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 $$\delta *f\in L_p$$.