Proving Convolution in R^n using Isometric Isomorphism and Lp Spaces

[hatsoff]

Homework Statement

Prove the following: If \delta\in L_1(\mathbb{R}^n) and f\in L_p(\mathbb{R}^n) then the convolution \delta * f\in L_p(\mathbb{R}^n) with \lVert \delta * f\rVert_p\leq\lVert\delta\rVert_1\lVert f\rVert_p.

Homework Equations

We use the natural isometry (or isometric isomorphism, if you like) h\mapsto\lambda_h between L_q and L_p^*, where \frac{1}{p}+\frac{1}{q}=1, and where we define each \lambda_h by \lambda_h(f)=\int fh.

The Attempt at a Solution

Well I can show that \lVert (\delta * f)h\rVert_1\leq\lVert\delta\rVert_1 \lVert f\rVert_p\lVert h\rVert_q. Supposedly I can use this along with the natural isometry between L_q and L_p^* to finish the proof. But I don't see how that natural isometry is applicable.

Any help would be much appreciated. Thanks!

 

[MathematicalPhysicist]

I think you need to use here Minkowski inequality theorem, and the fact that dirac-delta measure is invriant under raising its power.

http://www.math.duke.edu/~wka/math204/conv.12.4.pdf
it's in page 2, but there's a misprint, and the LHS, the aboslute value integrand should be raised to power of p.

 

[hatsoff]

Thanks, I appreciate the link, and indeed that is a very nice proof of the theorem in question. However, I'm looking to finish the particular approach I was given.

Basically, I have to show, using the fact that for all h\in L_q we have

\lVert (\delta * f)h\rVert_1\leq\lVert\delta\rVert_1 \lVert f\rVert_p\lVert h\rVert_q

and also using the natural isometric isomorphism between Lp* and Lq, that the \delta *f\in L_p.