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## Homework Statement

If [itex]0 < \alpha < n[/itex], define an operator [itex]T_{\alpha}[/itex] on function on [itex]\mathbb{R}^n[/itex] by

[tex]T_{\alpha}f(x) = \int |x-y|^{-\alpha}f(y)dy[/tex]

Then prove that [itex]T_{\alpha}[/itex] is weak type [itex](1,(n-\alpha )^{-1})[/itex] and strong type (p,r) with respect to Lebesgue measure on [itex]\mathbb{R}^n[/itex], where [itex]1 < p < n\alpha ^{-1}[/itex] and [itex]r^{-1} = p^{-1} -\alpha n^{-1}[/itex].

## Homework Equations

Let X be a set, [itex]\mu[/itex] a measure on this set. For [itex]0 < q < \infty[/itex] define the

**L**of a function [itex]g : X \to \mathbb{C}[/itex] with respect to [itex]\mu[/itex] to be:

^{q}norm[tex]||g||_q = \left (\int _X |g|^qd\mu \right )^{1/q}[/tex]

Define the

**weak L**of such a function to be:

^{q}norm[tex][g]_q = \left [\mbox{sup} _{\beta > 0}(\beta ^q\mu \{ x : |g(x)| > \beta \})\right ]^{1/q}[/tex]

Define the space of functions [itex]\mathbf{L^q(\mu )}[/itex] to be the set of function with finite L

^{q}norm. Define the space of functions [itex]\mathbf{weak\ L^q(\mu )}[/itex] to be those function with finite weak L

^{q}norm.

An operator T is

**sublinear**if |T(f + g)|

__<__|Tf| + |Tg| and |T(cf)| = c|Tf| for every function in the domain of T (which is some vector space of functions). A sublinear operator T is

**strong type (a,b)**if [itex]L^a(\mu )[/itex] is contained in its domain, T maps [itex]L^a(\mu )[/itex] into [itex]L^b(\mu )[/itex], and there exists C > 0 such that [itex]||Tf||_b \leq C||f||_a[/itex] for all f in [itex]L^a(\mu )[/itex]. A sublinear operator T is

**weak type (a,b)**if [itex]L^a(\mu )[/itex] is contained in its domain, T maps [itex]L^a(\mu )[/itex] into [itex]weak\ L^b(\mu )[/itex], and there exists C > 0 such that [itex][Tf]_b \leq C||f||_a[/itex] for all f in [itex]L^a(\mu )[/itex].

As this is real analysis, there are few relevant equations, instead there are inequalities. They include:

Holder's inequality

Minkowski's inequality

Chebyshev's inequality

Minkowski's inequality for integrals

The Riesz-Thorin Interpolation Theorem

The Marcinkiewicz Interpolation Theorem

and a few other propositions and lemmas that I would take too long to write out.

## The Attempt at a Solution

I've only started on the "weak type" part of the problem, and I've only gotten as far as writing out what I need to prove in terms of the definitions. Then I guess I have to find one of the inequalities in my book and find some non-obvious way to apply it which ends up giving the right answer, but I have no clue really of what to do. So this is all I have:

I need to find C > 0 such that

[tex]\left [\mbox{sup} _{\beta > 0} \left (\beta ^{(n-\alpha )^{-1}}m\{ x : |\int f(y)|x-y|^{-\alpha }dy| > \beta \}\right ) \right ]^{n-\alpha } \leq C\int |f|[/tex]

I've determined that this is equivalent to proving:

[tex]\left [\mbox{sup} _{\beta > 0} \left (\beta ^{(n-\alpha )^{-1}}m\{ x : \int |f(y)||x-y|^{-\alpha }dy > \beta \}\right ) \right ]^{n-\alpha } \leq C\int |f|[/tex]

but I don't know if that's any use. Help!

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