Let $ f: mathbb {R} ^ {N} to mathbb {R} $ be the initial data of the following linear heat equation:

begin {align *} tag {1.1}

begin {cases}

partial_ {t} u (t, x) = Delta u (t, x) quad x in mathbb {R} ^ {N}, t> 0 \

u (0, x) = f (x) quad x in mathbb {R} ^ {N}

end {cases}

end {align *}

Then we leave $ G_ {t} (x): = frac {1} {(4 pi t) ^ {N / 2}} exp big ( frac {- | x | ^ {2}} {4t} big) $ and we can write the solution of (1.1) as follows

begin {align *} tag {1.2}

u (t, x): = int _ { mathbb {R} ^ {N}} G_ {t} (x-y) f (y) dy

end {align *}

I want to prove the following theorem using Young's inequality for convolution.

**Theorem ($ L ^ {p} -L ^ {q} $ estimates)**

Let $ u $ to be the solution of (1.1) with the initial data $ f $. For $ 1 leq q leq p leq infty $, there is a constant $ C = C (p, q, N) $ such as

begin {align *} tag {1.3}

| partial_ {x_ {j}} u (t) | _ {p} leq Ct ^ {- frac {N} {2} big ( frac {1} {q} – frac {1} {q} big) + frac {1} {2} } || f || _ {q} quadri = 1,2, …, N, , t> 0

end {align *}

begin {align *} tag {1.4}

| partial_ {t} u (t) | _ {p} leq Ct ^ {- frac {N} {2} big ( frac {1} {q} – frac {1} {q} big) +1} || f || _ {q} quadri = 1,2, …, N, , t> 0

end {align *}

In addition, for higher derivatives, there is a constant $ C = C (p, q, N, k, alpha) $ such as

begin {align *} tag {1.5}

|| partial_ {t} ^ {k} partial_ {x} ^ { alpha} u (t) || _ {p} leq Ct ^ {- frac {N} {2} big ( frac {1} {q} – frac {1} {p} large) + k + frac {| alpha |} {2}} | f | _ {q}

end {align *}

or $ k in mathbb {N} $ and $ alpha $ is a multi-index $ ( alpha_ {1}, alpha_ {2}, …, alpha_ {N}) $ with $ | alpha | = alpha_ {1} + alpha_ {2} + … + alpha_ {N} $.

For convenience, I will also include the following theorem.

**Theorem (Young Inequality)**

Let $ 1 leq p, q, r leq infty $ such as $ frac {1} {p} = frac {1} {r} + frac {1} {q} -1 $. Then for everything $ h in L ^ {r} ( mathbb {R} ^ {N}) $ and $ f in L ^ {q} ( mathbb {R} ^ {N}) $, we have $ h * f in L ^ {p} ( mathbb {R} ^ {N}) $ and

begin {align *} tag {Y-in}

| h * f | _ {p} leq | h | _ {r} | f | _ {q}

end {align *}

This is my attempt until here:

First, we will prove (1.3) and (1.4). Observe that we have the following calculations

begin {align *}

partial_ {x_ {j}} u & = partial_ {x_ {j}} G_ {t} * f \

partial_ {t} u & = partial_ {t} G_ {t} * f

end {align *}

It means that

begin {align *} tag {1.6a}

| partial_ {x_ {j}} G_ {t} (x) | & = bigg | (4 pi t) ^ {- N / 2} big ( frac {-2x_ {j}} {4t} big) exp big ( frac {- | x | ^ {2}} {4t } big) bigg | \

& leq (4 pi t) ^ {- N / 2} frac {| x |} {2t} exp big ( frac {- | x | ^ {2}} {4t} big)

end {align *}

and

begin {align *} tag {1.6b}

| partial_ {t} G_ {t} (x) | & = (4 pi t) ^ {- N / 2} bigg | – frac {N} {2} (4 pi t) ^ {- 1} 4 pi + frac {| x | ^ {2}} {4t ^ {2}} bigg | exp big ( frac {- | x | ^ {2}} {4t} big) \

& = 4 pi (4 pi t) ^ {- N / 2-1} bigg | – frac {N} {2} + frac {| x | ^ {2}} {4t} bigg | exp big ( frac {- | x | ^ {2}} {4t} big) \

& leq 4 pi (4 pi t) ^ {- N / 2-1} bigg ( frac {N} {2} + frac {| x | ^ {2}} {4t} bigg exp big ( frac {- | x | ^ {2}} {4t} big)

end {align *}

Then we start by examining the case $ p = infty $ and $ q = $ 1. From (1.6a), (1.6b) and $ z = frac {x} {2 sqrt {t}} $we see that

begin {align *}

| partial_ {x_ {j}} G_ {t} (x) | & leq frac {(4 pi t) ^ {- N / 2}} { sqrt {t}} | z | exp (- | z | ^ {2}) \

| partial_ {x_ {j}} G_ {t} | _ { infty} & leq (4 pi t) ^ {- N / 2- frac {1} {2}} frac {1} { sqrt {2e}}

end {align *}

and

begin {align *}

| partial_ {t} G_ {t} (x) | & leq 4 pi (4 pi t) ^ {- N / 2 -1} bigg ( frac {N} {2} + | z | ^ {2} bigg) exp (- | z | ^ {2}) \

| partial_ {t} G_ {t} | _ { infty} & leq 4 pi (4 pi t) ^ {- N / 2 -1} ( frac {N} {2} + frac {1} {e})

end {align *}

To choose $ C (p, q, N) = max bigg { frac {(4 pi) ^ {- N / 2- frac {1} {2}} { sqrt {2e}}, 4 pi (4 pi) ^ {- N / 2 -1} ( frac {N} {2} + frac {1} {e}) bigg $we obtain (1.3) and (1.4) for the case $ p = infty $ and $ q = $ 1 since (Y-in) implies

begin {align *}

| partial_ {x_ {j}} u (t) | _ { infty} & leq | partial_ {x_ {j}} G_ {t} | _ { infty} | f | _ {1} \

| partial_ {t} u (t) | _ { infty} & leq | partial_ {t} G_ {t} | _ { infty} | f | _ {1}

end {align *}

Then we can consider another case. From (1.6a) and (1.6b), we can again obtain

begin {align *}

| partial_ {x_ {j}} G_ {t} (x) | ^ {r} & leq frac {2 ^ {r}} {(4 pi t) ^ {Nr / 2}} big | frac {| x |} {4t} big | ^ {r} exp big ( frac {- | x | ^ {2} r} {4t} big) \

| partial_ {x_ {j}} G_ {t} | _ {r} ^ {r} & leq frac {2 ^ {r}} {(4 pi t) ^ {Nr / 2}} int _ { mathbb {R} ^ {N}} t ^ {- r / 2} (2 sqrt {t}) ^ {N} | z | ^ {r} exp (- | z | ^ {2} r) dz \

| partial_ {x_ {j}} G_ {t} | _ {r} ^ {r} & leq C_ {1} t ^ {- frac {N} {2} (r-1) – frac {r} {2}} int _ { mathbb {R } ^ {N}} | z | ^ {r} exp (- | z | ^ {2} r) dz \

| partial_ {x_ {j}} G_ {t} | _ {r} & leq C_ {A} t ^ {- frac {N} {2} (1- frac {1} {r}) – frac {1} {2}} \

| partial_ {x_ {j}} G_ {t} | _ {r} & leq C_ {A} t ^ {- frac {N} {2} ( frac {1} {q} – frac {1} {p}) – frac {1} {2 }}

end {align *}

and similarly

begin {align *}

| G_ {t} (x) | _ {r} & leq (4 pi t) ^ {- N / 2 -1} 4 pi bigg { frac {N} {2} | exp (- frac {- | , cdot , | ^ {2}} {4t}) | _ {r} + | frac {- | , cdot , | ^ {2}} {4t} exp (- frac {- | , cdot , | ^ {2}} {4t}) | _ {r} bigg) \

& leq C_ {B} t ^ {- frac {N} {2} -1+ frac {N} {2r}} \

& = C_ {B} t ^ {- frac {N} {2} ( frac {1} {q} – frac {1} {p}) – 1}

end {align *}

Using (Y-in), again, we obtain (1.3) and (1.4) for the other case of $ p $ and $ q $.

Now, my question is how to proceed to get the upper derivative case in (1.5)? I've tried using induction and strong induction but it does not work very well with manual calculation. Any help or suggestion allowing me to move to (1.5) is very appreciated!

Thank you!