel Riesz
Marcel Riesz ( hu, Riesz Marcell ; 16 November 1886 – 4 September 1969) was a Hungarian mathematician, known for work on summation methods, potential theory, and other parts of analysis, as well as number theory, partial differential equations ...
asserts that singular integral operators that are continuous for the norm are also continuous in the norm for and that the operator norms vary continuously with .
Bochner's proof for Hilbert transform on the circle
Once it is established that the operator norms of the Hilbert transform on are bounded for even integers, it follows from the
Riesz–Thorin interpolation theorem and duality that they are bounded for all with and that the norms vary continuously with . Moreover, the arguments with the Poisson integral can be applied to show that the truncated Hilbert transforms are uniformly bounded in operator norm and converge in the strong operator topology to .
It is enough to prove the bound for real trigonometric polynomials without constant term:
Since ''f'' + ''iHf'' is a polynomial in without constant term
Hence, taking the real part and using
Hölder's inequality:
So the M. Riesz theorem follows by induction for an even integer and hence for all with .
Cotlar's proof for Hilbert transform on the line
Once it is established that the operator norms of the Hilbert transform on are bounded when is a power of 2, it follows from the
Riesz–Thorin interpolation theorem and duality that they are bounded for all with and that the norms vary continuously with . Moreover, the arguments with the Poisson integral can be applied to show that the truncated Hilbert transforms are uniformly bounded in operator norm and converge in the strong operator topology to .
It is enough to prove the bound when ''f'' is a Schwartz function. In that case the following identity of Cotlar holds:
In fact, write ''f'' = ''f''
+ + ''f''
− according to the eigenspaces of ''H''. Since ''f'' ± ''iHf'' extend to holomorphic functions in the upper and lower half plane, so too do their squares. Hence
(Cotlar's identity can also be verified directly by taking Fourier transforms.)
Hence, assuming the M. Riesz theorem for ,
Since
for sufficiently large, the M. Riesz theorem must also hold for .
Exactly the same method works for the Hilbert transform on the circle. The same identity of Cotlar is easily verified on trigonometric polynomials ''f'' by writing them as the sum of the terms with non-negative and negative exponents, i.e. the eigenfunctions of . The bounds can therefore be established when is a power of 2 and follow in general by interpolation and duality.
Calderón–Zygmund method of rotation
The method of rotation for Riesz transforms and their truncations applies equally well on spaces for . Thus these operators can be expressed in terms of the Hilbert transform on and its truncations. The integration of the functions from the group or into the space of operators on is taken in the weak sense:
where ''f'' lies in and lies in the
dual space with . It follows that Riesz transforms are bounded on and that the differences with their truncations are also uniformly bounded. The continuity of the norms of a fixed Riesz transform is a consequence of the
Riesz–Thorin interpolation theorem.
Pointwise convergence
The proofs of pointwise convergence for Hilbert and Riesz transforms rely on the
Lebesgue differentiation theorem
In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages taken about the point. The theorem is named for ...
, which can be proved using the
Hardy-Littlewood maximal function. The techniques for the simplest and best-known case, namely the Hilbert transform on the circle, are a prototype for all the other transforms. This case is explained in detail here.
Let ''f'' be in L
p(T) for ''p'' > 1. The Lebesgue differentiation theorem states that
for almost all ''x'' in T. The points at which this holds are called the Lebesgue points of ''f''. Using this theorem it follows that if ''f'' is an integrable function on the circle, the Poisson integral ''T
rf'' tends pointwise to ''f'' at each
Lebesgue point of ''f''. In fact, for ''x'' fixed, ''A''(''ε'') is a continuous function on . Continuity at 0 follows because ''x'' is a Lebesgue point and elsewhere because, if ''h'' is an integrable function, the integral of , h, on intervals of decreasing length tends to 0 by
Hölder's inequality.
Letting ''r'' = 1 − ''ε'', the difference can be estimated by two integrals:
The Poisson kernel has two important properties for ''ε'' small
The first integral is bounded by ''A''(''ε'') by the first inequality so tends to zero as ''ε'' goes to 0; the second integral tends to 0 by the second inequality.
The same reasoning can be used to show that ''T''
1 − ε''Hf'' – ''H''
''ε''''f'' tends to zero at each Lebesgue point of ''f''. In fact the operator ''T''
1 − ''ε''''Hf'' has kernel ''Q
r'' + ''i'', where the conjugate Poisson kernel ''Q''
''r'' is defined by
Hence
The conjugate Poisson kernel has two important properties for ε small
Exactly the same reasoning as before shows that the two integrals tend to 0 as ε → 0.
Combining these two limit formulas it follows that ''H''
ε''f'' tends pointwise to ''Hf'' on the common Lebesgue points of ''f'' and ''Hf'' and therefore almost everywhere.
Maximal functions
Much of the ''L
p'' theory has been developed using maximal functions and maximal transforms. This approach has the advantage that it also extends to L
1 spaces in an appropriate "weak" sense and gives refined estimates in ''L
p'' spaces for ''p'' > 1. These finer estimates form an important part of the techniques involved in
Lennart Carleson
Lennart Axel Edvard Carleson (born 18 March 1928) is a Swedish mathematician, known as a leader in the field of harmonic analysis. One of his most noted accomplishments is his proof of Lusin's conjecture. He was awarded the Abel Prize in 2006 fo ...
's solution in 1966 of
Lusin's conjecture that the Fourier series of L
2 functions converge almost everywhere. In the more rudimentary forms of this approach, the L
2 theory is given less precedence: instead there is more emphasis on the L
1 theory, in particular its measure-theoretic and probabilistic aspects; results for other ''L
p'' spaces are deduced by a form of
interpolation between L
1 and L
∞ spaces. The approach is described in numerous textbooks, including the classics and . Katznelson's account is followed here for the particular case of the Hilbert transform of functions in L
1(T), the case not covered by the development above.
F. Riesz
Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a Hungarian people, HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, ...
's proof of convexity, originally established by
Hardy, is established directly without resorting to
Riesz−Thorin interpolation.
If ''f'' is an L
1 function on the circle its maximal function is defined by
''f''* is finite almost everywhere and is of weak L
1 type. In fact for λ > 0 if
then
where ''m'' denotes Lebesgue measure.
The Hardy−Littlewood inequality above leads to a proof that almost every point ''x'' of T is a
Lebesgue point of an integrable function ''f'', so that
In fact, let
If ''g'' is continuous, then the ''ω''(''g'') =0, so that ''ω''(''f'' − ''g'') = ''ω''(''f''). On the other hand, ''f'' can be approximated arbitrarily closely in L
1 by continuous ''g''. Then, using
Chebychev's inequality,
The right-hand side can be made arbitrarily small, so that ω(''f'') = 0 almost everywhere.
The Poisson integrals of an L
1 function ''f'' satisfy
It follows that ''T''
''r'' ''f'' tends to ''f'' pointwise almost everywhere. In fact let
If ''g'' is continuous, then the difference tends to zero everywhere, so Ω(''f'' − ''g'') = Ω(''f''). On the other hand, ''f'' can be approximated arbitrarily closely in L
1 by continuous ''g''. Then, using
Chebychev's inequality,
The right-hand side can be made arbitrarily small, so that Ω(''f'') = 0 almost everywhere. A more refined argument shows that convergence occurs at each Lebesgue point of ''f''.
If ''f'' is integrable the conjugate Poisson integrals are defined and given by convolution by the kernel ''Q''
''r''. This defines ''Hf'' inside , ''z'', < 1. To show that ''Hf'' has a radial limit for almost all angles, consider
where ''f''(''z'') denotes the extension of ''f'' by Poisson integral. ''F'' is holomorphic in the unit disk with , ''F''(''z''), ≤ 1. The restriction of ''F'' to a countable family of concentric circles gives a sequence of functions in L
∞(T) which has a weak ''g'' limit in L
∞(T) with Poisson integral ''F''. By the L
2 results, ''g'' is the radial limit for almost all angles of ''F''. It follows that ''Hf''(''z'') has a radial limit almost everywhere. This is taken as the definition of ''Hf'' on T, so that ''T''
''r''''H'' f tends pointwise to ''H'' almost everywhere. The function ''Hf'' is of weak L
1 type.
The inequality used above to prove pointwise convergence for L
''p'' function with 1 < ''p'' < ∞ make sense for L
1 functions by invoking the maximal function. The inequality becomes
Let
If ''g'' is smooth, then the difference tends to zero everywhere, so ω(''f'' − ''g'') = ''ω''(''f''). On the other hand, ''f'' can be approximated arbitrarily closely in ''L''
1 by smooth ''g''. Then
The right hand side can be made arbitrarily small, so that ''ω''(''f'') = 0 almost everywhere. Thus the difference for ''f'' tends to zero almost everywhere. A more refined argument can be given to show that, as in case of ''L''
''p'', the difference tends to zero at all Lebesgue points of ''f''. In combination with the result for the conjugate Poisson integral, it follows that, if ''f'' is in L
1(T), then ''H''
''ε''''f'' converges to ''Hf'' almost everywhere, a theorem originally proved by Privalov in 1919.
General theory
introduced general techniques for studying singular integral operators of convolution type. In Fourier transform the operators are given by multiplication operators. These will yield bounded operators on L
2 if the corresponding multiplier function is bounded. To prove boundedness on L
''p'' spaces, Calderón and Zygmund introduced a method of decomposing L
1 functions, generalising the
rising sun lemma of
F. Riesz
Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a Hungarian people, HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, ...
. This method showed that the operator defined a continuous operator from L
1 to the space of functions of weak L
1. The
Marcinkiewicz interpolation theorem and duality then implies that the singular integral operator is bounded on all L
''p'' for 1 < ''p'' < ∞. A simple version of this theory is described below for operators on R. As showed, results on R can be deduced from corresponding results for T by restricting the multiplier to the integers, or equivalently periodizing the kernel of the operator. Corresponding results for the circle were originally established by Marcinkiewicz in 1939. These results generalize to R
''n'' and T
''n''. They provide an alternative method for showing that the Riesz transforms, the higher Riesz transforms and in particular the Beurling transform define bounded operators on L
''p'' spaces.
Calderón-Zygmund decomposition
Let ''f'' be a non-negative integrable or continuous function on
'a'',''b'' Let ''I'' = (''a'',''b''). For any open subinterval ''J'' of
'a'',''b'' let ''f''
''J'' denote the average of , ''f'', over ''J''. Let α be a positive constant greater than ''f''
''I''. Divide ''I'' into two equal intervals (omitting the midpoint). One of these intervals must satisfy ''f''
''J'' < α since their sum is 2''f''
''I'' so less than 2α. Otherwise the interval will satisfy α ≤ ''f''
''J'' < 2α. Discard such intervals and repeat the halving process with the remaining interval, discarding intervals using the same criterion. This can be continued indefinitely. The discarded intervals are disjoint and their union is an open set Ω. For points ''x'' in the complement, they lie in a nested set of intervals with lengths decreasing to 0 and on each of which the average of ''f'' is bounded by α. If ''f'' is continuous these averages tend to , ''f''(''x''), . If ''f'' is only integrable this is only true almost everywhere, for it is true at the
Lebesgue points of ''f'' by the
Lebesgue differentiation theorem
In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages taken about the point. The theorem is named for ...
. Thus ''f'' satisfies , ''f''(''x''), ≤ ''α'' almost everywhere on Ω
''c'', the complement of Ω. Let ''J''
''n'' be the set of discarded intervals and define the "good" function ''g'' by
By construction , ''g''(''x''), ≤ 2''α'' almost everywhere and
Combining these two inequalities gives
Define the "bad" function ''b'' by ''b'' = ''f'' − ''g''. Thus ''b'' is 0 off Ω and equal to ''f'' minus its average on ''J''
''n''. So the average of ''b'' on ''J''
''n'' is zero and
Moreover, since , ''b'', ≥ ''α'' on Ω
The decomposition
is called the Calderón–Zygmund decomposition.
Multiplier theorem
Let ''K''(''x'') be a kernel defined on R\ such that
exists as a
tempered distribution
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
for ''f'' a
Schwartz function. Suppose that the Fourier transform of ''T'' is bounded, so that convolution by ''W'' defines a bounded operator ''T'' on L
2(R). Then if ''K'' satisfies Hörmander's condition
then ''T'' defines a bounded operator on L
''p'' for 1 < ''p'' < ∞ and a continuous operator from L
1 into functions of weak type L
1.
In fact by the Marcinkiewicz interpolation argument and duality, it suffices to check that if ''f'' is smooth of compact support then
Take a Calderón−Zygmund decomposition of ''f'' as above
with intervals ''J''
''n'' and with ''α'' = ''λμ'', where ''μ'' > 0. Then
The term for ''g'' can be estimated using
Chebychev's inequality:
If ''J''* is defined to be the interval with the same centre as ''J'' but twice the length, the term for ''b'' can be broken up into two parts:
The second term is easy to estimate:
To estimate the first term note that
Thus by Chebychev's inequality:
By construction the integral of ''b
n'' over ''J
n'' is zero. Thus, if ''y
n'' is the midpoint of ''J
n'', then by Hörmander's condition:
Hence
Combining the three estimates gives
The constant is minimized by taking
The Markinciewicz interpolation argument extends the bounds to any ''L''
''p'' with 1 < ''p'' < 2 as follows. Given ''a'' > 0, write
where ''f''
''a'' = ''f'' if , ''f'', < ''a'' and 0 otherwise and ''f''
''a'' = ''f'' if , ''f'', ≥ ''a'' and 0 otherwise. Then by Chebychev's inequality and the weak type ''L''
1 inequality above
Hence
By duality
Continuity of the norms can be shown by a more refined argument
or follows from the
Riesz–Thorin interpolation theorem.
Notes
References
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*{{citation, last=Zygmund, first= Antoni, authorlink=Antoni Zygmund, title=Intégrales singulières, series=Lecture Notes in Mathematics, volume= 204, publisher=Springer-Verlag, year= 1971
Operator theory
Harmonic analysis
Singular integrals