Hölder Inequality with Variable Index

Transcription

1 Theoretical Mathematics & Alications, vol.4, no.3, 204, 9-23 ISSN: (rint), (online) Scienress Ltd, 204 Hölder Ineuality with Variable Index Richeng Liu Abstract By using the Young ineuality, we considered the classic Hölder ineuality extended it to the case that the conjugate indices were, essentially bounded Lebesgue measurable functions. Mathematics Subject Classification: 26A42; 26D07 Keywords: Young ineuality; Hölder ineuality; essentially bounded Lebesgue measurable functions Introduction We know, Hölder ineuality [] is very imortant in mathematical analysis real analysis, has a very wide range of alications in differential euations many other branches of mathematics, in articular in estimates of solutions for artial differential euations. It is no exaggeration to say that if there Northeast Petroleum University; Mathematics College; Deartment of Information Comuting Sciences; No.49 azhan Road; High-tech Develoment Zone; Daing 6338, China. -mail: richengliu@sina.com Article Info: Received: March 2, 204. Revised: May 7, 204. Published online : August 3, 204.

2 20 Hölder Ineuality with Variable index is no Hölder ineuality, then it would be imossible for us to solve many artial differential euations. The classic Hölder ineuality holds for the conjugate indices, being constants. While in alication, we will also meet the case that the indices are the variables, for examle, when we make estimate for solutions of artial differential euations with x ( ) Lalace oerator (extended as Lalace oerator), the classic Hölder ineuality it will become helless. So we need to extend the classic Hölder ineuality to the case for the conjugate indices, being variable. In this aer, we shall extend the classic Hölder ineuality to the case that the conjugate indices, were essentially bounded Lebesgue measurable functions. 2 Preliminary Notes N R be a Lebesgue measurable set L( ) be the set of all Lebesgue measurable functions on. x ( )( > ), ( )( ) x > denote the Lebesgue measurable functions which are almost everywhere bounded on. = ess x ( ), 2 = esssu x ( ), = ess x ( ), 2 = esssu x ( ), where ess esssu denote the essential imum the essential suremum of f( x) L( ) on, defined as ess f( x) : = su f( x), esssu f( x) : = su f( x), f( x) L( ). m= 0 x\ m= 0 x\ Lemma 2.. or ess esssu, there exist null subsets (.. i e m = 0 m = 0) such that This can be seen in [2]. f \ ( x x ) ess f ( x = ), su f ( x x ) esssu f ( x = ). \

3 Richeng Liu 2 3 Main Results Theorem 2.. Suose x ( ) x ( ) are almost everywhere conjugate on, i.e. + = aeon... If the two constants α β satisfying the x ( ) x ( ) following conditions f x x = f x x f x x ( x) / α ( x) / ( x) / 2 [ ( ) d ] max{[ ( ) d ],[ ( ) d ] } then gx x = gx x gx x, ( x) / β ( x) / ( x) / 2 [ ( ) d ] max{[ ( ) d ],[ ( ) d ] } f xgx ( ) ( ) L( ), ( ) / ( ) / ( ) ( ) d 2 ( ) x α x β f xgx x f x d x] [ gx ( ) d x]. () Proof. Owing to Lemma 2., there exist null subsets, that x \ K,, x ( ) = ess x ( ), su x \ K x ( ) = esssu x ( ) ; x ( ) = ess x ( ), su x \ K x ( ) = esssu x ( ). x \ K such S = K K, then S is a null set. By the definitions of ess esssu, we have x \ S x ( ) = ess x ( ), su x \ S x ( ) = esssu x ( ) ; x \ S x ( ) = ess x ( ), su x \ S x ( ) = esssu x ( ). Owing to the act that integral value is identically eual to 0, we assume that the essential imum the essential suremum of f( x) L( ) on are the suremum imum of f( x) L( ) on resectively, for simlicity. Write =, ( x) f( x) dx By Young ineuality, we have ( ) gx ( ) x dx =.

4 22 Hölder Ineuality with Variable index further then ( x) ( x) f( xgx ) ( ) f( x) gx ( ) +, / ( x) / ( x) x ( ) x ( ) f( xgx ) ( ) ( ) f( x) ( ) gx ( ) x ( ) x ( ) = or t / ( x) ( x) / ( x) ( x) +. (2) 2 / t / ( x) ( ) = max( ), (3) x. By noticing that we get = or t 2 / ( x) / ( x) ( ) = ( ) ( ), (/ t) / ( x) ( ) = max( ), (4) x. By virtues of (2) (4), we obtain f( xgx ) ( ) ( ) f( x) ( ) gx ( ) Uon integrating (5) about x on we get / t ( x) (/ t) ( x) +. (5) (/ t) / t f( xgx ) ( ) dx 2. (6) Kee in mind the assignments for t in (6), we obtain the desired result. t This comleted the roof of the theorem.

5 Richeng Liu 23 References [] N. L. Carothers, Real Analysis. Cambridge University Press, Cambridge, [2] Walter Rudin, unctional Analysis, Second edition, McrawHill, New York, 990.