## Boundedness of fractional integration operator

Suppose is locally integrable, define the fractional integration operator

,

Suppose and , then

where and . If , then is a weak type .

Suppose is radially decreasing function, then

when .

Write

By the lemma,

By holder inequality, if

If

with .

Combing the preceeding result,

for

Choose ,

When , maximal function is bounded from to ,

When , maximal function is weak (1,1),

When the domain is not but a bounded one, we can obtain the boundedness of .

Suppose , . We have is bounded from to with

,

Loukas Grafakos: Modern fourier analysis.

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