A Few Remarks about the Hilbert Transform

Proof.

m

Hv= f&Hw= g& : w i ,

(4.2)

i=1

where g # L (R) and w i # L qi (R), 1
|



|/ S Hv| q1 dx<, by (4.2) and Holder's inequality.

&

Also, since q 1 >1,

|

 &



|(1&/ S ) Hv| q1 dx : k=0

|

|Hv(x)| q1 dx [x # R : 12k+1  |Hv(x)| 12k ]



: k=0

1 2

kq1

{

meas x # R : |Hv(x)|  

2A &v& L1( R ) : k=0

1

1 2

k+1

=

k

\2 + <, by (1.2). (q1 &1)

Hence Hv # L q1(R) and therefore g # X 0, by (4.2). Hence Hv # X 0 and so v, and hence u, is in X 0 . Hence f =Hu # X 0. This completes the proof. K We close this section by noting, from Kober's result [11] and Theorem 6, that Hu=v, u # L 1(R), v # L 1(R) implies that   & u(x) dx= 0=  & v(x)dx.

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J. F. TOLAND

5. A NON-LINEAR DIVERSION Some important equations in continuum mechanics associated with nonlinear Neumann boundary-value problems in the upper half-plane can be written in the form (5.1)

H(u$)=F(u)

for a (locally) absolutely continuous function u : R  R with u$ # X. Here F is a continuous function with the property there exists :>0 such that both the sets [t>0: |F(t)| :] and [t<0 : |F(t)| :] have infinite measure.

=

(P)

The following result is both elementary and useful. Theorem 9. Suppose that u is a solution of (5.1) with u$ # X. If F satisfies (P) then u # L (R) and u$ # X 0 . Proof.

For t # R let g(t)=

|

t

F :(s) ds 0

where for s # R, F :(s)=

:

{0

if if

|F(s)| : |F(s)| <:.

Then | g(t)|   as |t|  , by property (P). Since F(u)=H(u$) and u$ # X, meas[x : |F(u(x))| :]< and therefore F :(u) # L 1(R) & L (R) is a function which is non-zero only on a set of finite measure. It follows easily that g b u has bounded variation on compact intervals and hence [8, 18.37] g(u) is (locally) absolutely continuous. Now | g(u(x))| =

}|

x

0

}

g$(u( y)) u$( y) dy =

}|

x

}

F :(u( y)) u$( y) dy C,

0

where C is a constant independent of x. This follows by Holder's inequality because u$ #  1p< L p(R) and F :(u) #  1p L p(R). Hence u # L (R), since | g(t)|   as |t|  . That u$ # X 0 now follows from Corollary 8. K In the PeierlsNabarro and BenjaminOno equations, F(u)=sin u and u 2 &u, respectively (see [22]). Both of these functions F satisfy (P). In

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167

HILBERT TRANSFORM

Theorem 9, u itself need not be in any L p(R), 1p<, as the PeierlsNabarro example shows. There u(x)=2 tan &1(x). Let K denote the convex set of all odd functions u on R with u0 and u(x)x non-increasing on (0, ). The following result indicates briefly how the inverse of the operator u [ H(u$) maps Y & K into itself. Theorem 10. Suppose f # K & Y, u$ # X and Hu$= f. Then u # K & X 0. Proof. This is immediate from Theorem 2, Hardy's inequality, and Corollary 8. K It is clear that if F # K and F is increasing on (0, ) then the operator u [ F(u) maps K into itself. In such circumstances equation (5.1) may be regarded as a fixed-point problem on K & X 0.

6. THE HILBERT TRANSFORM OF REGULAR DISTRIBUTIONS Pandey [15] defined the Hilbert transform of a distribution as follows. Consider H(D) as a topological space with the topology of D induced by H so that H : D  H(D) is a homeomorphism. The dual space of H(D) (the space of continuous linear functionals on H(D)) is what Pandey calls the space of ultradistributions, which he denotes by H$(D). After a discussion of H(D) and H$(D) he gives the following definition of the Hilbert transforms of a distribution. [15] For f # D$, Hf is the ultradistribution defined by

Definition.

( Hf, ,) =&( f, H,)

for all , # H(D).

For an ultradistribution g, Hg is the distribution defined by (Hg, ) =&( g, H)

for all  # D.

Here ( , ) denotes the duality bracket defined on T $_T, where T is a topological space and T$ its dual. Pandey notes that H(Hf )=&f

and

H(Hg)=&g,

for f # D$,

g # H$(D).

Also, when u # L p(R), 1
|



u(x) ,(x) dx,

, # D,

&

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(6.1)

168

J. F. TOLAND

he observes that the ultradistribution Hf u is given by ( Hf u , ) =

|



Hu(x) (x) dx, &

where Hu is defined classically by (1.1). Here we give an integral representation for the ultradistribution Hf u when u # L 1(R) using the > integral. First we need to extend the > integral to functions defined on all of R. For convenience, say that a sequence [ \ N ]/D is admissible if (i)

0\ N (x)1, x # R;

(ii)

[&\$N & L( R ) ], [&\"N & L( R ) ] and [&\$N & L1( R ) ] are bounded;

(iii)

\ N (x)=1, x # [ &N, N].

For f: R  R write

Definition.

>

|

f (x) dx=I R

if, and only if, for every admissible sequence [\ N ]/D I= lim > N

|

\ N (x) f (x) dx.

R

Clearly, because of (iii), this coincides with the previous definition of the > integral when f has compact support and, because of Lemma 3,

|



f (x) dx=>

&

|

f (x) dx

for all

f # L 1(R).

R

Now note that if u # L 1(R) then f u # D$ may be defined using the formula (6.1); alternatively an ultradistribution f u may be defined by ( f u , ) =

|



u(x) (x) dx,

 # H (D).

&

Recall that H(D)/L p(R),

1
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(6.2)

169

HILBERT TRANSFORM

Theorem 11. For u # L 1(R), ( Hf u , ) =&>

|

Hu(x) (x) dx,

 # H(D)

R

and ( Hf u , ,) =&>

|

Hu(x) ,(x) dx,

, # D.

R

Proof. Let [ \ N ] be an admissible sequence and let  # H(D)/L p(R), 1
(6.3)

as k  , since H : L p(R)  L p(R) is continuous. Moreover, from the admissibility of [ \ N ] and Lemma 1, there exists a constant K independent of k such that &H( \ Nk )& L( R ) &H& L( R ) +K && Lp( R )

for all k # N.

(6.4)

Therefore for u # L 1(R), Theorem 5 gives >

|

R

\ Nk(x) (x) Hu(x) dx= & &

|



&

|

u(x) H( \ Nk )(x) dx



u(x) H()(x) dx

(6.5)

&

by (6.3), (6.4) and the Dominated Convergence Theorem, since  is smooth. Since every admissible sequence [ \ N ] has a subsequence for which (6.5) holds, it follows that for every admissible sequence >

|

\ N (x) (x) Hu(x) dx  & R

|



u(x) H()(x) dx. &

Therefore, by definition, for  # H(D) and u # L 1(R), >

|

R

(x) Hu(x) dx=&

|



u(x) H(x) dx.

&

The proof of the formula for the distribution Hf u is identical.

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K

170

J. F. TOLAND

7. A GENERALISED FOURIER TRANSFORM In this section the goal is to show that when u # L 1(R) the Fourier transform of Hu, calculated using the > integral, is a multiplication operator. This is immediate from Theorem 5 and the definitions once some technicalities have been dealt with. We use the definition of Fourier transform from [18, p. 28]. Suppose \ # D and let &( y)=

\( y)&\(0) , y

y{0, y=0.

=\$(0),

It follows from the Mean value Theorem that &&& L( R ) &\$& L( R ) . From first principles, 1 &$(0)= \"(0) 2

and

&$( y)=

\(0)&\( y)+ y\$( y) , y2

y{0.

(7.1)

Hence by the Mean Value Theorem &&$& L( R ) &\"& L( R ) and it follows from (7.1) that for M>0 &&$& L1( R ) 2M &\"& L(&M, M) +

|

| y| M

\$( y) dy+ y

} }

|

2 &\& L( R ) dy y2 | y| M

2M &\"& L(&M, M) +M &1(&\$& L1( R ) +4 &\& L( R ) ). Let [ \ N ] be an admissible sequence and for k # R"[0] let  kN( y)=\ N ( y) e iky, Lemma 12.

y # R.

There is a constant C such that for k{0 &H kN & L( R ) 1+

C |k|

for all N # N

and H kN(x)  &m(k) e ikx

as

N  ,

where m(k)=i sign(k).

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(7.2)

171

HILBERT TRANSFORM

Proof.

By definition

1 H kN(x)= lim ? =z0

|

\ N (x& y) e ik(x& y) dy y | y| =

=

e ikx lim ? =z0

=

e ikx lim ? =z0

\ N (x& y) e &iky dy y = | y| = &1

|

{

|

= | y| = &1

+\ N (x) lim =z0

ikx

=

e ?

{

1 ik

|

|

+

\

\ N (x& y)&\ N (x) &iky e dy y

+

e

= | y| = &1

&iky

y

dy

=

&$N, x ( y)(e &iky &1) dy&i?\ N (x) sign(k)

&

=

(see [21, Article 290] for example), where \ N (x& y)&\ N (x) , y & N, x ( y)= &\$N (x),

{

y{0 y=0.

Since 0\ N (x)1 and \ N (x)  1 as N  , it will suffice to show that

|



|&$N, x ( y)| dy is bounded independent of x and N

(7.3)

&

and

|



|&$N, x ( y) dy|  0

as

N

(7.4)

&

for each x # R. Clearly (7.3) is ensured by taking M=1 and \( y)=\ N (x& y), for any x and N, in (7.2) and using the fact that [ \ N ] is admissible. Let x # R and M>0. Then for N |x| +M & N, x ( y)=0

for all y # [ &M, M]

and hence, by (7.2),

|



&

|&$N, x ( y)| dy

1 (&\$N & L1( R ) +4 &\ N & L( R ) ) M

for all NM+ |x|. Since [ \ N ] is admissible, this ensures (7.4), and the proof is complete. K

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172

J. F. TOLAND

Theorem 13. For u # L 1(R) and k{0

|

> Proof. k{0,

e iky Hu( y) dy=i sign(k) u7 (k2?). R

Let [ \ N ] be an admissible sequence. Then by Theorem 5, for

>

|

|

\ N ( y) e iky Hu( y) dy=& R

 i sign(k)

|



 &

u( y) H kN( y) dy as N  ,

e iky u( y) dy

&

by Lemma 12 and the Dominated Convergence Theorem, =m(k) u7 (k2?). Therefore, by the definition of u 7 in [18, p. 28], >

|

e iky Hu( y) dy=m(k) u7 (k2?), R

as required. K This result is the exact analogue for Hilbert transforms of Kolmogorov's result [24, Chap. VII, (4.3)] for Fourier series and enables one to reconcile the meaning of the statements ``f # L 1(R) and Hf # L 1(R)'' based on the classical formula and the alternative interpretation based upon Fourier transforms [18, p. 221]. Theorem 14. Suppose that u and v are in L 1(R). Then Hu=v if, and only if, mu7 =v7 almost everywhere. Proof. Suppose that mu7 =v7 . Let [, n ] be a sequence of regularising kernels as in the proof of Theorem 7. Then , n V u # L 1(R) & L (R)/L 2(R) and , n V u  u in L 1(R) as n  . Hence, by (2.5) and (2.6) for each n # N, (H(, n V u))7 =m,n7 u7 =(, n V v)7 . Therefore for n # N, H(, n V u)=, n V v

in L 2(R).

Now , n V u  u in L 1(R), a subsequence of [, nk V v] converges to v almost everywhere. Therefore H(u)=v since H(, nk V u)  H(u) in measure.

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HILBERT TRANSFORM

173

Now suppose that H(u)=v. Then the Fourier transform of H(u), calculated using the > integral, coincides with the Fourier transform of v in the usual sense, because, for any f # L 1(R),

|



&

f (x) dx=>

|

f (x) dx.

R

Hence mu7 =v7 , by the preceding theorem. This completes the proof.

K

ACKNOWLEDGMENT Some of these results were considered during a visit to the University of Sydney, Australia. I am grateful to my host, Professor E. N. Dancer, for his hospitality and for his interest, which stimulated some of the questions addressed here. Also, I must thank Professor A. Carbery of Edinburgh University for casting his expert eye over an earlier draft and making a number of extremely helpful suggestions.

REFERENCES 1. C. J. Amick and J. F. Toland, On periodic water waves and their convergence to solitary waves in the long-wave limit, Philos. Trans. Roy. Soc. London Ser. A 303 (1981), 633669. 2. N. K. Bary, ``A Treatise on Trigonometric Series,'' Pergamon, Oxford, 1964. 3. T. J. Boks, Sur le rapport entre les methodes d'integration de Riemann et de Lebesgue, Rend. Circ. Math. Palermo 45 (1921), 211264. 4. A. Denjoy, Sur l'integration Riemannienne, C.R. Acad. Sci. Paris 169 (1919), 219221. 5. D. Gilbarg and N. S. Trudinger, ``Elliptic Partial Differential Equations of Second Order,'' 2nd ed., Springer-Verlag, Berlin, 1983. 6. R. Henstock, ``The General Theory of Integration,'' Oxford Univ. Press, Oxford, 1991. 7. R. Henstock, personal communication, 1995. 8. E. Hewitt and K. Stromberg, ``Real and Abstract Analysis,'' Springer-Verlag, Berlin, 1965. 9. E. Hille and J. D. Tamarkin, On the absolute integrability of Fourier transforms, Fund. Math. 25 (1935), 329352. 10. S. Kempisty, Sur l'integral (A) de M. Denjoy, C.R. Acad. Sci. Paris 185 (1927), 749751 and 192 (1931), 11861189. 11. H. Kober, A note on Hilbert's operator, Bull. Amer. Math. Soc. 48 (1942), 421427. 12. S. Koizumi, On the Hilbert transform, I, II, Fac. Hokkaido Univ. Ser. I 14 (1959), 153224 and 15 (1960), 93130. 13. A. N. Kolmogorov, ``Fundamental Concepts in the Theory of Probability,'' United Scientific and Technical Press, 1936. [In Russian] 14. B. F. Logan, The Hilbert transform of a function having a bounded integral and a bounded derivative, SIAM J. Math. Anal. 14 (1983), 247248 and 845. 15. J. N. Pandey, The Hilbert transform of a Schwartz distribution, Proc. Amer. Math. Soc. 89 (1983), 8690. 16. W. Rudin, ``Principles of Mathematical Analysis,'' McGrawHill, New York, 1964. 17. W. Rudin, ``Real and Complex Analysis,'' 3rd ed., McGrawHill, New York, 1986. 18. E. M. Stein, ``Singular Integrals and Differentiability Properties of Functions,'' Princeton Univ. Press, Princeton, NJ, 1970.

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19. E. M. Stein and G. Weiss, ``Introduction to Fourier Analysis on Euclidean Spaces,'' Princeton Univ. Press, Princeton, NJ, 1971. 20. E. C. Titchmarsh, ``Introduction to the Theory of Fourier Integrals,'' Oxford Univ. Press, Oxford, 1948. 21. I. Todhunter, ``A Treatise on Integral Calculus,'' Macmillan, 1921. 22. J. F. Toland, The PeierlsNabarro and BenjaminOno equations, J. Funct. Anal. 145 (1997), 136150. 23. K. Yoneda, On generalised (A)-integrals, Math. Japon. 18 (1973), 149167. 24. A. Zygmund, ``Trigonometric Series, I, II,'' Cambridge Univ. Press, Cambridge, UK, 1959.

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