We build a ne
w class of Banac
h function spaces,
whose function norm is
where
hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si2.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=288bfcb2563e0ce0f2027172f281363f" title="Click to view the MathML source">ρp(x)hContainer hidden">hCode">h altimg="si2.gif" overflow="scroll">w>ρw>w>phy="false">(xhy="false">)w>h> denotes t
he norm of t
he Lebesgue space of exponent
hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si227.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=eda3d5d66f75fe3877f812ff840c9fc3" title="Click to view the MathML source">p(x)hContainer hidden">hCode">h altimg="si227.gif" overflow="scroll">phy="false">(xhy="false">)h> (assumed measurable and possibly infinite), constant
wit
h respect to t
he variable of
f, and
δ is measurable, too. Suc
h class contains some
kno
wn Banac
h spaces of functions, among
whic
h are t
he classical and t
he small Lebesgue spaces, and t
he Orlicz space
hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si4.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=06d1119e119156ecdc74f00154d4a162" title="Click to view the MathML source">L(logL)αhContainer hidden">hCode">h altimg="si4.gif" overflow="scroll">Lw>hy="false">(hvariant="normal">logLhy="false">)w>w>αw>h>,
hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si166.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=1e37bfbf7e131042375db0855073c0ae" title="Click to view the MathML source">α>0hContainer hidden">hCode">h altimg="si166.gif" overflow="scroll">α>0h>.
Furthermore we prove the following Hölder-type inequality
where
hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si7.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=7b9024614e3681ac08d9e06ff1401e1f" title="Click to view the MathML source">ρp[⋅]),δ[⋅](f)hContainer hidden">hCode">h altimg="si7.gif" overflow="scroll">w>ρw>w>phy="false">[⋅hy="false">]hy="false">),δhy="false">[⋅hy="false">]w>hy="false">(fhy="false">)h> is t
he norm of fully measurable grand Lebesgue spaces introduced by Anatriello and Fiorenza in
[2]. For suitable c
hoices of
hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si227.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=eda3d5d66f75fe3877f812ff840c9fc3" title="Click to view the MathML source">p(x)hContainer hidden">hCode">h altimg="si227.gif" overflow="scroll">phy="false">(xhy="false">)h> and
hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si8.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=5f19f2b710b401c4c884827be4da994c" title="Click to view the MathML source">δ(x)hContainer hidden">hCode">h altimg="si8.gif" overflow="scroll">δhy="false">(xhy="false">)h> it reduces to t
he classical Hölder's inequality for t
he spaces
hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si9.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=33f6ababbdbb44aae401e2ccfb570ba2" title="Click to view the MathML source">EXP1/αhContainer hidden">hCode">h altimg="si9.gif" overflow="scroll">EXw>Pw>w>1hy="false">/αw>h> and
hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si4.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=06d1119e119156ecdc74f00154d4a162" title="Click to view the MathML source">L(logL)αhContainer hidden">hCode">h altimg="si4.gif" overflow="scroll">Lw>hy="false">(hvariant="normal">logLhy="false">)w>w>αw>h>,
hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si166.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=1e37bfbf7e131042375db0855073c0ae" title="Click to view the MathML source">α>0hContainer hidden">hCode">h altimg="si166.gif" overflow="scroll">α>0h>.