Lipschitz integral condition

A restriction on the behaviour of increase of a function in an integral metric. A function $ f $ in a space $ L _ {p} ( a , b ) $ with $ p \geq 1 $ satisfies the Lipschitz integral condition of order $ \alpha > 0 $ on $ [ a, b ] $ with constant $ M > 0 $ if

$$ \tag{* } \left \{ \int\limits _ { a } ^ { b- } h | f ( x + h ) - f ( x) | ^ {p} \ d x \right \} ^ {1/p} \leq M h ^ \alpha $$

for all $ h \in ( 0 , b - a ) $. In this case one writes $ f \in \mathop{\rm Lip} _ {M} ( \alpha , p ) $, $ f \in H _ {p} ^ \alpha ( M) $ or $ f \in \mathop{\rm Lip} ( \alpha , p ) $, $ f \in H _ {p} ^ \alpha $. For the case of a periodic function (with period $ b - a $) the Lipschitz integral condition is defined similarly, only in inequality (*) the upper limit of integration $ b - h $ must be replaced by $ b $.

References

[1]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
[2]	S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[3]	O.V. Besov, V.P. Il'in, S.M. Nikol'skii, "Integral representations of functions and imbedding theorems" , Wiley (1978) (Translated from Russian)