There is a main obstacle to improve the kakeya conjecture,remain in dimension 3,and the result established by Tomas Wolff in 1995 is almost the best result in even until now.the result establish by Katz and Tao can be view as a corollary of Wolff’s X-ray estimate.

For ,for :

..

is varise in all cylinders with length 1.radius .axis in the direction.

..

varise in cylinders contains x,length 1,radius .

Keeping this two maximal function in mind,we give the statement of the Kakeya maximal function conjecture:

Where or .

Because we have the obviously estimate:

.

.

So by the Riesz-Thorin interpolation we have:

. (*)

for .the task is establish (*) for as large as posible in the range.

for the 2 dimension case,the result is well know.the key estimate is:

for case,the main result of Wolff is:

hold for . or .

Now we sketch the proof.

prove cases together.

We can make some reduction:

the first one is we can assume the sup of is in a fix compact set.

the second is instead of consider ,we can consider .

where varies in all cylinder with radius ,length 1,axis with a fix direction.

the first reduction is obvious(why?)

the second reduction rely on a observe:

this is just finite cover by rotation of the coordinate and triangle inequality.

now we begin to establish a frame and put the two situations into it.

Let be all line in .

then is a dim manifold.

is the line parallel to .and the middle point is original.

.

Wolf axiom:

metric space.

...

for certain .

. is given.and is compact.

for all .

If then we define by

.

Property (**):

If . is a 2-plane.containing and if and if is a subset of and for each j,there is with and .then