AN ESTIMATE OF THE INTEGRAL [xxx] WITH HOLDER CONTINUOUS FUNCTIONS a v(x) AND v(x)
In the present paper we establish the estimate b j u'(x)v{x) dx\ < K{b a) assuming that the given integral exists, u(x) e Lipa, v(x) e Lipp (0 < Q < 1, 1-Q < ß <1); the constant K depends on q, ß, max|t;(x)| and the Holder constants of the functions u{x) and ?;(z). AHS Subject Classification: 26D15; 26D10 If two functions u(x) and v{x) satisfy the Holder inequality respectively with exponents a and ß, where 0 < a,ß < 1 and a + ß > 1, then the existence of the Riemann-Stieltjes integral b J V du a is well known.It was established in the papers of L.C.Young [3] and V.Kondurar [1]. V.I.Macaev and M.Z.Solomjak [2] considered the weak Riemann-Stieltjes in2 0 8 Umanskiy tegral 1 j f d g 0 and obtained an estimate of the corresponding integral sums when the functions f(x) and g{x) satisfy appropriate conditions (see theorem 1 below). In this paper we will obtain an estimate of the Riemann integral 0 j u'{x)v{x) dx with Holder continuous functions u{x) and v(x) satisfying some additional conditions (see theorem 2).This estimate can be used in the qualitative theory of partial differential equations for the exploration of removable singularities for Holder continuous solutions of peurtial differential equations with Holder continuous coefficients. Let C'(c,d) be the Space of functions f(x) having continuous first order derivative f'(x) for all ie(c,d) .Denote by Lipa[c,d] (0<q<1) the class of functions f(x) (x e [c,d]) satisfying the Holder inequality |/(a;)-/(j/)| <const|x-j/r for all x,j/e[c,rf].The Holder constant of a function f{x)eLipa[c,d] is defined as usual by Laif) = sup — r — T ^ . F y\ Let Li [0,1] denote the Space of the Lebesgue integrable functions f{t) for which 1 ii/ik = / \m\dt<^. 0 For /(t)eLi[0,1] consider l r wi[f-,h]= snp [ \f(t + r)-f(t)\dt {0<h<l) 0<r<h J 0 and for. 0 < a < 1 define 1 dh if this (Lebesgue) integral exists.The following theorem was proved in [2] (see theorem 1, page 518). b An estimate of the integral / u ' ( x ) v { x ) dx 2 0 9 THEOREM 1 (V.I.Macaev and M.Z.Solomjak). Suppose /(i)6Li[0,1] and M,a(/)<oo for some number 0<a<l.Let ff(t)eLipi_a[0,1]. For an arbitrary set of points 0 = to < < • • • < tn = 1 denote R= m^{tk-tk-i). l<K<.n Then the following limit exists Jmdt K-^O^i tk-tk-l tk-l and it is independent of the choice of points {tt}.In addition S satisfies Using this result we will prove the following theorem. Suppose that a function u(x)eC"(c,d) n Lipa[c,d\ and a function v{x)eLipß[c,d], where 0 < q < 1 , 0 < / ? < l a n d a + /3>l.Let V = sup |»j(x)| ie(c,<i] and [a, b] C {c,d) be an arbitrary interval such that 0 < 6 a < l.Then i j u'(x)v(x)dx\ ^Kib-a)" a with K = K{a, ß, K, Lß{v)) > 0. PROOF: For any xe[a,6] let t = {x a)/{b a). Then x = x{t) = a + t{b a) with [0,1]. Consider the functions f(t) = v{x{t))=v{a + t(b-a)) and g{t) = u(x{t)) = u{a + t{b a)). where t 6 [0,1]. Then g{t) e C"[0,1], u'(x{t)) = g'it)/ib a) and therefore 6 1 Ju'(x)v(x)dx = Jg'{t)f(t)dt. a 0 Consider an arbitrary set of points 0 = to < ti < • • • < t„ = 1 and denote