We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness
ω
α
(
f
,
t
)
q
\omega _\alpha (f,t)_q
and
ω
β
(
f
,
t
)
p
\omega _\beta (f,t)_p
for
0
>
p
>
q
≤
∞
0>p>q\le \infty
. A similar problem for the generalized
K
K
-functionals and their realizations between the couples
(
L
p
,
W
p
ψ
)
(L_p, W_p^\psi )
and
(
L
q
,
W
q
φ
)
(L_q, W_q^\varphi )
is also solved.
The main tool is the new Hardy–Littlewood–Nikol’skii inequalities. More precisely, we obtained the asymptotic behavior of the quantity
sup
T
n
‖
D
(
ψ
)
(
T
n
)
‖
q
‖
D
(
φ
)
(
T
n
)
‖
p
,
0
>
p
>
q
≤
∞
,
\begin{equation*} \sup _{T_n} \frac {\Vert \mathcal {D}(\psi )(T_n)\Vert _q}{\Vert \mathcal {D}({\varphi })(T_n)\Vert _p},\qquad 0>p>q\le \infty , \end{equation*}
where the supremum is taken over all nontrivial trigonometric polynomials
T
n
T_n
of degree at most
n
n
and
D
(
ψ
)
,
D
(
φ
)
\mathcal {D}(\psi ), \mathcal {D}({\varphi })
are the Weyl-type differentiation operators.
We also prove the Ulyanov and Kolyada-type inequalities in the Hardy spaces. Finally, we apply the obtained estimates to derive new embedding theorems for the Lipschitz and Besov spaces.