1488 CHAPTER 45. TRACES OF SOBOLEV SPACES

Lemma 45.3.1 Let g = f ∗h where f ∈ L1 (R) ,h ∈ Lp (R) , and f ,h are all Borel measur-able, p≥ 1. Then g ∈ Lp (R) and

||g||Lp(R) ≤ || f ||L1(R) ||h||Lp(R)

Proof: First of all it is good to show g is well defined. Using Minkowski’s inequality(∫ (∫|h(t− s) f (s)|ds

)p

dt)1/p

≤∫ (∫

|h(t− s)|p | f (s)|p dt)1/p

ds

=∫| f (s)|

(∫|h(t− s)|p dt

)1/p

ds

= || f ||L1 ||h||Lp

Therefore, for a.e. t, ∫|h(t− s) f (s)|ds =

∫|h(s) f (t− s)|ds < ∞

and so for all such t the convolution f ∗h(t) makes sense. The above also shows

||g||Lp ≡(∫ ∣∣∣∣∫ f (t− s)h(s)ds

∣∣∣∣p dt)1/p

≤ || f ||L1 ||h||Lp

and this proves the lemma.The following is a very interesting inequality of Hardy Littlewood and Pólya.

Lemma 45.3.2 Let f be a real valued function defined a.e. on [0,∞) and let α ∈ (−∞,1)and

g(t) =1t

∫ t

0f (ξ )dξ (45.3.6)

For 1≤ p < ∞ ∫∞

0tα p |g(t)|p dt

t≤ 1

(1−α)p

∫∞

0tα p | f (t)|p dt

t(45.3.7)

Proof: First it can be assumed the right side of 45.3.7 is finite since otherwise there isnothing to show. Changing the variables letting t = eτ , the above inequality takes the form∫

∞

−∞

eτ pα |g(eτ)|p dτ ≤ 1(1−α)p

∫∞

−∞

eτ pα | f (eτ)|p dτ

Now from the definition of g it follows

g(eτ) = e−τ

∫ eτ

−∞

f (ξ )dξ

= e−τ

∫τ

−∞

f (eσ )eσ dσ