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Abstract
We study membership in Lipschitz classes for a class of α-harmonic functions in the open unit disc
in the complex plane. From earlier work by Olofsson and Wittsten we know that such an α-harmonic function u is the α-harmonic Poisson integral
of its boundary value function f on the unit circle
. We determine when the Poisson integral
belongs to a Lipschitz class
for the unit disc.
AMS SUBJECT CLASSIFICATIONS:
1. Introduction
Let be the open unit disc in the complex plane, let
be the unit circle, and denote by
the standard complex partial derivatives, where z=x + iy is the complex coordinate. A chief concern of this paper is the homogeneous second order partial differential equation
(1)
(1) where
(2)
(2) is a certain weighted Laplacian with weight parameter
. The restriction
on the weight parameter is needed for Equation (Equation1
(1)
(1) ) to have a sufficiently rich supply of solutions with well-behaved boundary behaviour.
A solution u of Equation (Equation1(1)
(1) ) is called an α-harmonic function. Such a function u is smooth in
but can be rather wild near the boundary
. Notice that
is one-quarter of the usual Laplacian and that a 0-harmonic function is a harmonic function in
in the usual sense. Notice also that the differential operator
has a certain singular or degenerate behaviour on the unit circle
when
. The study of weighted Laplacians of the above form (Equation2
(2)
(2) ) goes back to Garabedian [Citation1] and has attracted quite some interest recently.
Of particular interest is the Dirichlet boundary value problem
(3)
(3) for α-harmonic functions. Here
is a distribution on
and the boundary condition in (Equation3
(3)
(3) ) is interpreted in a distributional sense that
in
, where
(4)
(4) for
. From Olofsson and Wittsten [Citation2] it is known that a twice continuously differentiable function
in
solves the Dirichlet problem (Equation3
(3)
(3) ) if and only if it has the form of a Poisson integral
where
(5)
(5) is the α-harmonic Poisson kernel,
for
in accordance with (Equation4
(4)
(4) ) and
is convolution in
. For an integrable function
on
the above Poisson integral has the form
(6)
(6) where the function
is as in (Equation5
(5)
(5) ). Notice that
is the usual Poisson integral representation of a harmonic function in
. The Poisson integral representation
leads to improved convergence to boundary values (in norm as well as non-tangentially) of solutions of (Equation3
(3)
(3) ) provided the boundary data
has some appropriate regularity, see [Citation2–4] for results.
Let us recall the notion of Lipschitz continuity. For a bounded subset E of the complex plane and parameter we denote by
the set of complex-valued functions f on E such that
for some finite constant
. The class
is commonly referred to as the Lipschitz (or Hölder) class for the set E of order β. The space
is often equipped with the norm
and is then a commutative Banach algebra under pointwise multiplication of functions (see for instance Sherbert [Citation5]). It is well-known that every function f in
extends uniquely to a continuous function on the closure
of E in such a way that
. It is also well-known that
if
. The family of spaces
forms a much natural scale to measure smoothness of continuous functions.
A main purpose of this paper is to study membership in Lipschitz classes for α-harmonic functions. Observe that implies that
, where
A fundamental question is to determine when the Poisson integral
of a function
belongs to the space
. Notice that this latter question is an extension problem in the sense that we ask when the function
extends to an α-harmonic function u in the class
. We shall here solve this problem in full generality. The solution we propose depends on the individual parameters
and
as well as on spectral properties of the function
.
Our results contrast the situation in the category of continuous functions where one has that the Poisson integral extends to a continuous function on the closed disc
if
is a continuous function on
and
(see Olofsson and Wittsten [Citation2, Section 5]). The study of Lipschitz conditions
goes back to classical work of Hardy and Littlewood [Citation6, Section 5] in the case of analytic functions.
An integrable function on
is identified with the distribution
on
defined by
where
is the distributional pairing and
is the space of indefinitely differentiable test functions on
. The Fourier coefficients of a function
are defined by
for integers
, and similarly for distributions using the distributional pairing (see for instance [Citation2] for details). The conjugate function of a distribution
is the distribution
defined by the condition that
where
is the n-th Fourier coefficient of f,
for
and
. A much influential introduction to Fourier analysis is Katznelson [Citation7].
Let us first recall the situation in the classical case of usual harmonic functions in (
). If
and
, then
. Furthermore, a harmonic function u in
belongs to the class
if and only if it has the form
for some function
such that
(see Theorem 6.7). These results are well-known and follow quite easily from the work of Hardy and Littlewood [Citation6, Section 5] on analytic functions mentioned above. Recall here that the conjugate function
is a bounded linear operator on the Lipschitz space
for
(see Zygmund [Citation8, Theorem III.13.29]). Results of this latter type often go under the name of Privalov's theorem.
Let us next consider the case when and
are such that
. For such parameters we show that
belongs to
if
(see Theorem 3.2). This generalizes a classical result from the previous paragraph as well as a recent result of Li and Wang [Citation9] for
to the bigger parameter range considered; a less precise result can be found inChen [Citation10]. In fact, as we shall see later on, the inequality
defines the exact parameter range for the implication
to hold true.
Lipschitz continuity of a continuously differentiable function
in
is inferred from a gradient growth estimate of the form
(7)
(7) (see Theorem 3.1). This observation goes back to classical work of Hardy and Littlewood [Citation6] and has been elaborated on by Gehring and Martio [Citation11] and others. In the context of the previous paragraph we establish (Equation7
(7)
(7) ) by careful estimations using the Poisson integral representation
(see Theorems 2.5 and 2.10). Notably, we show that
(8)
(8) for
with
provided
(see Theorem 2.13).
It is an important observation that the combination of partial derivatives yields an estimate (Equation8
(8)
(8) ) which is valid without any extra restriction on β more than what is previously stated. In fact, this more general validity of (Equation8
(8)
(8) ) compared to (Equation7
(7)
(7) ) is quite natural from the point of view of convolution structure in (Equation6
(6)
(6) ) and an interpretation of the differential operator
as angular derivative. The growth estimate (Equation8
(8)
(8) ) is established for
with a fairly economical constant
using a precise result on the
-means of the function
from Olofsson and Wittsten [Citation2, Section 2] later refined in Olofsson [Citation3] (see Theorem 2.4). We mention that inequality (Equation8
(8)
(8) ) generalizes a recent result by Chen [Citation10, Lemma 3.2].
Let us next consider the case when and
are such that
. For such parameters we show that if u is α-harmonic and
, then u is analytic in
(see Theorem 5.4). A closely related result is that, in this parameter range, the Poisson integral
belongs to
if and only if
and
for n<0 (see Theorem 5.5). This latter result puts earlier examples of Li and Wang [Citation9] and Chen [Citation10] in a precise context.
The results in the previous paragraph are proved using properties of the homogeneous expansion of α-harmonic functions. Let and set
for positive integers
, where Γ is the standard Gamma function. We now introduce the functions
for
. From Olofsson and Wittsten [Citation2, Section 5] we know that the Poisson integral
has the series expansion
(9)
(9) for
.
We derive a system of two first-order partial differential equations in satisfied by the function
(see Proposition 4.2). This system of partial differential equations is shown to characterize the function
in the punctured disc
up to an additive term of the form
(see Theorem 4.3). As an application of this development we show that
for
(see Theorem 4.7). This latter formula refines an earlier result from Olofsson and Wittsten [Citation2, Theorem 1.9]. As a consequence we have that
if and only if
(see Lemma 5.1). Combined with some homogeneity type of arguments this leads to the results about Lipschitz continuity of α-harmonic functions in the case of big smoothness
discussed above.
Let us now consider the case when and
are such that
. The analysis of this case is the main technical contribution of this paper. We shall need the Fourier multiplier
defined by
for
, where
is the n-th Fourier coefficient of f. Denote also by
the space of essentially bounded measurable functions on
normed by the essential supremum
In this parameter range, we show that the function
belongs to the class
if and only if
is such that
(see Theorems 6.3 and 6.5).
The constraint on a Lipschitz function (
) that
belongs to
is quite demanding. An interesting example is provided by the Weierstrass function
which has the properties that
and
(see discussion following formula (Equation49
(49)
(49) )).
It is an important observation that the function above is a constant multiple of a hypergeometric function. Assuming that
belongs to the class
, we prove necessity of the condition
by a differentiation type argument using asymptotic properties of hypergeometric functions (see Theorem 6.3).
Differentiating (Equation9(9)
(9) ) using the above mentioned partial differential equations for the
's we show that
for
, where
(see Lemma 6.4). This latter formula suggests an obvious estimation of
. Combined with (Equation8
(8)
(8) ) this yields that
satisfies a growth estimate of the form (Equation7
(7)
(7) ) if
is such that
(see Theorem 6.5).
In conclusion we have shown that an inequality of the form (Equation7(7)
(7) ) is also necessary for Lipschitz continuity
of an α-harmonic function u. We point out that such an implication requires interior rigidity of the function u and is not true in general (see Proposition 3.5).
For further results on Lipschitz continuity of analytic functions in we mention Dyakonov [Citation12]. Recent papers dealing with aspects of α-harmonic functions not quoted above are [Citation4, Citation13–17]. We mention also a related interesting paper by Borichev and Hedenmalm [Citation18].
The author thanks Kevin Klein and Jens Wittsten for useful discussions.
2. Bounds for partial derivatives
In this section, we prove some growth bounds for partial derivatives of α-harmonic Poisson integrals. Applications of these bounds appear in later sections.
We first consider the -derivative.
Lemma 2.1
Let . Let
and set
. Then
for
with
and
.
Proof.
The interpretation of the Poisson integral (Equation6(6)
(6) ) as solution of the Dirichlet problem (Equation3
(3)
(3) ) leads to the property that
for
. Let
and notice that
for
. Differentiating we have that
for
. The choice
gives the conclusion of the lemma.
We now calculate .
Lemma 2.2
Let and let
be as in (Equation5
(5)
(5) ). Then
for
.
Proof.
From formula (Equation5(5)
(5) ) we have that
for
. Differentiating we have that
for
. A simplification of terms now leads to the conclusion of the lemma.
From Lemma 2.2 we have that
(10)
(10) for
.
The following lemma is well-known but included here for the sake of convenience.
Lemma 2.3
Let . Then
for
.
Proof
Sketch of proof
First show that
for x>0. A homogeneity argument then leads to the conclusion of the lemma.
The interest in Lemma 2.3 comes from the fact that the power function operates on metrics: If d is a metric on a set X, then so is the function
for
.
Recall the standard Gamma function
For the sake of easy reference we recall the following result from [Citation2, Section 2].
Theorem 2.4
Let and set
. Then
for
. Furthermore,
For a detailed discussion of Theorem 2.4 and its relation to hypergeometry we refer to [Citation3].
We next estimate the -derivative of the Poisson integral. We denote by
the space of continuous functions on
.
Theorem 2.5
Let and
be such that
. Let
be such that
where C is a finite positive constant. Set
. Then
for
where
is as in Theorem 2.4.
Proof.
Let with
and
. Recall Lemma 2.1. By the triangle inequality we have that
By the assumption on f we have that
where the last equality follows by a change of variables. We next invoke Lemma 2.2 to the extent that
(11)
(11) for
with
.
We shall now estimate the factor in the integrand in (Equation11
(11)
(11) ). By Lemma 2.3 and the triangle inequality we have that
By a geometric consideration we see that
for 0<r<1 and
.
We now return to the estimation of . By (Equation11
(11)
(11) ) and the result of the previous paragraph we have that
for
, where
. An application of Theorem 2.4 now yields the conclusion of the theorem.
We now turn our attention to the ∂-derivative of the Poisson integral.
Lemma 2.6
Let . Let
and set
. Then
for
.
Proof.
The proof of the lemma is analogous to that of Lemma 2.1 and is therefore omitted.
Recall the partial fraction decomposition
(12)
(12) of the classical Poisson kernel.
We now calculate .
Lemma 2.7
Let and let
be as in (Equation5
(5)
(5) ). Then
for
.
Proof.
Recall formulas (Equation5(5)
(5) ) and (Equation12
(12)
(12) ). Differentiating we have that
for
, where the last equality is straightforward to check.
It is evident that the quotient belongs to
for every
with
.
Lemma 2.8
We have that
for every
.
Proof.
Let . The line segment
between
and the point 1 is parametrized by
We have that
Observe that
is the disjoint union of line segments
as above. Thus
for
. This completes the proof of the lemma.
We remark in passing that
if and only if
.
We now estimate .
Proposition 2.9
Let and let
be as in (Equation5
(5)
(5) ). Then
for
. The constant
is best possible.
Proof.
From Lemma 2.7 we have that
for
, where
Thus
(13)
(13) for
.
It remains to calculate the supremum . By the triangle inequality we have that
. From Lemma 2.8 we have that
for every
. Maximizing over
we conclude that
This completes the proof of the proposition.
We now return to the α-harmonic Poisson integral.
Theorem 2.10
Let and
be such that
. Let
be such that
where C is a finite positive constant. Set
. Then
where
is as in Theorem 2.4.
Proof.
The proof parallels that of Theorem 2.5. Let with
and
. Recall Lemma 2.6. By the triangle inequality we have that
By the assumption on f we have that
where the last equality follows by a change of variables. We next invoke Proposition 2.9 to the extent that
(14)
(14) for
with
. Formula (Equation14
(14)
(14) ) should be compared with formula (Equation11
(11)
(11) ) in the proof of Theorem 2.5. We now proceed as in the proof of Theorem 2.5 to arrive at the conclusion of the theorem.
We next calculate the angular derivative of .
Theorem 2.11
Let and let
be as in (Equation5
(5)
(5) ). Then
for
.
Proof.
From Lemma 2.7 we have that
for
. Using also Lemma 2.2 we have that
(15)
(15)
for
, where the last equality is straightforward to check.
We now turn our attention to the leftmost factor in (Equation15(15)
(15) ). By a partial fraction decomposition we have that
for
. In view of the partial fraction formula (Equation12
(12)
(12) ) for the classical Poisson kernel we thus have that
(16)
(16) for
.
We return to the leftmost factor in (Equation15(15)
(15) ). We have that
(17)
(17) for
, where the last equality again follows by (Equation12
(12)
(12) ).
Recall formula (Equation15(15)
(15) ). From (Equation16
(16)
(16) ) and (Equation17
(17)
(17) ) we have that
for
. We next rewrite the leftmost factor so that
for
. This completes the proof of the theorem.
Theorem 2.11 leads to an important inequality for the angular derivative of the Poisson kernel.
Corollary 2.12
Let and let
be as in (Equation5
(5)
(5) ). Then
for
.
Proof.
Recall Theorem 2.11. The result follows by the triangle inequality.
The point of Corollary 2.12 is that the angular derivative of behaves somewhat better than
or
(see formula (Equation10
(10)
(10) ) or Proposition 2.9).
Theorem 2.13
Let and
. Let
be such that
where C is a finite positive constant. Set
. Then
where
is as in Theorem 2.4.
Proof.
The proof parallels that of Theorem 2.5. Let with
and
. Let
be the angular derivative. By Lemmas 2.1 and 2.6 we have that
By the triangle inequality we have that
By the assumption on f we have that
where the last equality follows by a change of variables. We next invoke Corollary 2.12 to the extent that
(18)
(18) for
with
.
As in the proof of Theorem 2.5 we have that for 0<r<1 and
. From (Equation18
(18)
(18) ) we have that
An application of Theorem 2.4 now yields the conclusion of the theorem.
It is important to notice that Theorem 2.13 carries no additional assumption on the Lipschitz parameter (compare Theorems 2.5 and 2.10). The case
of Theorem 2.13 is closely related to Chen [Citation10, Lemma 3.2].
3. Lipschitz continuity from gradient growth
It is well-known that Lipschitz continuity of a function
is inferred from a gradient growth estimate of the form (Equation7
(7)
(7) ). This section pertains to a discussion of this aspect of the theory.
Let . We shall consider the quantity
where the infimum is taken over all piecewice continuously differentiable curves γ from
to
in
and the integration is with respect to arclength. It is straightforward to check that the function
is a metric on
.
It is evident that is usual Euclidean distance. Furthermore, conformal invariance leads to the formula
for
, where
(see for instance [Citation19, Theorem 2.2]). The function
is known as the hyperbolic metric for
, whereas the function
is known as the pseudo hyperbolic metric for
.
We shall make use of the result that
(19)
(19) for
, where
is a finite positive constant. A careful analysis shows that the constant in (Equation19
(19)
(19) ) can be chosen of the form
, where C is an absolute constant. For proofs of (Equation19
(19)
(19) ) we refer to the original papers Gehring and Martio [Citation11, Section 2] or Hardy and Littlewood [Citation6, Section 5].
Theorem 3.1
Let be such that
(20)
(20) for some
and C>0. Then
and
for
where
is as in (Equation19
(19)
(19) ).
Proof.
Let . By the fundamental theorem of calculus we have that
where γ is a piecewise
-curve from
to
(see for instance Gamelin [Citation20, Section III.2]). By the triangle inequality we have that
where the last inequality follows by (Equation20
(20)
(20) ). Passing to an infimum over curves γ we have that
where the last equality follows by (Equation19
(19)
(19) ). This completes the proof of the theorem.
The interest in Theorem 3.1 is when the function u has some interior rigidity.
Theorem 3.2
Let and
be such that
. Let
and set
. Then
.
Proof.
Recall Theorems 2.5 and 2.10. The result now follows from Theorem 3.1.
The case of analytic functions merits particular mention.
Theorem 3.3
Let and
. Let
be such that
for n<0. Then
belongs to
.
Proof.
Recall Theorem 2.13. The spectral assumption means that in
. The result now follows from Theorem 3.1.
Remark 3.4
We point out that Theorem 3.3 can be regarded as a special case of Theorem 3.2. Indeed, if u is as in Theorem 3.3, then for all
and an application of Theorem 3.2 with
yields that
.
Theorem 3.3 goes back to Hardy and Littlewood [Citation6, Section 5].
Let us revisit a classic function
(21)
(21) from calculus. Notice that
. Repeated differentiation shows that
for
, where
and
for
. The function
is a rational function. From the vanishing of
at infinity we see that there is an estimate of the form
(22)
(22) for
, where
is a finite positive constant.
We next construct an example where Theorem 3.1 is of less interest.
Proposition 3.5
Let and let w be a positive function on
. Then there exists a function
such that
and
Proof.
We shall make use of the functions
for
, where f is as in (Equation21
(21)
(21) ). Notice that
for
. An application of Lemma 2.3 shows that
(23)
(23) for
. By differentiation we have that
(24)
(24) In particular, we have that
for
and
. Moreover, an induction using (Equation22
(22)
(22) ) and (Equation24
(24)
(24) ) shows that
(25)
(25) for
and
, where
is a finite positive constant.
Let us now construct a function u with the desired properties. Let be a sequence of points in the interval
with accumulation points
and 1 only. Let
be a sequence of positive numbers such that
as
. We consider a function u of the form
(26)
(26) where the
's are as above. From the bound (Equation23
(23)
(23) ) we see that the Lipschitz norms
for
are uniformly bounded. By completeness of the space
this yields that
.
We next show that . Let
be a compact set and
an integer. Let 0<r<1 be such that
. Since the sequence
accumulates only at the points
and 1 there exists
such that
for k>N. From the bound (Equation25
(25)
(25) ) we have that
for k>N. This proves that
. Varying the compact K and the integer
, we see that the series (Equation26
(26)
(26) ) defining u is convergent in
.
By differentiation, we see that . The final conclusion of the proposition is satisfied provided
rapidly enough.
We mention that growth estimates of the form (Equation25(25)
(25) ) appear in the definition of symbol classes for pseudo differential operators (see Hörmander [Citation21, Chapter 18]).
4. The homogeneous expansion
In this section, we discuss some generalities related to the homogeneous expansion of α-harmonic functions. Applications of this material appear in later sections.
Let us recall the Beta function defined by
for a,b>0. It is well-known that
where Γ is the Gamma function (see [Citation22, Section 1.1]).
Let and set
for
, where B is the Beta function. Clearly,
as follows by a passage to the limit.
We now introduce the functions
(27)
(27) for
. From [Citation2, Section 1] we know that a complex-valued function u in
is α-harmonic if and only if it has a series expansion of the form
(28)
(28) for some sequence of complex numbers
such that
(29)
(29) Furthermore, the series expansion (Equation28
(28)
(28) ) is convergent in
provided (Equation29
(29)
(29) ) holds.
We proceed to study the functions in some more detail. The integral formula
(30)
(30) extends the function
to an analytic function in the slit plane
. Here the power is defined in the usual way using a logarithm which is real on the positive real axis.
Lemma 4.1
Let . Let
be as in (Equation30
(30)
(30) ) for some
. Then
for
.
Proof.
Differentiating under the integral sign in (Equation30(30)
(30) ) we have that
for
. An integration by parts now gives that
for
, where the last equality again follows by (Equation30
(30)
(30) ). This yields the conclusion of the lemma.
Lemma 4.1 leads to differential equations for the 's.
Proposition 4.2
Let . Let
be as in (Equation27
(27)
(27) ) for some
. Then
for
.
Proof.
Recall formula (Equation27(27)
(27) ). Differentiating we have that
for
. We now use Lemma 4.1 to conclude that
for
. This yields the first equation in the proposition.
Recall formula (Equation27(27)
(27) ). Differentiating we have that
for
. We now use Lemma 4.1 to conclude that
for
, where the last equality again follows by (Equation27
(27)
(27) ). This yields the second equation in the proposition.
Let us digress on the differential equations from Proposition 4.2.
Theorem 4.3
Let and
. Then a function
satisfies the differential equations
(31)
(31) for
if and only if it has the form
(32)
(32) for some
.
Proof.
Assume first that satisfies (Equation31
(31)
(31) ). In view of Proposition 4.2, the first equation in (Equation31
(31)
(31) ) implies that
where f is analytic in
. Now the second equation in (Equation31
(31)
(31) ) and Proposition 4.2 implies that
Solving for f we see that
for some constant
.
Conversely, using Proposition 4.2 it is straightforward to check that every function u of the form (Equation32(32)
(32) ) satisfies (Equation31
(31)
(31) ).
Following earlier practice from [Citation2–4], a function u in is called homogeneous of order
with respect to rotations if it has the property that
for
. Notice that the function
is homogeneous of order
.
Lemma 4.4
Let be homogeneous of order
. Then
for
.
Proof.
Assume first that . Since
is homogeneous of order
it has the form
for some
. Differentiating we see that
for
. By these two differentiation formulas we see that
for
. This yields the conclusion of the lemma for
.
If is homogeneous of order
, then its complex conjugate
has homogeneity
. The case n<0 thus follows by complex conjugation.
Let and consider its homogeneous parts defined by
(33)
(33) for
. Notice that the function
is homogeneous of order n with respect to rotations and that
for
. From a classical result of Fejér we have that
(34)
(34) with convergence in the space
, where the
's are as in (Equation33
(33)
(33) ) (see Katznelson [Citation7, Section I.2]). We shall refer to the expansion (Equation33
(33)
(33) )–(Equation34
(34)
(34) ) as the homogeneous expansion of u.
The differential equations (Equation31(31)
(31) ) in Theorem 4.3 exhibit certain structure owing to homogeneity.
Theorem 4.5
Let . Then a function
satisfies the differential equation
(35)
(35) for
if and only if it is homogeneous of order n with respect to rotations.
Proof.
By Lemma 4.4 we know that satisfies (Equation35
(35)
(35) ) if u is homogeneous of order n with respect to rotations. Assume next that
is a general solution of (Equation35
(35)
(35) ). We consider the k-th homogeneous part
of u defined as in (Equation33
(33)
(33) ). Let
be the angular derivative. Differentiating under the integral in (Equation33
(33)
(33) ) we have that
for
. Since u satisfies (Equation35
(35)
(35) ) we conclude that
for
, where the last equality follows by (Equation33
(33)
(33) ). Also, since
is homogeneous of order k, we have by Lemma 4.4 that
. We conclude that
if
. By (Equation34
(34)
(34) ) we have that
is homogeneous of order n.
It is straightforward to check that a function is homogeneous of order m=−n<0 with respect to rotations if and only if it has the form
(36)
(36) for some function
.
Let us return to Equation (Equation31(31)
(31) ).
Theorem 4.6
Let and
. Then a function
satisfies Equation (Equation31
(31)
(31) ) if and only if it has the form (Equation36
(36)
(36) ) for some function
such that
(37)
(37) for 0<x<1.
Proof.
A Gauss elimination shows that (Equation31(31)
(31) ) is equivalently stated saying that
(38)
(38) for
. By Theorem 4.5, the latter equation in (Equation38
(38)
(38) ) is equivalent to
having the form (Equation36
(36)
(36) ) for some function
. For such u, a differentiation shows that
for
. In view of this latter differentiation formula, we see that the first equation in (Equation38
(38)
(38) ) is equivalent to (Equation37
(37)
(37) ). This yields the conclusion of the theorem.
Notice that the choice in (Equation37
(37)
(37) ) gives by (Equation36
(36)
(36) ) rise to the α-harmonic function
.
Theorem 4.7
Let . Let
be as in (Equation27
(27)
(27) ) for some
. Then
for
.
Proof.
We shall exhibit a solution of Equation (Equation31(31)
(31) ). An interesting solution f of (Equation37
(37)
(37) ) is the function
for 0<x<1. The change of variables
leads to the formula
(39)
(39) for 0<x<1. We observe also that
where the last equality follows by a standard formula for the Beta function.
We now consider the function u given by (Equation36(36)
(36) ) with f as in (Equation39
(39)
(39) ). Since f satisfies (Equation37
(37)
(37) ), we have from Theorem 4.6 that u satisfies Equation (Equation31
(31)
(31) ). An application of Theorem 4.3 shows that the function u has the form
for some
. Observe that
Thus
for
. Solving for
, we arrive at the conclusion of the theorem.
Theorem 4.7 displays interesting structure in the function . Notice that the factor
is a polynomial.
We mention that an earlier version of Theorem 4.7 goes back to Olofsson and Wittsten [Citation2, Theorem 1.9]. The case n=2 of Theorem 4.7 has appeared recently in Chen [Citation10, Example 2.1].
5. The case of big smoothness
In this section, we discuss Lipschitz continuity of an α-harmonic function u in the case when
and
. We refer to this parameter range as the case of big smoothness.
Recall that the space is a commutative Banach algebra under pointwise multiplication of functions. We shall use that the algebra
is inverse closed in the sense that
if
and
for
(see Sherbert [Citation5, Proposition 1.7]).
Lemma 5.1
Let and
. Let
be an integer. Then
if and only if
.
Proof.
From Theorem 4.7 we have that
(40)
(40) for
. We set
and
(41)
(41) Observe that
for
.
Assume now that . By the algebra property of
we have from (Equation40
(40)
(40) ) that
. Since
is inverse closed we can divide by p to conclude that
. This yields that
.
Assume next that . From Proposition 4.2 we have that the function
satisfies an estimate of the form (Equation20
(20)
(20) ). An application of Theorem 3.1 yields that
.
Remark 5.2
Let p be as in (Equation41(41)
(41) ). The fact that p is a polynomial makes division by p a simpler operation not dependent on the full strength of commutative Banach algebras (see end of Section 4).
Remark 5.3
As a byproduct from the proof of Lemma 5.1 we have the norm bound that for
if
and
.
We now return to the study of α-harmonic functions.
Theorem 5.4
Let and
be such that
. Let u be an α-harmonic function such that
. Then u is analytic in
.
Proof.
We consider the series expansion (Equation28(28)
(28) ) for u. We shall show that
for n<0. Consider the n-th homogeneous part
of u for
. The assumption that
gives that
for
. Indeed, since
for
by assumption, we have by the triangle inequality that
for
.
We shall next evaluate the constraint . Let
. By (Equation28
(28)
(28) ) and homogeneity we have that
for
, where the last equality follows by an orthogonality property of exponential monomials. Observe that the change of order of integration and summation is permitted by uniform convergence. Thus
for
.
Since by assumption, we have from Lemma 5.1 that
for
. Now, since
belongs to
by the results of the previous two paragraphs, we conclude that
for n<0. From (Equation28
(28)
(28) ) we then have that
is analytic in
.
We point out that Theorem 5.4 generalizes an earlier result by Olofsson and Wittsten [Citation2, Theorem 1.11] to the scale of spaces .
A variant of Theorem 5.4 goes as follows.
Theorem 5.5
Let and
be such that
. Let
. Then
if and only if
for n<0.
Proof.
Assume first that belongs to
. By Theorem 5.4 we conclude that u is analytic in
. By passage to boundary values we see that
for n<0.
Conversely, assume that for n<0. Then
is analytic in
and Theorem 3.3 yields that
.
6. The case of critical smoothness
In this section, we discuss Lipschitz continuity of an α-harmonic function u in the case when
and
. We refer to this parameter range as the case of critical smoothness.
We shall need some more properties of the functions defined in (Equation30
(30)
(30) ) for
and
. By differentiation we see that
(42)
(42) for
.
Lemma 6.1
Let . Let
be as in (Equation30
(30)
(30) ) for some
. Then
is totally monotone on the interval
in the sense that
for x<1 and
.
Proof.
The conclusion of the lemma is evident from formula (Equation42(42)
(42) ).
The study of the functions belongs to the subject of hypergeometry. We observe that
(43)
(43) where
is a standard hypergeometric function. In fact, formula (Equation42
(42)
(42) ) is a specialization of the Euler integral formula
valid when c>b>0 (see [Citation22, Theorem 2.2.1]).
The study of asymptotic behaviour of hypergeometric functions is a classical topic which goes back to Gauss. We shall use the result that
(44)
(44) when c<a+b (see [Citation22, Theorem 2.1.3]). The asymptotic result (Equation44
(44)
(44) ) belongs to the realm of the famous Gauss summation formula.
Lemma 6.2
Let . Let
be as in (Equation30
(30)
(30) ) for some
. Then
Proof.
We consider the derivative . From formula (Equation43
(43)
(43) ) above we have that
By (Equation44
(44)
(44) ) we have that
The conclusion of the lemma now follows from L'Hospital's rule.
We shall consider the Fourier multiplier defined by
for
. From Stirling's formula we have that
(45)
(45) asymptotically in the sense that the quotient tends to 1 as
(see [Citation22, Section 1.4]).
Recall that the space is the dual of
and has as such a natural weak
topology. A sequence
in
converges to a function f in
in the weak
topology of
if and only if
and
for every
. It follows by a well-known diagonal procedure that every bounded sequence in
has a weak
convergent subsequence. An interesting discussion of Fourier coefficients for linear functionals is found in Katznelson [Citation7, Section I.7]; see [Citation2, Section 4] for information on a related notion of relative completeness.
Theorem 6.3
Let and set
. Let
be such that
. Then
.
Proof.
Since and
we have from Theorem 3.2 that
belongs to
. Set
. By assumption we have that
. Since also v=0 on
by construction, there exists a finite positive constant C such that
(46)
(46) for
.
We consider now the functions
for 0<r<1. Clearly
for 0<r<1. Furthermore, from (Equation46
(46)
(46) ) we have that
for 0<r<1. The functions
thus form a bounded sequence in
.
We now turn our attention to Fourier coefficients. From (Equation9(9)
(9) ) we have that
(47)
(47) for
. This latter series expansion (Equation47
(47)
(47) ) makes evident that
for
. Let
. Again from (Equation47
(47)
(47) ) we have that
for 0<r<1. By Lemma 6.2 we have that
for
.
From the result of the previous paragraph we know that the limit
exists for every
. Since the sequence
is bounded in
, it follows that the limit
exists in the weak
topology of
. We conclude that
belongs to
and
, where C is as in (Equation46
(46)
(46) ).
The Fourier multiplier is naturally derived from the binomial series:
(48)
(48) Let
and consider the anti-analytic function
From the binomial series (Equation48
(48)
(48) ) we have that
for
. From here we see that
where
in
is the distributional boundary limit of the function h,
is as in (Equation4
(4)
(4) ) and
is convolution in
.
The Fourier coefficients of a Lipschitz function decay as
as
(see Zygmund [Citation8, Theorem II.4.7]). An interesting example is provided by the Weierstrass function
(49)
(49) where b>1 is an integer. It is known that
for
(see Zygmund [Citation8, Theorem II.4.9]).
Let and set
. Using (Equation45
(45)
(45) ) and (Equation49
(49)
(49) ) it is straightforward to check that the Fourier coefficients of the distribution
are not vanishing at infinity. By Riemann–Lebesgue lemma we conclude that
for such parameters. The conclusion
in Theorem 6.3 is thus quite demanding of a Lipschitz function
.
Lemma 6.4
Let . Let
and set
. Then
for
.
Proof.
Recall the homogeneous expansion of the Poisson integral (Equation9(9)
(9) ). Differentiating we have that
for
. We now use the first of the differential equations in Proposition 4.2 to conclude that
for
, where the last equality is straightforward to check. Since
for
, this yields the conclusion of the lemma.
We now return to the study of Lipschitz classes.
Theorem 6.5
Let and set
. Let
be such that
. Then the function
belongs to
.
Proof.
We shall apply Theorem 3.1. Recall Lemma 6.4. A standard estimation of gives that
for
. Recall Theorem 2.13. An application of Theorem 3.1 now yields the conclusion of the theorem.
We denote by the distributional derivative of
. Recall that
for
.
Lemma 6.4 is accompanied with the following lemma.
Lemma 6.6
Let . Let
and set
. Then
for
.
Proof.
Recall the homogeneous expansion of the Poisson integral (Equation9(9)
(9) ). Differentiating we have that
for
. We now use the second of the differential equations in Proposition 4.2 to conclude that
for
, where the last equality is straightforward to check. In view of the series expansion (Equation9
(9)
(9) ), this yields the conclusion of the lemma.
Let . The conjugate function of f is the distribution
defined by the condition that
(50)
(50) where
for
and
. From (Equation50
(50)
(50) ) we have that the α-harmonic functions
and
are such that the function g=u+iv is analytic in
and
.
The following result is well-known but included here for the sake of completeness.
Theorem 6.7
Let and set
. Then
if and only if
.
Proof.
Assume first that . We set
. Since
we have that
for
, where C is a finite positive constant. This latter Lipschitz condition gives that the partial derivatives of u are uniformly bounded in
. Since v is a harmonic conjugate for u we conclude that v also has uniformly bounded partial derivatives in
. By Theorem 3.1 we have that
. Passing to the boundary limit of v we see that
.
Assume next that . Now the function
is analytic in
and
. By Theorem 3.3 we have that
. Similarly, the function
is anti-analytic in
and
. An application of Theorem 3.3 gives that
. We now conclude that
.
The Fourier multiplier is naturally expressed in terms of the conjugate function. It is straightforward to check that
(51)
(51) where the prime indicates distributional derivative.
A classical result is that the kernel of the distributional derivative
is the space of constant functions (see Hörmander [Citation23, Theorem 3.1.4]). It is also well-known that the distributional derivative maps the Lipschitz space
onto
.
Proposition 6.8
Let . Then
if and only if
.
Proof.
Recall formula (Equation51(51)
(51) ). If
, then
and a differentiation gives that
. Conversely, if
, then
which yields that
.
Disclosure statement
No potential conflict of interest was reported by the author.
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