On integral bounded variation

Abstract

In this paper we will investigate the concept of the q-integral p-variation introduced in 1970's by Terehin. This kind of integral variation has been mainly used, until now, to describe the regularity of functions in L p in terms of the L q -norm (for q > p).In the present paper we show that the class of functions of bounded q-integral p-variation appears a very nice tool to work with integral operators, including these corresponding to Riemann-Liouville fractional integration. We also give the acting conditions for autonomous superposition operators and prove some existence theorems for Hammerstein integral equation in this space.

Rate of convergence of exponential type operators related to $$p\left( x\right) =2x^{3/2}$$ for functions of bounded variation

“…Remark 2 One may extend the present result to generalized bounded variations of integral type (see [4]) and for BV ( p) and ϕ BV discussed in [12]. This may be treated as further research in this direction.…”

Hardy’s inequalities, Hilbert inequalities and fractional integrals on function spaces of $q$-integral $p$-variation

“…It satisfies (3.1) with θ = α and Notice that the mapping properties for Riemann-Liouville integrals on I p,q (R + ) were given in [9,Section 6]. The result in [9,Theorem 5] shows that R α is a bounded operator from Lebesgue spaces to I p,q (R + ). Thus, Theorem 3.4 gives a complementary result for Riemann-Liouville integrals on I p,q (R + ).…”

“…Furthermore, for any θ ∈ R, t θ g(t) ∈ I θ p,q (R + ). The above example is a particular case of [9,Proposition 2].…”