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The existence of an analytic operator-valued function space integral as an      S ( L p , L p ′ ) \mathcal {S}({L_p},{L_{p’}})       theory      ( 1 ≤  p ≤  2 ) (1 \leq p \leq 2)       has been established for certain functionals involving the Lebesgue measure. Recently, Johnson and Lapidus proved the existence of the integral as an operator on      L 2 {L_2}       for certain functionals involving any Borel measure. We establish the existence of the integral as an operator from      L p {L_p}       to      L p ′ ( 1 &gt; p &gt; 2 ) {L_{p’}}\;({1 &gt; p &gt; 2} )       for certain functionals involving some Borel measures.

Analytic operator valued function space integrals as an ℒ(ℒ_{𝓅},ℒ_{𝓅’}) theory