As the linear subspace \(S_{p}^{(q)}\subset L_{p}(M,d\mu ),\) embedded \(L_{q}(M,d\mu )\), \(q >p >1\), is closed in \(\;L_{p}(M,d\mu ),\) one can define the identity embedding mapping \[J_{p}^{(q)}:S_{p}^{(q)}\subset L_{p}(M,d\mu ) \rightarrow L_{q}(M,d\mu ). \label{1a}\tag{1}\] If a sequence \(\{f_{n}:n\in \mathbb{N}\}\) \(\subset S_{p}^{(q)}\) converges in \(S_{p}^{(q)}\subset L_{p}(M,d\mu )\) to an element \(f\rightarrow\) \(S_{p}^{(q)}\hookrightarrow\) \(L_{p}(M,d\mu )\) with respect to the norm on \(\;L_{p}(M,d\mu )\) and simultaneously its image \(\{J_{p}^{(q)}f_{n}:n\in \mathbb{N}\}\) \(\subset S_{p}^{(q)}\subset L_{q}(M,d\mu )\) converges to an element \(g\in L_{q}(M,d\mu )\subset L_{p}(M,d\mu )\) with respect to the norm on\(\;L_{q}(M,d\mu ),\) one can identify these limiting functions \(f\sim g\) almost everywhere. Really, since \((M,d\mu )\subset L_{p}(M,d\mu )\hookrightarrow L_{q}(M,d\mu ),q>p\;>1,\) from the estimations \[\begin{array}{c} ||\;f-g||_{p}\leq ||f-f_{n}||_{p}+||g-f_{n}||_{p}\leq \\ \leq ||f-f_{n}||_{p}+||\left( g-f_{n}\right) ||_{p}\leq \\ \leq ||f-f_{n}||_{p}+||\left( g-J_{p}^{(q)}f_{n}\right) ||_{q}\;\overset{ n\rightarrow \infty }{\rightarrow }0 \end{array} \label{1b}\tag{2}\] one obtains that \(f\sim g\) almost everywhere and the image \(J_{p}^{(q)}\) \((S_{p}^{(q)})\subset L_{q}(M,d\mu )\) is closed in \(L_{q}(M,d\mu ),q>p>1.\) The latter, owing to the Banach closed graph theorem [18]–[22], gives rise to the existence of such a positive constant \(K_{p,q}<\infty\) that \[||\;f||_{q}\leq K_{p,q}\text{ }||f||_{p} \label{2}\tag{3}\] for arbitrary \(f\in S_{p}^{(q)}\subset L_{p}(M,d\mu )\hookrightarrow L_{q}(M,d\mu ),q>p>1\). Remark also, that the following estimations \[||f||_{2}\leq ||\;f||_{q}=||J_{p}^{(q)}f||_{q}\leq K_{p,q}\text{ } ||f||_{p}<\infty \label{3}\tag{4}\] hold for any \(f\in S_{p}^{(q)}\subset L_{p}(M,d\mu )\;\hookrightarrow L_{q}(M,d\mu ),q>p>2,\) easily following from the Young inequality. \(\Box\)

If \(1<p\leq 2,\) from the Young inequality \[||f||_{p}\leq \;||f||_{2} \label{S1a}\tag{5}\] for any \(f\in S_{p}^{(q)}\subset L_{p}(M,d\mu )\hookrightarrow L_{q}(M,d\mu )\) one obtains inequality (?? ) \(\;\)for the bounded \(\;\) \(K_{p,q}>0.\) If \(p>2,\) then one can make use of the following inequality: \[||f||_{p}\leq ||f||_{2^{m}(p-2)+2}^{\frac{\;2^{m}(p-2)+2}{2^{m}p}}\text{ } ||f||_{2}^{\frac{2(2^{m}-1)\;}{2^{m}p}}, \label{S2}\tag{6}\] which holds for any \(f\in S_{p}^{(q)}\subset L_{p}(M,d\mu )\) and arbitrary natural \(m\in \mathbb{N}.\) Now having put, by definition, \(q=2+(p-2)2^{m}\; >p>1(\neq 2),m\in \mathbb{N},\) the inequality (6 ) jointly with that of (?? ) gives rise to the searched estimation (?? ), where the constant \(K_{p,q(m)}=K_{p,q}^{\frac{(q-2)p}{2(q-p)}}\) \(>0\;\)is bounded, thus proving the lemma. \(\Box\)

(Proof of Theorem 5). Based on the lemmas above, one can proceed to proving Theorem 5. First we can observe that inequality (?? ) \(\;\) can be estimated, owing to the classical Young inequality, from the below as \[|l_{\varphi }(f)|\text{ }\leq K_{p,q(m)}\;\;||f||_{2} \label{S2a}\tag{7}\] by means of a bounded linear functional \(l_{\varphi }:(S_{p}^{(q)};||\cdot |_{p}|)\) \(\rightarrow \mathbb{R}\;\) on the Banach subspace \((S_{p}^{(q)};||\cdot ||_{q}),\) where \(l_{\varphi }(f)=(\varphi |f):=\int_{M}\varphi fd\mu\) for some \(\varphi \in (S_{p}^{(q)};||\cdot ||_{q})^{\prime }\simeq\) \((S_{p}^{(q)};||\cdot ||_{\tilde{q}}),\) \(1/\tilde{q }+1/q=1,\) under the constraint \(\;||\varphi ||_{\tilde{q}}=1.\) Taking inequality (7 ) and the evident embedding condition \((S_{p}^{(q)};||\cdot ||_{2})\subset (S_{p}^{(q)};||\cdot ||_{\tilde{q}}),\) one can calculate that \[\sup_{||f||_{2}\neq 0}\frac{|l_{\varphi }(f)|}{||f||_{2}}=||\varphi ||_{2}\leq K_{p,q(m)}. \label{S2b}\tag{8}\] If now to choose an orthonormal basis \(\;\Phi =\{\varphi _{1},\varphi _{2},...,\varphi _{N}\}\subset (S_{p}^{(q)};||\cdot ||_{2})\;\)for some \(N\in \mathbb{N},\) \((\varphi _{j}|\varphi _{k})=\) \(\int_{M}\) \(\varphi _{j}\varphi _{k}d\mu =\) \(\delta _{jk},||\varphi _{j}||_{2}=1,j,k=\overline{1,N},\) one can observe that a function \(\varphi _{a}:=\langle a|\varphi \rangle _{N}=\) \(\sum_{j=1}^{N}a_{j}\varphi _{j}\) \(\in\) \((S_{p}^{(q)};||\cdot ||_{2})\) has the norm \[||\varphi _{a}||_{2}=\left( \sum_{j=1}^{N}|a|^{2}\right) ^{1/2}=|a|_{N}, \label{S2bb}\tag{9}\] where the vector \(\varphi :=(\varphi _{1},\varphi _{2},...,\varphi _{N})^{\intercal }\in \left( S_{p}^{(q)}\right) ^{N}\) and took \(a\in \mathbb{ E}^{N},\) as an arbitrary vector. Having substituted the value of the norm ( 9 ) into (8 ), one obtains the inequality \[|a|_{N}\text{ }\leq K_{p,q(m)}, \label{S3}\tag{10}\] which should be combined with the imposed above condition \(||\varphi _{a}||_{ \tilde{q}}=1.\) Taking into account that \[||\varphi _{a}||_{\tilde{q}}^{\tilde{q}}=\int_{M}|\langle a|\varphi \rangle _{N}|^{\tilde{q}}d\mu =|a|_{N}^{\tilde{q}}\int_{M}|\langle \xi |\varphi \rangle _{N}|^{\tilde{q}}d\mu =1, \label{S2c}\tag{11}\] where \(a\in \mathbb{E}^{N}\;\)and \(\xi :=a/|a|_{N}\) \(\in \mathbb{S} ^{N-1},|\xi |_{N}=1,\;\) we can get rid of the spherical variables \(\xi \in\) \(\mathbb{S}^{N-1},\) if to apply to the above norm equality (11 ) the averaging method [23] over the unit sphere \(\mathbb{S} ^{N-1}.\;\;\)Namely, by integrating it with respect to the spherical measure \(d\omega _{N}(\xi ),\xi \in \mathbb{S}^{N-1}:\) \[|a|_{N}^{\tilde{q}}\int_{\mathbb{S}^{N-1}}d\omega _{N}(\xi )\int_{M}|\langle \xi |\varphi \rangle _{N}|^{\tilde{q}}d\mu =|a|_{N}^{\tilde{q}}\text{ } |||\varphi |_{N}||_{\tilde{q}}^{\tilde{q}}\frac{2\sqrt{\pi ^{N-1}}\Gamma ( \frac{\tilde{q}+1}{2})\Gamma (\frac{N}{2})}{\Gamma (\frac{N+\tilde{q}}{2})} =\;\omega _{N}, \label{S3a}\tag{12}\] we can equivalently obtain from (12 ) that \[\text{ }|a|_{N}=\frac{1}{|||\varphi |_{N}||_{\tilde{q}}}\left( \frac{\sqrt{ \pi }\Gamma (\frac{N+\tilde{q}}{2})}{\;\Gamma (\frac{\tilde{q}+1}{2})\Gamma ( \frac{N}{2})}\right) ^{1/\tilde{q}} \label{A0}\tag{13}\] where we denoted by \(\omega _{N}=\frac{2\sqrt{\pi ^{N}}}{\Gamma (N/2)}\) the surface of the \((N-1)-\) dimnsional sphere \(\mathbb{S}^{N-1}.\) \(\;\)Since the norm \(|||\varphi |_{N}||_{\tilde{q}}\leq |||\varphi |_{N}||_{2}=\left( \int_{M}\langle \varphi |\varphi \rangle _{N}d\mu \right) ^{1/2}=N^{1/2},\) the equality (13 ) jointly with the condition (10 ) yields the final numerical estimation \[\;\frac{1}{N}\left( \frac{\sqrt{\pi }\Gamma (\frac{N+\tilde{q}}{2})}{\Gamma ( \frac{\tilde{q}+1}{2})\Gamma (\frac{N}{2})}\right) ^{2/\tilde{q}}\leq K_{p,q(m)}^{2}, \label{A0a}\tag{14}\]

whose left hand side is bounded for those integers \(N\in \mathbb{N},\) which ensure the embedded subspace \(S_{p}^{(q)}\subset L_{p}(M;d\mu )\hookrightarrow\) \(L_{q}(M;d\mu )\;\;\) at given \(\tilde{q}=\) \(q/(q-1),\) \(q=2+(p-2)2^{m}\;\) \(>\) \(p(\neq 2),\;m\in \mathbb{N},\) to be finite dimensional, that is \(\;\mathrm{dim}\) \(S_{p}^{(q)}=N<\infty .\)

Regarding the critical case \(q=\infty\), since the representation (7 ) is not more acceptable, we need to consider that the linear bounded functional used there should be replaced by the following natural expression: \[|l_{x}(f)|\text{ }\leq K_{p,\infty }||f||_{2} \label{B1}\tag{15}\] for any \(f\in S_{p}^{(\infty )}\subset L_{\infty }(M;d\mu )\;\)and \(\;x\in M,\) where the value \(l_{x}(f):=f(x)\in \mathbb{R}.\) Having calculated the value \[\sup_{||f||_{2}\neq 0}\frac{|l_{x}(f)|}{||f||_{2}}=||l_{x}||\text{ }\leq K_{p,q}\; \label{B2}\tag{16}\] and using the Riesz representation theorem for the functional \(l_{x}:(S_{p}^{(\infty )};||\cdot ||_{2})\rightarrow \mathbb{R}\) on the Hilbert subspace \((S_{p}^{(\infty )};||\cdot ||_{2})\subset (L_{2}(M;d\mu );||\cdot ||_{2}),\) there exists for any \(x\in M\) such a function \(g_{x}\in\) \((S_{p}^{(\infty )};||\cdot ||_{2})\) that \(l_{x}(f)=(g_{x}|f)\;\)and \(||l_{x}||\) \(=||g_{x}||_{2}\) for all \(f\in (S_{p}^{(\infty )};||\cdot ||_{2}).\) If now \(\Phi _{p}^{(\infty )}:=\{\varphi _{1},\varphi _{2},...,\varphi _{N},...\}\) \(\subset (S_{p}^{(\infty )};||\cdot ||_{2})\;\) is a complete orthonormal set of functions, that is \(||\varphi _{j}||_{2}=1,(\varphi _{j}|\varphi _{k})=\) \(\int_{M}\) \(\varphi _{j}\varphi _{k}d\mu =\) \(\delta _{jk},j,k\in \mathbb{N},\) the related Parceval equality \[||g_{x}||_{2}^{2}=\sum_{j\in \mathbb{N}}|(g_{x}|\varphi _{j})|^{2}=\sum_{j\in \mathbb{N}}|\varphi _{j}(x)|^{2} \label{B3}\tag{17}\] combined with the inequality (16 ) yields the next one: \[\sum_{j\in \mathbb{N}}|\varphi _{j}(x)|^{2}\leq K_{p,\infty }^{2}, \label{B4}\tag{18}\] which holds for any \(x\in M.\) Having integrated the obtained inequality ( 18 ) over the whole space \(M,\) we obtain that \[\mathrm{card}\text{ }\Phi _{p}^{(\infty )}=N\leq K_{p,q}^{2} \label{B5}\tag{19}\] for some \(N=\dim S_{p}^{(\infty )},\) thus proving the theorem. \(\Box\)

As a closed subspace \(S_{p}^{(c)}\subset L_{p}(0,1;d\mu )\hookrightarrow C([0,1];\mathbb{R})\) of continuous functions on the interval \([0,1]\) can be closely identically embedded into the Banach space \(C([0,1];\mathbb{R}),\) from the Banach closed mapping theorem [20], [22] one derives the existence of such a constant \(K_{q}>1\;\)that the corresponding embedding operator \(J_{p}^{(c)}:S_{p}^{(c)}\subset L_{p}(0,1;d\mu )\hookrightarrow C([0,1];\mathbb{R})\) is bounded, that is \[||J_{p}^{(c)}f||_{\infty }\leq K_{p,c}||f||_{p} \label{S14}\tag{20}\] for any \(f\in S_{p}^{(c)}\subset (C([0,1];\mathbb{R}),||\cdot ||_{\infty }).\) Remark now that if \(1<p\leq 2,\) then the inequality \(||f||_{p}\leq ||f||_{2}\) holds for all \(f\in S_{p}^{(c)}\subset C([0,1];\mathbb{R})\hookrightarrow L_{p}(0,1;d\mu ).\) If \(p>2,\) we can observe that \(|f|^{p}\leq ||f||_{\infty }^{p-2}|f|^{2}\) for any \(f\in S_{p}^{(c)}\subset C([0,1];\mathbb{R} )\hookrightarrow L_{p}(0,1;d\mu ),\) whence by integration over the interval \([0,1]\) one easily obtains that \(||f||_{p}\leq ||f||_{\infty }^{\frac{p-2}{p} }||f||_{2}^{\frac{2}{p}}.\) Substituting the latter inequality into ( 20 ), we obtain that \(\;||f||_{\infty }\leq K_{q,c}^{p/2}||f||_{2},\) what jointly with the evident inequality \(||f||_{2}\leq ||f||_{\infty }\) gives rise to the dual inequality \[||f||_{2}\leq ||f||_{\infty }\leq K_{p,c}^{p/2}||f||_{2} \label{S15}\tag{21}\] for all \(f\in S_{p}^{(c)}\subset L_{p}(0,1;d\mu )\hookrightarrow C([0,1]; \mathbb{R}).\) As above, define for any \(t\in \lbrack 0,1]\) a bounded linear functional \(l_{t}:\) \((S_{p}^{(\infty )};||\cdot ||_{2})\) \(\rightarrow \mathbb{R}\) on the Hilbert subspace \((S_{p}^{(c)};||\cdot ||_{2}),\) such \(\;\) that \(l_{t}(f)=f(t)\in \mathbb{R},\) which allows owing to the Riesz theorem the representation \(l_{t}(f)=(g_{t}|f)\) for all \(f\in (S_{p}^{(\infty )};||\cdot ||_{2}),\) where \(g_{t}\in\) \((S_{p}^{(\infty )};||\cdot ||_{2})\) and \(||l_{t}||\) \(=||g_{t}||_{2}.\) If now \(\Phi _{p}^{(c)}:=\{\varphi _{1},\varphi _{2},...,\varphi _{N},...\}\) \(\subset (S_{p}^{(c)};||\cdot ||_{2})\;\) is a complete orthonormal set of functions, that is \(||\varphi _{j}||_{2}=1,(\varphi _{j}|\varphi _{k})=\) \(\int_{M}\) \(\varphi _{j}\varphi _{k}d\mu =\) \(\delta _{jk},j,k\in \mathbb{N},\) the related Parceval equality \[||g_{t}||_{2}^{2}=\sum_{j\in \mathbb{N}}|(g_{t}|\varphi _{j})|^{2}=\sum_{j\in \mathbb{N}}|\varphi _{j}(t)|^{2} \label{S15aa}\tag{22}\] jointly with the inequality (16 ) gives rise to the inequality \[\sum_{j\in \mathbb{N}}|\varphi _{j}(t)|^{2}\leq K_{p,c}^{2}, \label{S15ab}\tag{23}\] which holds for any \(t\in M.\) Integration of the obtained above inequality (23 ) over the whole interval \([0,1]\) yields the constraint \[\mathrm{card}\text{ }\Phi _{p}^{(c)}=N\leq K_{p,q}^{2} \label{S15bb}\tag{24}\] for some \(N=\dim S_{p}^{(c)},\) thus proving the proposition. \(\Box\)