We build a new
class of Banach function spaces, whose function norm is
class="formula" id="fm0010">
where
class="mathmlsrc">class="formulatext stixSupport mathImg" 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)class="mathContainer hidden">class="mathCode"> denotes
the norm of
the Lebesgue space of exponent
class="mathmlsrc">class="formulatext stixSupport mathImg" 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)class="mathContainer hidden">class="mathCode"> (assumed measurable and possibly infinite), constant with respect to
the variable of
f, and
δ is measurable, too. Such
class contains some known Banach spaces of functions, among which are
the classical and
the small Lebesgue spaces, and
the Orlicz space
class="mathmlsrc">class="formulatext stixSupport mathImg" 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)αclass="mathContainer hidden">class="mathCode">,
class="mathmlsrc">class="formulatext stixSupport mathImg" 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>0class="mathContainer hidden">class="mathCode">.
Furthermore we prove the following Hölder-type inequality
class="formula" id="fm0020">
where
class="mathmlsrc">class="formulatext stixSupport mathImg" 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)class="mathContainer hidden">class="mathCode"> is
the norm of fully measurable grand Lebesgue spaces introduced by
Anatriello and Fiorenza in
[2]. For suitable choices of
class="mathmlsrc">class="formulatext stixSupport mathImg" 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)class="mathContainer hidden">class="mathCode"> and
class="mathmlsrc">class="formulatext stixSupport mathImg" 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)class="mathContainer hidden">class="mathCode"> it reduces to
the classical Hölder's inequality for
the spaces
class="mathmlsrc">class="formulatext stixSupport mathImg" 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/αclass="mathContainer hidden">class="mathCode"> and
class="mathmlsrc">class="formulatext stixSupport mathImg" 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)αclass="mathContainer hidden">class="mathCode">,
class="mathmlsrc">class="formulatext stixSupport mathImg" 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>0class="mathContainer hidden">class="mathCode">.