Hi.
The Hilbert transform is given by
I would like
(a) if (is continuous and compactly supported), then exists and is continuous, and
(b) .
I have a tentative proof of the first; I really need help on the second.
Proof of (a).
Existence of Integral.
Let f be continuous and compactly supported. By changing variables, etc., we can consider only the existence of
.
Let (so that ) to obtain
.
Claim. If and , then exists.
From this, we can conclude that the principal value integral exists near the pole ( ) and from compact support, that it exists away from the pole ( ) and existence follows.
Proof of Claim.
The integrand is bounded for ; it is also bounded as because for some so that we have and we can apply l'Hôpital's rule to conclude the limit is finite (is there an easier way to show this limit is finite?). Hence the integral exists.
Continuity.
Let and be disjoint balls containing x and z, respectively, so the last integral is bounded iff is bounded. The integrals are similar so we consider only the first. Setting , we see g is continuous on and the integral in question is , so we have that it is finite and remains finite as , so that we have continuity (this argument is rough; will touch up later; also, better arguments appreciated!)
I know this is sloppy; sorry. Is this correct? and can anyone point me in the right direction on (b)? Thanks a whole lot—I really appreciate it. _(_ _)_
(p.s. is there an \inline{...} command? Integrals look kind of ugly in the middle of lines of text... thanks)
Edit. I proved (b) for all simple functions by direct computation on an indicator function and then used linearity. Since simple functions are dense in (say) , I should have (b) for all of , correct? Thanks.