- #1
hatsoff
- 20
- 3
Homework Statement
Prove the following: If [tex]\delta\in L_1(\mathbb{R}^n)[/tex] and [tex]f\in L_p(\mathbb{R}^n)[/tex] then the convolution [tex]\delta * f\in L_p(\mathbb{R}^n)[/tex] with [tex]\lVert \delta * f\rVert_p\leq\lVert\delta\rVert_1\lVert f\rVert_p[/tex].
Homework Equations
We use the natural isometry (or isometric isomorphism, if you like) [tex]h\mapsto\lambda_h[/tex] between [tex]L_q[/tex] and [tex]L_p^*[/tex], where [tex]\frac{1}{p}+\frac{1}{q}=1[/tex], and where we define each [tex]\lambda_h[/tex] by [tex]\lambda_h(f)=\int fh[/tex].
The Attempt at a Solution
Well I can show that [tex]\lVert (\delta * f)h\rVert_1\leq\lVert\delta\rVert_1 \lVert f\rVert_p\lVert h\rVert_q[/tex]. Supposedly I can use this along with the natural isometry between [tex]L_q[/tex] and [tex]L_p^*[/tex] to finish the proof. But I don't see how that natural isometry is applicable.
Any help would be much appreciated. Thanks!