Functions with different strong and weak 𝐹-variations

Abstract

This paper shows by example how different the strong Φ \Phi -variation can be from the weak Φ \Phi -variation. Let Φ \Phi be a convex function on [ 0 , ∞ ) [0,\infty ) with Φ ( 0 ) = 0 \Phi (0) = 0 . A continuous function f f on [ a , b ] [a,b] is of bounded strong Φ \Phi -variation if sup Σ Φ ( | f ( x i ) − f ( x i − 1 ) | ) > ∞ \sup \Sigma \Phi (|f({x_i}) - f({x_{i - 1}})|) > \infty for the partitions of [ a , b ] [a,b] . Since inf Σ Φ ( | f ( x i ) − f ( x i − 1 ) | ) = 0 \inf \Sigma \Phi (|f({x_i}) - f({x_{i - 1}})|) = 0 if lim x → 0 x − 1 Φ ( x ) = 0 {\lim _{x \to 0}}{x^{ - 1}}\Phi (x) = 0 , the weak Φ \Phi -variation is defined as inf Σ Φ ( ω ( f ; x i − 1 , x i ) ) \inf \Sigma \Phi (\omega (f;{x_{i - 1}},{x_i})) , where ω ( f ; c , d ) \omega (f;c,d) is the oscillation of f f on [ c , d ] [c,d] . Of special interest is the case Φ ( x ) = x p , p ⩾ 1 \Phi (x) = {x^p},p \geqslant 1 , in terms of which strong and weak variation dimensions are defined. They are denoted by dim s ( f ) {\dim _{\text {s}}}(f) and dim w ( f ) {\dim _{\text {w}}}(f) , respectively. By a lemma of Goffman and Loughlin, the Hausdorff dimension d d of the graph of f f provides a lower bound for dim w ( f ) : 1 / ( 2 − d ) ⩽ dim w ( f ) {\dim _w}(f):1/(2 - d) \leqslant {\dim _w}(f) . A Lipschitz condition of order a provides an upper bound for dim s ( f ) : dim s ( f ) ⩽ 1 / α {\dim _s}(f):{\dim _s}(f) \leqslant 1/\alpha . Besicovitch and Ursell showed that 1 ⩽ d ⩽ 2 − α 1 \leqslant d \leqslant 2 - \alpha and gave examples to show that d d can take on any value in this interval. It turns out that these examples provide the extreme cases for variation dimensions; i.e., dim w ( f ) = 1 / ( 2 − d ) {\dim _w}(f) = 1/(2 - d) and dim s ( f ) = 1 / α {\dim _s}(f) = 1/\alpha .