Why is AF so slow in Live View on SLRs ? On compacts, the AF works very smoothly and quickly.
On a DSLR the mirror is up when the camera is in live view, blocking off the AF sensors which are located away from the lens-sensor axis. AF in live view is usually handled with contrast detection, which is much more processor intensive. Compacts are designed to focus when in live view mode, so their mechanism is optimised for that particular setup, which is the same when shooting video or stills.
A typical compact camera uses a smaller sensor, which allows for shorter focal lengths and deeper depth of field for similar field of view and f-number. Therefore, a compact camera has to move its lens much less and also needs less precision for achieving acceptable focus.
Also, expectations for compact camera focusing speed are lower than for dSLR, so slowness is less noticeable.
In Live View the mirror is folded up, so there is nothing that sends any light to the regular focus sensors.
Instead the focusing is done by analysing the details of the image before and after moving the focus a bit (as analysing a single image doesn't tell you if the focus is right, or if it is behind or in front of the current setting). This means that the camera has to adjust the focus back and forth a lot to find where there is most detail.
Simply, the image sensor is used to do the focusing, and as it's not constructed specifically for that, it gives much less usable information than the regular focus sensors.
Compacts are always in live view mode, so to speak, so they are constructed with that in mind. They may also have other focus sensors than the image sensor.