Inequalities for derivatives of functions in the spaces L _p

We obtain a new sharp inequality for the local norms of functions x ∈ L _{∞, ∞} ^r(R), namely,

$$\frac{1} {{b - a}}{\int\limits_a^b {{\left| {x^{\prime} {\left( t \right)}} \right|}^{q} dt \leq \frac{1} {\pi }{\int\limits_0^\pi {{\left| {\upvarphi _{{r - 1}} {\left( t \right)}} \right|}^{q} dt{\left( {\frac{{{\left\| x \right\|}_{{L_{\infty } {\left( {\mathbf{R}} \right)}}} }} {{{\left\| {\upvarphi _{r} } \right\|}_{\infty } }}} \right)}} }} }^{{\frac{{r - 1}} {r}q}} {\left\| {x^{{{\left( r \right)}}} } \right\|}^{{\frac{q} {r}}}_{\infty } ,\quad r \in {\mathbf{N}},$$

where φ _r is the perfect Euler spline, on the segment [a, b] of monotonicity of x for q ≥ 1 and for arbitrary q > 0 in the case where r = 2 or r = 3.

As a corollary, we prove the well-known Ligun inequality for periodic functions x ∈ L _∞ ^r, namely,

$${\left\| {x^{{{\left( k \right)}}} } \right\|}_{q} \leq \frac{{{\left\| {\upvarphi _{{r - k}} } \right\|}_{q} }} {{{\left\| {\upvarphi _{r} } \right\|}^{{{1 - k} \mathord{\left/ {\vphantom {{1 - k} r}} \right. \kern-\nulldelimiterspace} r}}_{\infty } }}{\left\| x \right\|}^{{{1 - k} \mathord{\left/ {\vphantom {{1 - k} r}} \right. \kern-\nulldelimiterspace} r}}_{\infty } {\left\| {x^{{{\left( r \right)}}} } \right\|}^{{k \mathord{\left/ {\vphantom {k r}} \right. \kern-\nulldelimiterspace} r}}_{\infty } ,\quad k,r \in {\mathbf{N}},\quad k < r,\quad 1 \leq q < \infty ,$$

for q ∈ [0, 1) in the case where r = 2 or r = 3.