Orlicz-Lorentz Gauge functional inequalities for some integral operators
aa r X i v : . [ m a t h . F A ] F e b ORLICZ-LORENTZ GAUGE FUNCTIONAL INEQUALITIES FOR POSITIVEINTEGRAL OPERATORS
RON KERMAN AND SUSANNA SPEKTOR
Abstract.
Let f ∈ M + ( R + ), the class of nonnegative, Lebesgure-measurable functions on R + = (0 , ∞ ). We deal with integral operators of the form( T K f )( x ) = Z R + K ( x, y ) f ( y ) dy, x ∈ R + , with K ∈ M + ( R ).We are interested in inequalities ρ (( T K f ) ∗ ) ≤ Cρ ( f ∗ ) , in which ρ and ρ are functionals on functions h ∈ M + ( R + ), and h ∗ = µ − h ( t ) , t ∈ R + , with µ h ( λ ) = |{ x ∈ R + : h ( x ) > λ }| . Specifically, ρ and ρ are so-called Orlicz-Lorentz functionals of the type ρ ( h ) = ρ Φ ,u ( h ) = inf ( λ > Z R + Φ (cid:18) h ( x ) λ (cid:19) u ( x ) dx ≤ ) , h ∈ M + ( R + );here Φ( x ) = R x φ ( y ) dy , φ an increasing mapping from R + onto itself and u ∈ M + ( R + ).2020 Classification: 46B06, 60C05Keywords: Integral operator, Orlicz-Lorentz gauge functional Introduction
Let K ∈ M + ( R ), the class of nonnegative Lebesgue-measurable functions on R = R + × R + , R + = (0 , ∞ ).We consider positive integral operators T K defined at f ∈ M + ( R n + ) by( T K f )( x ) = Z R + K ( x, y ) f ( y ) dy, x ∈ R + . We are interested in Orlicz gauge functionals ρ and ρ on M + ( R + ) for which ρ (( T K f ) ∗ ) ≤ Cρ ( f ∗ ) , (1)where C > f .The function f ∗ in (1) is the nonincreasing rearrangement of f on R + , with f ∗ ( t ) = µ − f ( t ) , where µ f ( λ ) = |{ x : f ( x ) > λ }| . The gauge functional ρ is given in terms of an N -functionΦ( x ) = Z x φ ( y ) dy, x ∈ R + , φ being a nondecreasing function mapping R + onto itself, and u a locally-integrable (weight)function in M + ( R + ). Specifically, the gauge functional ρ = ρ Φ ,u is defined at f ∈ M + ( R + ) by ρ Φ ,u ( f ) = inf { λ > Z R + Φ (cid:18) f ( x ) λ (cid:19) u ( x ) dx ≤ } . Thus, in (1), ρ = ρ Φ ,u and ρ = ρ Φ ,u . The gauge functionals in (1) involving rearrangementsare referred to as Orlicz-Lorentz functionals.The inequality (1) is shown to follow from ρ ( T L f ∗ ) ≤ Cρ ( f ∗ ) , f ∈ M + ( R + ) , (2)in which L = L ( t, s ) , s, t ∈ R + is the so-called iterated rearrangement of K ( x, y ), whichrearrangement is nonincreasing in each of s and t .Using the concept of the down dual of an Orlicz-Lorentz gauge functional the inequality (2)is reduced to a gauge functional inequalitiy for general functions in Theorem 2.4. Sufficientconditions for stronger integral inequalities likeΦ − Z R + Φ ( cw ( x )( T f )( x )) t ( x ) dx ! ≤ Φ − Z R + Φ ( u ( y ) f ( y )) v ( y ) dy ! , (3)with c > f ∈ M + ( R + ) then serve for (2) and hence for (1).The integral inequality (3) is the same as the corresponding gauge functional inequality ρ Φ ,t ( wT f ) ≤ Cρ Φ ,v ( uf ) , when Φ ( s ) = s q and Φ ( s ) = s p , < p ≤ q < ∞ . The necessary and sufficient conditions arespelled out in this case.The operators T K are treated in Section 2. One can prove similar theorems for such operatorson M + ( R n ). We have preferred to work in the context of one dimension where we believe theresults are simpler and more elegant.Section 3 presents examples either illustrating our results or comparing them to previous work.As well we consider integral operators with kernels that are homogeneous of degree − K ∈ M ( R + ) can beobtained from ours using the fact that such a K is the difference of two positive kernels, namely, K = K + − K − , where K + ( x, y ) = max[ K ( x, y ) ,
0] and K − ( x, y ) = max[ − K ( x, y ) , Positive integral operators on M + ( R + )As a first step in our study of (1) we focus on the related inequality ρ ( T K f ∗ ) ≤ Cρ ( f ∗ ) , f ∈ M + ( R + ) . (4) Theorem 2.1.
Fix K ∈ M + ( R ) and let Φ and Φ be N -functions, with Φ (2 t ) ≈ Φ ( t ) , t ≫ .Given weight functions u , u ∈ M + ( R + ) , R R + u = ∞ , one has (4) for ρ i = ρ Φ i ,u i , i = 1 , , if ρ Ψ ,u ( Sg/U ) ≤ Cρ Ψ ,u ( g/u ) , g ∈ M + ( R +) , (5) ( Sg )( x ) = Z x ( T ′ K g )( y ) dy = Z x Z ∞ K ( z, y ) g ( z ) dzdy,U ( x ) = Z x u , and Ψ i ( t ) = Z t φ − i , i = 1 , . RLICZ-LORENTZ GAUGE FUNCTIONAL INEQUALITIES FOR POSITIVE INTEGRAL OPERATORS 3
Proof.
The identity Z R + gT K f ∗ = Z R + f ∗ T ′ K g readily yields that (4) holds if and only if( ρ ′ ) d ( T ′ K g ) ≤ Cρ ′ ( g ) , where ρ ′ ( g ) = ρ Ψ ,u ( g/u )(6)and ( ρ ′ ) d ( h ) = ρ Ψ ,u (cid:18)Z x h/U ( x ) (cid:19) , h ∈ M + ( R + ) . (7)For (7), see [3, Theorem 6.2]; (6) is straightforward. The proof is complete on taking h = T ′ K ( g ). (cid:3) To replace T K f ∗ in (4) by ( T K f ) ∗ we will require ρ (cid:18) t − Z t f ∗ (cid:19) ≤ Cρ ( f ∗ ) , f ∈ M + ( R + ) . For ρ = ρ Φ ,u Theorem 2.1 allows one to reduce this inequality to ρ Ψ ,u (cid:18)Z x g/U ( x ) (cid:19) ≤ Cρ Ψ ,u ( g/u ) , g ∈ M + ( R + )and another such inequality; here Ψ( x ) = R x φ − . As mentioned in the introduction, such gaugefunctional inequalities are implied by integral inequalities of the form (3). We combine theorems1.7 and 4.1 from Bloom-Kerman [1] to obtain next theorem. Theorem 2.2.
Consider K ( x, y ) ∈ M + ( R ) , which, for fixed y ∈ R + , increases in x and, forfixed x ∈ R + , decreases in y , and which satisfies the growth condition K ( x, y ) ≤ K ( x, z ) + K ( z, y ) , < y < z < x. (8) Let t, u, v and w be nonnegative, measurable (weight) functions on R + and suppose Φ and Φ are N -functions having complementary functions Ψ and Ψ , respectively, with Φ ◦ Φ − convex.Then, there exists c > such that Φ − Z R + Φ ( cw ( x )( T K f )( x )) t ( x ) dx ! ≤ Φ − Z R + Φ ( u ( y ) f ( y )) v ( y ) dy ! (9) for all f ∈ M + ( R + ) , if and only if there is a c > , independent of λ, x > , with Z x K ( x, y ) u ( y ) φ − (cid:18) cα ( λ, x ) K ( x, y ) λu ( y ) v ( y ) (cid:19) dy ≤ c − λ and (10) Z x u ( y ) φ − (cid:18) cβ ( λ, x ) λu ( y ) v ( y ) (cid:19) dy ≤ c − λ, where α ( λ, x ) = Φ ◦ Φ − (cid:18)Z ∞ x Φ ( λw ( y )) t ( y ) dy (cid:19) and β ( λ, x ) = Φ ◦ Φ − (cid:18)Z ∞ x Φ ( λw ( y ) K ( y, x )) t ( y ) dy (cid:19) . R. KERMAN AND S. SPEKTOR
In the case K ( x, y ) = χ (0 ,x ) ( y ) only the first of the conditions in (10) is required. Remark 2.3.
The integral inequality (9) with the kernel K of Theorem 2.2 replaced by any K ∈ M + ( R m + ) implies the norm inequality ρ Φ ,t ( wT K f ) ≤ Cρ Φ ,v ( uf ) , f ∈ M + ( R n + ) , C > . (11)Thus, in the generalization of (9), replace f by fCρ Φ ,v ( uf ) and suppose Φ i (1) = 1 , i = 1 , R R + Φ (cid:18) ufρ Φ ,v ( uf ) (cid:19) v ≤ , we get Z R + Φ (cid:18) wT K fCρ Φ ,v ( uf ) (cid:19) t ≤ , whence (11) holds.To replace T K f ∗ in (4) by ( T K f ) ∗ we will require ρ (cid:18) t − Z t f ∗ (cid:19) ≤ Cρ ( f ∗ ) , f ∈ M + ( R + ) . Conditions sufficient for the inequality to hold are given in
Theorem 2.4.
Let Φ be an N -function satisfying Φ(2 t ) ≈ Φ( t ) , t ≫ , and suppose u is weighton R + with R R + u = ∞ . Then, ρ Φ ,u (cid:18) t − Z t f ∗ (cid:19) ≤ Cρ Φ ,u ( f ∗ ) , f ∈ M + ( R + ) , (12) provided Z R + φ (cid:18) cα ( λ, x ) λ (cid:19) u ( y ) dy ≤ c − λ, with c > independent of λ, x ∈ R + , in which α ( λ, x ) = Z ∞ x Ψ( λ/U ( y )) u ( y ) dy and (13) φ − ( cβ ( λ, x ) /λ ) U ( y ) ≤ c − λ, with c > independent of λ, x ∈ R + , in which β ( λ, x ) = Z ∞ x Φ( λ/y ) u ( y ) dy. Proof.
In Theorem 2.1, take K ( x, y ) = χ (0 ,x ) ( y ) /x, Φ = Φ = Φ and u = u = u to get( Sg )( x ) = Z x Z ∞ y g ( z ) dzz = Z x g + x Z ∞ x g ( y ) dyy , whence (5) reduces to ρ Ψ ,u (cid:18)Z x g/U ( x ) (cid:19) ≤ Cρ Ψ ,u ( g/u ) and (14) ρ Ψ ,u (cid:18) x Z ∞ x g ( y ) dyy /U ( x ) (cid:19) ≤ Cρ Ψ ,u ( g/u ) . RLICZ-LORENTZ GAUGE FUNCTIONAL INEQUALITIES FOR POSITIVE INTEGRAL OPERATORS 5
The first inequality in (14) is a consequence of the modular inequality Z R + Ψ (cid:18) c Z x g/U ( x ) (cid:19) u ≤ Z R Ψ( g/u ) u, which, according [1, Theorem 4.1] holds if the first inequality in (13) does.Again, by duality, the second inequality in (14) holds when the modular inequality Z R + Φ (cid:18) c x Z x g (cid:19) u ≤ Z R + Φ( g/u ) u does, which inequality holds if and only if one has the second condition in (13). (cid:3) In Theorem 2.5 below we show the boundedness of T K f depends on that of T L f ∗ , wherethe kernel L is the iterated rearrangement of K considered in [2]. Thus, for each x ∈ R + , werearrange the function k x ( y ) = K ( x, y ) with respect to y to get ( k ∗ x )( s ) = K ∗ ( x, s ) = k s ( x ) andthen rearrange the function of x so obtained to arrive at ( K ∗ ) ∗ ( t, s ) = ( k ∗ s )( t ) = L ( t, s ). It isclear from its construction that K ( t, s ) is nonincreasing in each of s and t . Theorem 2.5.
Consider K ∈ M + ( R ) and set L ( t, s ) = ( K ∗ ) ∗ ( t, s ) , s, t ∈ R + . Suppose Φ and Φ are N -functions, with Φ (2 t ) ≈ Φ ( t ) , t ≫ , and let u and u be weight functions,with R R + u = ∞ . Then, given the conditions (13) for Φ = Φ and u = u one has ρ Φ ,u (( T K f ) ∗ ) ≤ Cρ Φ ,u ( f ∗ ) , provided ρ Φ ,u ( T L f ∗ ) ≤ Cρ Φ ,u ( f ∗ ) , f ∈ M + ( R + ) . Proof.
We claim ( T K f ) ∗∗ ( t ) ≤ ( T L f ∗ ) ∗∗ ( t ) , t ∈ R + , (15)in which, say, ( T K f ) ∗∗ ( t ) = t − R t ( T K f ) ∗ .Indeed, given E ⊂ R + , | E | = t , Z E T K f ≤ Z E Z R + K ∗ ( x, s ) f ∗ ( s ) ds = Z R + f ∗ ( s ) ds Z R + χ E ( x ) K ∗ ( x, s ) dx ≤ Z R + f ∗ ( s ) ds Z t L ( u, s ) du = Z t ( T L f ∗ )( u ) du. Taking the supremum over all such E, then dividing by t yields (15).Next, the inequality ρ Φ ,u (( T K f ) ∗∗ ) ≤ Cρ Φ ,u (( T L f ∗ ) ∗∗ )is equivalent to ρ Φ ,u (( T K f ) ∗ ) ≤ Cρ Φ ,u ( T L f ∗ ) , given (13) for Φ = Φ and u = u . For, in that case, ρ Φ ,u (( T K f ) ∗ ) ≤ ρ Φ ,u (( T K f ) ∗∗ ) ≤ ρ Φ ,u (( T L f ∗ ) ∗∗ ) ≤ Cρ Φ ,u ( T L f ∗ ) . The assertion of the theorem now follows. (cid:3)
R. KERMAN AND S. SPEKTOR
Theorem 2.6.
Let
K, L, Φ , Φ , u and u be as in theorem 2.5. Assume, in addition, that Φ (2 t ) ≈ Φ ( t ) , t ≫ , R R + u = ∞ and that conditions (13) hold for Φ = Φ , u = u . Then, ρ Φ ,u (( T K f ) ∗ ) ≤ Cρ Φ ,u ( f ∗ ) provided ρ Ψ ,u ( H f /U ) ≤ Cρ Ψ ,u ( f /u ) and (16) ρ Φ ,i u ◦ i ( H g ) ≤ Cρ Φ ,i u ◦ i ( g/i ) , f ∈ M + ( R + ) , where ( H f )( x ) = Z x M ( x, y ) f ( y ) dy and ( H g )( y ) = Z y M ( y, x ) g ( x ) dx, with i ( x ) = x − , M ( x, y ) = Z x L ( y, z ) dz and M ( y, x ) = Z x − L ( y − , z ) dz. Proof.
In view of Theorem 2.5 , we need only verify conditions (16) imply ρ Φ ,u ( T L f ∗ ) ≤ Cρ Φ ,u ( f ∗ ) , f ∈ M + ( R + ) . (17)Now, according to Theorem 2.1, (17) will hold if one has (5) with K = L . Again, (5) is equivalentto the dual inequality ρ Φ ,u ( S ′ g ) ≤ Cρ Φ ,u ( gU /u ) , where ( S ′ g )( y ) = Z ∞ (cid:20)Z x L ( y, z ) dz (cid:21) g ( x ) dx = (cid:18)Z y + Z ∞ y (cid:19) (cid:20)Z x L ( y, z ) dz (cid:21) g ( x ) dx = Z y (cid:20)Z x L ( y, z ) dz (cid:21) g ( x ) dx + Z ∞ y (cid:20)Z x L ( y, z ) dz (cid:21) g ( x ) dx. But, the operator Z ∞ y (cid:20)Z x L ( y, z ) dz (cid:21) g ( x ) dx has associate operator Z x (cid:20)Z x L ( y, z ) dzf ( y ) dy (cid:21) = ( H f )( x )and ρ Ψ ,u ( H f /U ) ≤ Cρ Ψ ,u ( f /u ) . Again, if there is to exist
C > ρ Φ ,u (cid:18)Z y (cid:20)Z x L ( y, z ) dz (cid:21) g ( x ) dx (cid:19) ≤ Cρ Φ ,u ( gU /u ) , (18)then, for such a C , Z ∞ Φ (cid:18)Z y (cid:20)Z x L ( y, z ) dz (cid:21) g ( x ) dx/Cρ Φ ,u ( gU /u ) (cid:19) u ( y ) dy ≤ RLICZ-LORENTZ GAUGE FUNCTIONAL INEQUALITIES FOR POSITIVE INTEGRAL OPERATORS 7 for all g ∈ M + ( R + ) , g = 0 a.e. That is, on making the changes of variables y → y − then x → x − , one will have Z ∞ Φ Z y "Z x − L ( y − , z ) dz g ( x − ) x dxx /Cρ Φ ,u ( gU /u ) ! u ( y − ) dyy ≤ . But, Z ∞ Φ ( gU /λu ) u = Z ∞ Φ ( g ◦ iU ◦ i/λu ◦ i ) i u ◦ i, so ρ Φ ,u ( gU /u ) = ρ Φ ,i u ◦ i ( g ◦ iU ◦ i/u ◦ i ) , whence (18) amounts to ρ Φ ,i u ◦ i ( H ( i g ◦ iU ◦ i/u ◦ i )) ≤ Cρ Φ ,i u ( g ◦ u ◦ iU ◦ i )or ρ Φ ,i u ◦ i ( H g ) ≤ Cρ Φ ,i u ◦ i ( g/i ) , since i g ◦ iU ◦ i/u ◦ i is arbitrary. (cid:3) We have to this point shown that the inequality (1) holds for ρ i = ρ Φ i ,u i , i =1 , , wheneverthe inequalities (16) hold for H and H . In Theorem 2.7 below we give four conditions which,together with (13) for Φ and Φ , guarantee (16).The kernel M ( x, y ) of the operator H is increasing in x and decreasing in y . Similarly, thekernel M ( y, x ) of H is increasing in y and decreasing in x . The operators H and H will beso-called generalized Hardy operators (GHOs) if their kernels satisfy the growth conditions M ( x, y ) ≤ M ( x, z ) + M ( z, y ) , y < z < x, and (19) M ( y, x ) ≤ M ( y, z ) + M ( z, x ) , x < z < y. Neither of the conditions in (19) are guaranteed to hold. They have to be assumed in theorem2.7 below so that we may apply Theorem 1.7 in [1] concerning GHOs. Theorem 3.1 in the nextsection gives a class of kernels for which (19) is satisfied.
Theorem 2.7.
Let
K, L, Φ , Φ , u , u , M , M , H and H be as in Theorem 2.6. Assume, inaddition, that Φ ◦ Φ − is convex and that M and M satisfy the growth conditions (19). Then,one has Z R + Φ ( T K f ) ∗ u ≤ C Z R + Φ ( f ∗ ) u , (20) provided Z x M ( x, y ) φ (cid:18) cα ( λ, x ) M ( x, y ) λ (cid:19) u ( y ) dy ≤ c − λ, Z x φ ( cβ ( λ, x )) U ( y ) dy ≤ c − λ, Z y M ( y, x ) φ − (cid:18) cα ( λ, y ) M ( y, x ) λu ( x − ) (cid:19) x − dx ≤ c − λ (21) and Z y φ − (cid:18) cβ ( λ, y ) λu ( x − ) (cid:19) x − dx ≤ c − λ. R. KERMAN AND S. SPEKTOR
Here, α ( λ, x ) = Ψ ◦ Ψ − (cid:18)Z ∞ x Ψ (cid:18) λU ( y ) (cid:19) u ( y ) dy (cid:19) ,β ( λ, x ) = Ψ ◦ Ψ − (cid:18)Z ∞ x Ψ (cid:18) λM ( y, x ) U ( y ) (cid:19) u ( y ) dy (cid:19) α ( λ, y ) = Φ ◦ Φ − (cid:18)Z ∞ y Φ ( λ ) x − u ( x − ) dx (cid:19) and β ( λ, y ) = Φ ◦ Φ − (cid:18)Z ∞ y Φ ( λM ( x, y )) x − u ( x − ) dx (cid:19) . Proof.
We prove the result involving H ; the proof for H is similar. Now, the norm inequalityfor H for holds if one has the integral inequalityΦ − Z R + Φ ( cH g ) y − u ( y − ) dy ! ≤ Φ − Z R + Φ ( y g ( y )) y − u ( y − ) dy ! . (22)Indeed, in the latter replace g ( y ) by g ( y ) /ρ Φ ,y − u ( y − ) ( y g ( y )) = λ , to get Z R + Φ (cid:18) cH gλ (cid:19) y − u ( y − ) dy ≤ Φ ◦ Φ − Z R + Φ (cid:18) y g ( y ) λ (cid:19) y − u ( y − ) dy ! ≤ Φ ◦ Φ − (1) = 1 , where we have assumed, without loss of generality, that Φ (1) = Φ (1) = 1. Hence, ρ Φ ,y − u ( y − ) ( H g ) ≤ Cλ = Cρ Φ ,y − u ( y − ) ( g/i ) . But, (22) is valid if and only if the third and fourth conditions in (22) hold. (cid:3) Examples
Theorem 3.1.
Let k be a nonnegative, nonincreasing function on R + . Then, the growth condi-tions (19) are satisfied for K ( x, y ) = k ( x + y ) , x, y ∈ R + .Proof. We observe that K ( x, y ) = L ( x, y ) since K decreases in each of x and y . So, M ( x, y ) = Z x k ( y + s ) ds and M ( x, y ) = Z x − k ( y − + s ) ds. Now, given y < z < x , M ( x, y ) = Z x k ( y + s ) ds = Z z k ( y + s ) ds + Z xz k ( y + s ) ds = M ( y, z ) + Z x − z k ( y + z + s ) ds ≤ M ( y, z ) + Z x k ( z + s ) ds = M ( y, z ) + M ( x, z ) . RLICZ-LORENTZ GAUGE FUNCTIONAL INEQUALITIES FOR POSITIVE INTEGRAL OPERATORS 9
Again, given x < z < y,M ( x, y ) = Z x − k ( y − + s ) ds = Z z − k ( y − + s ) ds + Z x − z − k ( y − + s ) ds = M ( z, y ) + Z x − − z − k ( y − + z − + s ) ds ≤ M ( z, y ) + Z x − k ( z − + s ) ds = M ( z, y ) + M ( x, z ) . (cid:3) Theorem 3.2.
Fix the indices p and q, < p ≤ q < ∞ , and suppose K ( x, y ) = k ( x + y ) , x, y ∈ R + , where k is nonnegative and nonincreasing on R + . Then, one has Z R + ( T K f ) q ! /q ≤ C Z R + f p ! /p , (23) with c > independent of f ∈ M + ( R + ) , if and only if cα ( x ) q − Z x M ( x, y ) q u ( y ) dy ≤ cβ ( x ) q − Z x u ( y ) dy ≤ cα ( y ) p ′ − Z y M ( y, x ) p ′ u ( x − ) − p ′ x − dx ≤ and cβ ( y ) p ′ − Z y u ( x − ) − p ′ x − dx ≤ . Here c > is independent of x and y . α ( x ) = (cid:18)Z ∞ x U ( y ) − p ′ u ( y ) dy (cid:19) q ′ /p ′ ,β ( x ) = Z ∞ x (cid:20) M ( x, y ) U ( y ) (cid:21) p ′ u ( y ) dy ! q ′ /p ′ ,α ( y ) = (cid:18)Z ∞ y x − u ( x − ) dx (cid:19) p/q and β ( y ) = (cid:18)Z ∞ y M ( y, x ) q x − u ( x − ) dx (cid:19) p/q . Proof.
The N -functions Φ ( t ) = t q and Φ ( t ) = t p , as well as the weights u and u satisfy theconditions required in theorem 2.7. According to Theorem 3.1, so do the kernels M and M of H and H , respectively. We conclude, then, that (20) holds for T K given (21), which in ourcase are the inequality (23) and the conditions (24). We observe that λ cancels out in the latterconditions and we are left with α (1 , x ) = α ( x ) , etc. (cid:3) We observe that if in Theorem 2.5, the weights v i ≡ , i = 1 ,
2, then the inequality ρ Φ (( T K f ) ∗ ) ≤ Cρ Φ ( f ∗ )is the same as ρ Φ ( T K f ) ≤ Cρ Φ ( f ) . In this context we consider an integral operator T K with K ( x, y ) = k ( p x + y ), where k ( t ) isnonincreasing, with k (1 / t ) ≤ Ck ( t ) , t ∈ R + , and R a k ( p x + y ) dy ≤ ∞ for all a ∈ R + .The growth condition on k ensures that1 C ( T K f )( x ) ≤ k ( x ) Z x f + Z ∞ x kf ≤ C ( T K f )( x ) , for C > f ∈ M + ( R + ) and t ∈ R + .Suppose now Φ ( t ) = t q and Φ ( t ) = t p , < p ≤ q < ∞ . According to Theorem 2.2, theinequality "Z R + ( k ( x ) Z x f ) q dx /q ≤ c "Z R + f p /p (25)holds if and only if x cα ( λ, x ) p ′ λ ! p ′ = Z x cα ( λ, x ) p ′ λ ! p ′ dy ≤ c − λ, where α ( λ, x ) = (cid:20)Z ∞ x ( λ, k ( y )) q dy (cid:21) p ′ /q = λ p (cid:20)Z ∞ x k ( y ) q dy (cid:21) p ′ /q , that is x (cid:18)Z ∞ x k ( y ) q dy (cid:19) p ′ /q ≤ c − p ′ , x ∈ R + . (26)Again, "Z R + (cid:18)Z ∞ x k p (cid:19) q dx /q ≤ C [ f p ] /p if and only if "Z R + (cid:18) k ( x ) Z x g (cid:19) p ′ dx /p ′ ≤ C "Z R + g q ′ /q or x (cid:20)Z ∞ x k ( y ) p ′ dy (cid:21) q/p ′ ≤ c, x ∈ R + . (27)Thus, (25) holds when and only (26) and (27) do.For example, if K ( x, y ) = x + y ) λ/ , max[1 /p ′ , /q ] < λ < , k ( x ) = x − λ , k ( y ) = y − λ andthe conditions become x p ′ (1 /q − λ ) ≤ c − p ′ and x q (1 /p ′ − λ ) ≤ c − q , which conditions are satisfied and only 1 q = λ − p ′ . RLICZ-LORENTZ GAUGE FUNCTIONAL INEQUALITIES FOR POSITIVE INTEGRAL OPERATORS 11
This kernel K does not satisfy the classical Kantorovic condition usually invoked to prove (25)for T K . Indeed, K ( x, y ) ≈ ( x + y ) − λ , whence, for, p > (cid:20)Z ∞ K ( x, y ) p ′ dy (cid:21) /p ′ ≈ x − λ +1 /p ′ and, therefore, (cid:20)Z ∞ (cid:20)Z ∞ K ( x, y ) p ′ dy (cid:21) q dx (cid:21) /q ≈ (cid:20)Z ∞ x ( − λ +1 /p ′ ) q dx (cid:21) /q = ∞ . Finally, R. O’Neil in [10] proved that, for K ∈ M + ( R ), one has, for each f ∈ M + ( R + ),1 x Z x ( T K f ) ∗ ( y ) dy ≤ Z ∞ K ∗ ( xy ) f ∗ ( y ) dy. See also Torchinsky [12]. Given K ( x, y ) = k ( p x + y ) as above, K ∗ ( t ) = k ( t / ), so the rightside of the O’Neil inequality is Z ∞ k ( √ xy ) f ∗ ( y ) dy. On the other hand, ( T K f ∗ )( x ) = Z ∞ k ( p x + y ) f ∗ ( y ) dy. We see that, if Φ (2 t ) ≈ Φ ( t ), for t ≫
1, the O’Neil inequality yields Z ∞ Φ (( T K f ∗ )( x )) dx ≤ Z ∞ Φ (cid:18)Z ∞ k ( √ xyf ∗ ( y )) dy (cid:19) dx. Whereas we have Z ∞ Φ (( T K f ∗ ) ∗ ( x )) dx = Z ∞ Φ (( T K f ∗ )( x )) dx = Z ∞ Φ (cid:18)Z ∞ k ( p x + y f ∗ ( y ) dy ) (cid:19) dx. Observing that k ( p x + y ) = k (cid:16)q x + y xy √ xy (cid:17) = k (cid:16)q yx + xy √ xy (cid:17) , we see our bound istighter. 4. Past results
As mentioned above,when Φ ( t ) = t q , Φ ( t ) = t p , p, q ∈ [1 , ∞ ], and u = u = 1, (1) becomesthe classical Lebesgue inequality (cid:20)Z ( T K f )( x ) q dx (cid:21) /q ≤ C (cid:20)Z f ( y ) p dy (cid:21) /p . (28)In the case K is homogeneous of degree −
1, that is, K ( λx, λy ) = λ − K ( x, y ) , λ, x, y ∈ R + , thewell-known result of Hardy-Littlewood-Polya in [7] asserts that (23) holds when p = q if andonly if Z R + K (1 , y ) y − /p dy < ∞ . More generally, we have
Theorem 4.1.
Let K ∈ M + ( R ) be homogeneous of degree − . Suppose given N -functions Φ and Φ and weights u and u , all satisfying the conditions of Theorem 2.2. Then, one has ρ Φ ,u ( T K f ) ≤ Cρ Φ ,u ( f ) , (29) with C > independent of f ∈ M + ( R + ) , provided Z R + K (1 , t ) h ( ρ Φ ,u , ρ Φ ,u )( t ) dt < ∞ . (30) In (30), h ( ρ Φ ,u , ρ Φ ,u )( t ) = inf { M > ρ Φ ,u ( f ( ts )) ≤ M ρ Φ ,u ( f ( s )) } . Proof.
We observe that K ( x, y ) = x − K (1 , y/x ) so( T K f )( x ) = x − Z R + K (1 , y/x ) f ( y ) dy = Z R + K (1 , t ) f ( tx ) dt. Again, the assumptions on the N -functions and weights guarantee that the ρ Φ i ,u i i = 1 , , areequivalent to so-called rearrangement-invariant norms on M + ( R + ). Theorem 3.1 in [3] thenensures that (30) implies (29). (cid:3) In the case Φ = Φ = Φ and u ( x ) = u ( x ) = u ( x ) = ( α + 1) x α , α > − , with Φ(2 x ) ≈ Φ( x ) , x ≫
1, one has, using methods of [3], h Φ ,u ( t ) ≈ sup s ∈ R + Φ − ( s − ( α +1) )Φ − ( s/t ) − ( α +1) , t ∈ R + , where h Φ ,u = h ( ρ Φ ,u,ρ Φ ,u ). When Φ ( x ) = Φ ( x ) = Φ( x ) = x p , p > h Φ ,u ( t ) = R s/t u ( y ) dy + ( s/t ) p R ∞ s/t u ( y ) y − p dy R s u ( y ) + s p R ∞ s u ( y ) y − p dy /p , t ∈ R + ;see [6].The classical mixed norm condition Z R + "Z R + K ( x, y ) p ′ dy q/p ′ dx < ∞ , p ′ = pp − , guaranteeing (1) for T K is due to Kantorovic [8]. In Walsh [13] such mixed norm conditionsinvolving so-called weak Lorentz norms followed by interpolation are shown to yield the sameinequality. This extended earlier work of Strichartz [11]. For a discussion of yet earlier work ofthis kind see [9].We have already introduced the generalized Hardy operators (GHOs)( T K f )( x ) = Z x K ( x, y ) f ( y ) dy, that is K ( x, y ) = K ( x, y ) χ (0 ,x ) ( y ), with K increasing in x , decreasing in y and satisfying thegrowth condition K ( x, y ) ≤ K ( x, z ) + K ( z, y ) , y < z < x, studied in Bloom-Kerman [1]. An exhaustive treatment of these and similar operators on mono-tone functions for all p, q ∈ R + is given in Gogatishvili-Stepanov [6]. RLICZ-LORENTZ GAUGE FUNCTIONAL INEQUALITIES FOR POSITIVE INTEGRAL OPERATORS 13
References [1] S. Bloom, R. Kerman, Weighted L Φ integral inequalities for operators of Hardy type, Studia Mathematica ,110(1) (1994), 35–52.[2] A.P Bolzinski, Multivariate rearrangements and Banach function spaces with mixed norms,
Trans. Amer.Math. Soc. , 2631(1) (1981), 149–167.[3] D. Boyd, The Hilbert transformation on rearangement invariant Banach spaces,
Thesis, University ofToronto , (1966).[4] A. Gogatishvili, R. Kerman, The rearrangement-invariant space Γ p, Φ , Positivity , 18(2) (2012), 319–345.[5] M.L. Goldman, R. Kerman, The dual of the cone of decreasing functions in a weighted Orlicz class and theassociate of an Orlicz-Lorentz space,
Differential Operators. Problems of Mathematical Education, Proc.Intern. Conf. Dedicated to the 75th Birthday of Prof. L. D. Kudrjavtsev (Moscow, 1998).[6] A. Gogatishvili, V.D. Stepanov, Reduction theorems for weighted integral inequalities on the cone of mono-tone functions,
Uspehi Math. Nauk. , 68(4/412) (2013), 3–68.[7] H.G. Hardy, J.E. Littlewood, G. Polya, Inequalities,
Cambridge Univ. Press. , Ney York, (1952).[8] L.V. Kantarovic, Integral operators,
Uspehi Math. Nauk. , 11(2/68) (1956), 3–29.[9] M.A. Krasnoselskii, P.P. Zabreyko, E.I. Pustylnik, P.E. Sobolevski, Integral operators in spaces of summablefunctions,
Springer
Dordrecht, Netherlands (1976).[10] R. O’Neil, Integral trransforms and tensor products on Orlicz spaces and L ( p, q ) spaces, J. Analyse , 21(1968), 1–276.[11] R.S. Strichartz, L p estimates for integral transforms, Trans Amer. Math. Sciences , 126 (1969), 33–50.[12] A. Torchinsly, Interpolation of operators and Orlicz classes,
Studia Math. , 59 (1976), 177–207.[13] T. Walsh, On L p estimates for integral transforms, Trans. Amer. Math. Soc. , 155 (1971), 195–215.
R. Kerman, Department of Mathematics and Statistics, Brock University, 1812 Sir Isaac BrockWay, St. Catharines, ON L2S 3A1
Email address : [email protected] S. Spektor, Department of Mathematics and Statistics Sciences, PSB, Sheridan College Instituteof Technology and Advanced Learning, 4180 Duke of York Blvd., Mississauga, ON L5B 0G5
Email address ::