On the Finite Dimensionality of Closed Subspaces in L p ( M , d µ ) ∩ L q ( M , d ν )

: Finding effective ﬁnite-dimensional criteria for closed subspaces in L p , endowed with some additional functional constraints, is a well-known and interesting problem. In this work, we are interested in some sufﬁcient constraints on closed functional subspaces, S p ⊂ L p , whose ﬁnite dimensionality is not ﬁxed a priori and can not be checked directly. This is often the case in diverse applications, when a closed subspace S p ⊂ L p is constructed by means of some additional conditions and constraints on L p with no direct exempliﬁcation of the functional structure of its elements. We consider a closed topological subspace, S ( q ) p , of the functional Banach space, L p ( M , d µ ) , and, moreover, one assumes that additionally, S ( q ) p ⊂ L q ( M , d ν ) is subject to a probability measure ν on M . Then, we show that closed subspaces of L p ( M , d µ ) ∩ L q ( M , d ν ) for q > max { 1, p } , p > 0 are ﬁnite dimensional. The ﬁnite dimensionality result concerning the case when q > p > 0 is open and needs more sophisticated techniques, mainly based on analysis of the complementary subspaces to L p ( M , d µ ) ∩ L q ( M , d ν ) .


Introduction
The problems, concerned with the finite dimensionality of closed functional subspaces in L p (in part, in L p (0, 1; C)), are of long-time interest in analysis, being related to their many applications in operator and approximation theories [1][2][3][4][5], in dynamical systems theory [6][7][8][9][10][11] and other applied fields.As an example, one can recall a central problem in Banach space theory to classify the complemented subspaces of L p up to isomorphism; the finite-dimensional analogue is to find for any given S p ⊂ L p a description of the finitedimensional spaces which are S p -isomorphic to S p -complemented subspaces of L p .These problems were thoroughly studied before [12], in particular finite-dimensional versions of this complemented subspaces of the L p problem, yet in both cases their classification is far from over.
It was observed that sometimes, the finite-dimensional version of an infinitedimensional problem leads to a theory which is much more interesting than the infinitedimensional theory.Here, one can recall the problem of describing the subspaces of L p which embed isomorphically into a "smaller" L p space; namely, the space l p , for which there is a fairly good answer [12].One can recall that density on a probability space, M, is a strictly positive measurable function h : M → R + for which hdµ = 1.Such a density h induces for fixed 0 < p < ∞ an isometry J (p) h from L p (M, dµ) onto L p (M, hdµ).The next result, due to D. Lewis [13,14], gives useful information about chosen a priori finite-dimensional subspaces S p ⊂ L p .Theorem 1.Let µ be a probability measure on M, and let S p be a N-dimensional subspace of L p (M, dµ), 0 < p < ∞, with full support.Then, there is a density h > 0 so that the image J Assuming that S p is already a subspace of L p for some finite dim S p ∈ N, one can randomly pick a few coordinates and hope that the natural projection onto these coordinates restricted to S p is a good isomorphism.If we do this with no additional preparation, this will not work.Indeed, the subspace S p may contain a vector with small support, say one of the unit vector basis elements of l N ∞ , in which case, the chance that a coordinate in its support is picked is small.Of course, if no such coordinate is picked, the said projection cannot be an isomorphism on S p .The point is that one wants to change S p first to another isometric copy of S p , in which each element of S p is spread out.This can be performed by a change of density.This method was used with other tools to produce the best known results.

Finite Dimensionality of Closed Subspaces in L p ∩ L q
As the imbedding structure of a priori taken finite-dimensional subspaces in L p is in many cases very important and instructive, nonetheless finding the effective criteria for closed subspaces in L p endowed with some additional functional constraints to be finite dimensional remains very important and hard both from theoretical and applied points of view.Below, we are interested in some sufficient constraints on functional closed subspaces S p ⊂ L p , whose finite dimensionality is not fixed a priori and cannot be checked directly.This is often the case in diverse applications, when a closed subspace S p ⊂ L p is constructed by means of some additional conditions and constraints on L p with no direct presentation of the functional structure of its elements.In particular, we consider a topological subspace, S (q) p , of the functional Banach space, L p (M, dµ), where µ is a probability measure on measurable space M.Moreover, one assumes that additionally, S (q) p ⊂ L q (M, dν) is subject to a probability measure ν on M.Then, we prove the following theorem first announced in [15].Theorem 2. Let a closed topological subspace S (q) p ⊂ L p (M, dµ) belong to L q (M, dν), q > max{1, p}, p > 0, where measures µ, ν are probabilistic and the measure µ is absolutely continuous with respect to the measure ν on M. Then the subspace S (q) Let us consider a closed topological subspace, S (q) p , of the functional Banach space, L p (M, dµ), where µ is a probability measure absolutely continuous with respect to the measure ν on M, and satisfies, in addition, the constraint S (q) p ⊂ L q (M, dν) subject to a probability measure ν on M. In order to state Theorem 2, formulated above, we need some lemmas.Lemma 1.For any q > p > 0, there exists a bounded positive constant K p,q , such that Proof.As the topological space , one can define the identical imbedding p with respect to the norm on L p (M, dµ) and simultaneously it converges to an element g ∈ L q (M, dν) with respect to the norm on L p (M, dµ), owing to the absolute continuity of the measure µ with respect to ν, one can identify these limits f ∼ g almost everywhere.Then, we enter into conditions of the Banach closed graph theorem [16][17][18] and can infer that there exists such a positive constant K < ∞ that for any f ∈ S (q) p ∩ L q (M, dν), where as usual, we denote It is easy to check, using the classical Young inequality, that giving rise to (1), where we take into account [19,20] that the Radon-Nikodym derivative based on the inequality (3), one can also easily obtain that for any f ∈ S (q) p ⊂ L p (M, dµ) ∩ L q (M, dν), if q > p > 1.Indeed, consider the next norm transformations, once more based on the Young inequality: Now, making use of the inequality (3), it ensues from (6) that which reduces, using (3) once more, in the inequality for all f ∈ S (q) p ⊂ L q (M, dν) ∩ L p (M, dµ), proving the lemma.
Proof of Theorem 2. As follows from Lemma 2, the closed subspace The latter makes it possible to reduce the finite dimensionality problem subject to the closed subspace S (q) p ⊂ L 1 (M, dν) ∩ L q (M, dν) to the one of the closed subspace S (q), * p ⊂ L ∞ (M, dν) ∩ L q(M, dν), following the Grothendieck [21] scheme.First, we observe that the embedding mapping S (q), * p ⊂ L q(M, dν) → S (q), * p ⊂ L ∞ (M, dν) is a closed operator, giving rise owing to the Banach closed operator theorem to the inequality ||g|| ∞ ≤ R||g|| q (14) for any g ∈ S (q), * p ⊂ L ∞ (M, dν) and some positive and bounded number R < ∞.Moreover, making use of the Young inequality, for any ∞ > q > 0 one can find such a positive constant R q < ∞ that ||g|| ∞ ≤ R q||g|| 2 (15) for any g ∈ S (q), * p ⊂ L ∞ (M, dν).Taking into account that, according to (14), any g ∈ S (q), * p ⊂ L 2 (M, dν)∩ L ∞ (M, dν), one can choose the finite dimensional subspace (10) such that the set of functions ψ := {ψ j ∈ S (q), * p : j = 1, N} can be ortonormal, that is for all j, k = 1, N. Let now Q ⊂ D 1 (0) be a countable everywhere dense subset of the unit disc D 1 (0) of the Euclidean space E N := (C N ; •|• ).Then, for every vector c ∈ D 1 (0), one finds that the function Taking into account the fact that the set Q ⊂ D 1 (0)is countable, one can find such a measurable subset M ⊂ M that the measure ν(M ) = 1 and |g c (u)| ≤ R q for all vectors c ∈ Q ⊂ D 1 (0) and all points u ∈ M .Since at a fixed point u ∈ M the mapping D Having integrated the inequality (19) over M M , one finds that N ≤ R 2 q < ∞.The latter means that dim S (q), * p ≤ max N < ∞, being equivalent to the condition that dim S (q) p ≤ max N < ∞, thus proving the theorem.

Conclusions
We studied a classical problem of finding finite-dimensional effective criteria for closed subspaces in L p , endowed with some additional functional constraints.We considered a closed topological subspace S (q) p of the functional Banach space L p (M, dµ) and, moreover, assumed that additionally, S (q) p ⊂ L q (M, dν) is subject to a probability measure ν on M.Then, we showed that closed subspaces of L p (M, dµ) ∩ L q (M, dν) for q > max{1, p}, p > 0, are finite dimensional, if the measures µ, ν are probabilistic on M and the measure µ is absolutely continuous with respect to the measure ν on M. The finite dimensionality result concerning the case when q > p > 0 is open and needs more sophisticated techniques, mainly based on analysis of the complementary subspaces to L p (M, dµ) ∩ L q (M, dν).
Author Contributions: The article was conceptualized by A.K.P. and the final manuscript preparation was done jointly by A.A.B. and A.K.P.All authors have read and agreed to the published version of the manuscript.
one can extend this function on the whole disc D 1 (0), obtaining the inequality|g c (u)| ≤ R q (18)already for all c ∈ D 1 (0) and u ∈ M .Making use of the arbitrariness of the vector c ∈ D 1 (0), it can be chosen as c := ψ(u) |ψ(u)| ∈ D 1 (0) ∩ S (q), * p, u ∈ M , giving rise to the following inequality: |ψ