Bull, Kyushu Inst. Tech.
(Math. Natur. Sci.) No. 39, 1992, pp. 1-7
ON HANNER'S INEQUALITIES IN Lp,q
By
Ken-ichi MIyAzAKI
(Received November 25, 1991)
O.Hanner proved in 1956 the following inequalities in L, with the norm IIfll. ==(f,'lf(t)lpdt)i/' for feL,, and he applied them to obtain the modulus of
convexity of L,.
Hanner's Theorem ([3]). For pÅr2 and for x,yEL. the following inequalities hold
(11x11p+ IlyII.)P + 1 llxII,- IlyIl,iP
(1)
i-l: llx+yIIfl + llx-yll; )- 2llxII;+211yll;•
For 1ÅqpÅq2 these inequalities hold in the reverse sense, and for p=2 all sides of (1) are equal.
The equality sign in the left hand side of (1) holds if and only if x(t) ==O or y(t) = O or there is a number a År O such that (x(t) - ay(t)) (x(t) + ay(t)) == O a.e., and the equality sign in the right hand side of (1) holds if and only if x(t)y(t) =O a.e..
The purpose of this paper is to give some results in the Lorentz space Lp,q-version of this theorem.
We adopt here the usual notations and notions for Lorentz spaces and the rearrangement of measurable functions (see [1], [4] and [5]).
Let f(C) be a complex-valued pt-measurable function on a measure space (9, pt). The distribution function f. and the non-increasing rearrangement f* off are respectively defined as follows.
f. (s) == pt({4:lf(4)l År s})
(2) for sGR' (the positive real axis) and
f"(t) -= inf {s: f. (s) S t}
for tER'.
The Lorentz space L,,,(S2, pt), 1 S.p ;.f co, 1 ;;q ;S oo, is defined as the space of all
equivalence classes of measurable functions f(C) such that the quasi-norm
2 Ken-ichi MiyAzAKi
(3) 11fllp,q ==
[ f,co {ti!pf* (t) }q dt/ t]'!q Åq co when i s q Åq oo ,
sup tiiPf"(t)Åq co when q= co.
t
In order to obtain the analogous inequalities of (1) in L,,,, we prepare some lemmas.
LEMMA 1 (see [5] and [4]). In this lemma, let f(C), g(4), f,(4) and g,(C) (i = 1, 2) be measurablefunctions on (S2, pt) which are non-negative valued and .finite a.e.. Then ( i ) f.(s) andf"(t) are non-increasing and continuous on the right for sER' and tER' respectively.
(ii) (f.).(t) =:f'(t) and (f')'(t) =f"(t) for tER".
L (iii) ijf(t) is non-increasing and continuous on the right a.e. on R', then f*(t)=f(t) a.e. on R+.
(iv) iff(4) S. g(4) a.e. on S2, then f.(s) ;S g.(s) for sER' andf"(t) =Åq g*(t) for tER'.
tN tN tN
(V) "i.M. J, "f(t)g*(t)dt S 3Um. J, f(t)g(t)dt S"i.m. J, f*(t)g*(t)dt, where *f(t) stands
fbr the non-decreasing rearrangement of f(t) on (O, N] which is dqfined similarly as f"(t) with reverse inequalities.
(vi) Let qÅrO then (fq)"(t)=(f"(t))q for tER'.
(vii) lf fi (4) + gi (4) S- f2 (4) + g2 (4) a. e. on 9, then (fi + gi)"(2t) S f2* (t) + g!(t)
and f,* (2t)+gf(2t) ;$ 2(f," (t) +g!(t)) for tER'.
PRooF. The assertions (i) through (v) are well known as proved in [5] and [4].
(vi) and (vii) may be shown from the definitions as follows. Let qÅrO, for sER' put s= sq,, then
(fq).(s) =- u({e: (f(c))q År s}) =- pa({4: f(4) År s,}) -= f. (s,).
Hence
(fq)*(t) - inf {s:(fq).(s) .Åq. t}
-= inf {sq, : f.(s,) s. t} -= (f*(t))q.
Next, for any si and s2GR+, by the assumption of (vii) we have
{4 : fi (O + gi (4) År si + s2} c {4 : f2 (4) År si} u {4 : g2 (4) År s2}•
Hence
on Hanner's lnequalities in L,,, 3
(fi + gi)* (Si + S2) S- f2*(Si) + g2*(S2)•
Therefore, if f2.(si) S. ti and g2.(s2) S t2, then (fi + gi)* (Si + S2) S- ti + t2•
Taking the infima of the left hand side first with respect to s= si + s2 and the right hand side next with respect to si and s2, we get
(fi + gi)" (ti + t2) ;El f2*(ti) + g:(t2) for ti, t2ER'.
Thus
(fi + gi)" (2t) 5 f2"(t) + g;(t) and
fi (2t) + gf(2t) S. 2(f, + g,)'(2t) S. 2(f,*(t) + g:(t)) for tER'.
The next two facts are shown in the proof of Theorem 1 in [3].
LEMMA 2. Let zi and z2 be complex numbers.
If 1 ÅqqÅq2 then the following inequalities (4) and (5) hold (4) , lz, + z,lq + lz, - z,lq l.lll (lz,l+lz,l)q +IIz,I- lz211q,
where the equality holds if and only if ziz2 IO and zi/z2 is real, or ziz2 = O, and
(s) 2Iz, lq+21z, lq ). Iz, +z, lq+lz, -z, lq
where the equality holds of and only if ziz2 = O•
If 2Åqq then the inequalities (4) and (5) hold in the reverse sense.
LEMMA 3. Let qÅr1 and let
(6) 4(u, v) == (ui/q + viiq)q +luiiq -viiqp for u).o andv)- O.
Then, for two positive valued functions f(t), g(t)ELi((O, oo)) we have (7) jl,co 4(f(t),g(t))dt ii 4(jl,cof(t)dt, jT,cog(t)dt) when iÅqqÅq2
and the inequality holds in the reverse sense when 2 Åq q.
Now we show some analogous inequalities of (1) for Lorentz spaces.
THEoREM. Let p and q be real numbers such that 1ÅqpÅq oo,1ÅqqÅq co and pIq. Let x(4) and y(4) be two elements of L,,,(9, pa) and we denote x*(t) and y"(t)
the non-increasing rearrangements of x and y respectively. Then the following
inequalities hold.
4 Ken-ichi MiyAzAKi
(8)
(9)
When 1ÅqqÅq 2,
When 2Åq q,
22'qip( II x" Ilq,',, + II y" ilS,q) ll; ll x* + y' 11S,q + ll x" - y" IIqp,q,
and in further restricted case q Åq p, the right hand side of this formula
lll; (11 x" IIp,q + ll y* 11p,q)q + l 11 x* llp,q - Il y" llp,q lq•2'q!"( 11 x" IIS,q + IIy" 11qp,q) ;-:i{ ll x* + y" ilqp,q + ll ix" - y" IIS,q and in more restric ted case 1 Åq p Åq q
:Sl (11 x* 11p,q + 11y" 11p,q)q + i II x* llp,q - 11 y' llp,q l"•
PRooF. Consider first the case of1ÅqqÅq2. Then by (5) of Lemma2
(lo) 2{x*(t)}q + 2{y*(t)}q ill {x*(t) + y* (t)}q + lx*(t) - y*(t)lq.
Hence by (ii), (vi) and (vii) of Lemma 1
22{(x*)q(t) + (y*)q(t)} lll; (x* + y*)q(2t) + ax* - y*l*)q(2t).
Multiply tqiP-' and integrate both sides of this inequality, then 22 [ f,co {tifpx*(t)}q dt/t + f,co {tilpy*(t)}q dt/t]
l 2-qlp[ f,co {(2t)ilp(x* + y*) (2t)}q dt/t + f,co {(2t)i!plx* - y*1* (2t)}q dt/t]
thus
22'q!'( ll x" 11S,q + ll Y* ll S,q) ll 11 x" + y* IIS,q + Ii x' - y* IIS,q•
Next we have to prove the right side inequality of (8). Consider
(ll x' llp,q + Il y* llp,q)q + 1 II )c* IIp,q w II y* IIp,qlq
= [( foco {tilpx*(t)}q dt/t)i!q + ( f,co itilpy*(t)}, dt/t)ilq]q
+ ( f,co {tiip x*(t)}q dt/t)`iq - ( jl,co {tiipy*(t)}, dt/t)iiq q
on Hanner's lnequalities in L.,, 5
Here, put
(11) - f(t) .= {tilp-'lqx*(t)}q and g(t) .. {tilp-ilqy*(t)}q
and apply Lemma 3 to these f, g, then the right hand side of this equation can be calculated as follows.
4( S,co f(t) dt, S,co g(t) dt)
S S,co 4(f(t), g(t))dt
= jl,co {ti /p (x* (t) + y* (t))}q dt/t + jl,co {ti /p i x* (t) - y* (t) i }q dt / t.
If we here restrict in the case ofqÅqp, then by (v) of Lemma1 .(,co {ti/pix*(t) - y*(t)i}qdt/t s. jl,co {typix* - y*i*(t)}qdt/t.
Thus
(ll X* iip,q + ll y* ilp,q)q + l ii x* 11p,q - li y* llp,qlq ;ll ll x* + y* ilS,q + il x" - y" liqp,q•
Let us last show that in the case of 2Åqq the proof can be proceeded in almost same way as above case using the reverse sense inequalities.
Since, in this case, because of Lemma 2
2{x*(t)}q + 2{y*(t)}q ;I$ {x*(t) + y*(t)}q + lx*(t) - y*(t)lq,
again using (vi), (vii) and (ii) of Lemma 1 we get
2{(x*)q(2t) + (y*)q(2t)} ;Is 2{(x* + y*)q(t) + qx* - y*I*)q(t)}.
Multiply tq/P - 1 and integrate both sides, then
2-q/P(ll x" Hq,,, + Il y" 11S,,) ;!ill 11 x* + y" llqp,q + Il x" - y" 11qp,q•
The last inequality of (9) can be proved again by Lemma 3 with (11) in the following way:
(II x* 11p,q + 11 y* 11p,q)q + l 11 x* 11p,q - ll y* llp,qlq
= 4( f,cof(t) dt, S,co g(t) dt)
6 Ken-ichi MiyAzAKi
l S,co 4(f(t), g(t))dt
= jl,co {tiip(x*(t) + y*(t))}qdt/t + jl,co {ti/pix*(t) - y*(t)i}qdt/t.
Here, restrict in the case pÅq q, then by (v) of Lemma 1 we have jl,co {tiipix*(t) - y*(t)i}qdt/t ;.iir jl,co {tiipix* - y*i*(t)}q dt/t
== 11 x' - y" 11S,q,
from which we get the last inequality of (9). This completes the proof.
REMARK 1. We notice that because of 11f'11,,, = 11fll,,, the terms 11x"ll,,, and 11y"ll.,, in the first sides and the last sides of both (8) and (9) can be written llxll,,, and llyll,,, respectively. Therefore it is better to be able to remove the * in both middle sides of (8) and (9).
REMARK 2. When p = q, for f(t)EL.,,((O, co))
ll f" 11 ;,p = f,co lf'(t) IP dt = f,co lf(t) IP dt == ll f ll S• Th us the proof of Theorem must be
modified as in Hanner's original way to get the Hanner's theorem. When 1Åqp=q S. 2, starting from
21x(t)IP + 21y(t)IP i-lr lx(t) + y(t)l" + lx(t) - y(t)IP instead of (10), we get directly
2(11xIIfl,. + lly11S.,) l-l:: 11x + yll;,. + llx - yll;,,•
Since
jl,co ix"(t) ' y*(t)ipdt = jl,co {ix" - y*i*(t)}pdt,
concerning the last inequality of (8), for 1 Åq p 5 2 without other additional restriction, we get
(11Xllp,p + 11Y11p,p)P + l llXIIp,p - 11Y11p,plP S- 11x+y11fl,. + llx-yHSI,p•
When 2 Åqp == q, all above inequalities hold in the reverse sense.
REMARK 3. Let Ke,,(Ao, Ai) and Je,,(Ao, Ai) be respectively the K and J method
on Hanner's lnequalities in L,,, 7
real interpolation spaces of the compatible quasi-mormed spaces Ao and Ai (see [1]). The motivation of this paper is to calculate the moduli of convexity of Ke,,(Ao,Ai) and Je,,(Ao,Ai) from those of Ao and Ai. This is an open problem (see [2] for other moduli).
Further it remains to complete the Theorem for L,,, in remaining p, q cases.
References
[ 1 ] J. Bergh and J. L6fstr6m, Interpolation spaces, Springer Verlag, Berlin Heidelberg New York, 1976.
