Functions of bounded second $$p$$ p -variation

Functions of Boundedkthp-Variation and Continuity Modulus

“…−1/ ( ; ) = ( −1/ ) if and only if ∈ . We extend the results proved in [1,9] to the case of functions of bounded th -variation. In Section 2 we prove some results which we will use in the proofs of our main theorems.…”

“…In the following theorem we prove the continuity at the end points of the scale determined by V ( ) , ( ) in the same manner as in [1,9]. ( ) , ( ) = V ( ) ,1/ ( ), and (ii) if ∈ ( ) , , for some > 0, then…”

“…We set ( ; ) = sup 0<ℎ≤ ‖Δ ℎ ‖ , which is the thmodulus of continuity; see [1,9]. By Hölder's inequality and Fubini's theorem…”

“…The above results motivated Barza and Silvestre to study the scale of spaces (2) , between the class (2) of functions of bounded second -variation and the Sobolev space 2 (see [9]) as an extension of [1].…”

“…Following [9,10], let > 1 and 0 ≤ ≤ 1/ . Let Π be a partition of the form 0 = 1,1 < 1,2 < ⋅ ⋅ ⋅ < 1, ≤ 1, +1 < ⋅ ⋅ ⋅ < 1,2 ≤ ⋅ ⋅ ⋅ ≤ 2, +1 < ⋅ ⋅ ⋅ ≤ ⋅ ⋅ ⋅ < 3,1 < ⋅ ⋅ ⋅ < ,1 < ⋅ ⋅ ⋅ < , ≤ , +1 < ⋅ ⋅ ⋅ < ,2 ≤ ⋅ ⋅ ⋅ ≤ ,1 < ⋅ ⋅ ⋅ < , ≤ , +1 < ⋅ ⋅ ⋅ < ,2 = 0 + 1.…”