Abstract Generated abstract
The paper introduces a family of function spaces defined through the variation of local polynomial approximation errors on cubes. It develops distribution estimates, embedding results, and a decomposition theorem showing that functions in these spaces can, outside sets of small measure, be represented by functions with controlled smoothness. The framework is then related to several classical spaces, including Sobolev, Lipschitz, Besov, Nikol’skii, and bounded variation spaces, yielding isomorphism or embedding statements under specified parameter conditions. An interpolation theorem is also proved for linear operators mapping from the scale of Lebesgue spaces into these approximation-defined spaces.
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UDC 517.51
MATHEMATICS
Yu. A. BRUDNYI
SPACES DEFINED BY THE VARIATION OF LOCAL APPROXIMATIONS
(Presented by Academician V. I. Smirnov on 4 VI 1969)
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A family of spaces defined by the variation of local approximations is obtained by means of a very simple construction. Nevertheless, among the members of this family there are a number of spaces that are constantly used in analysis (Lipschitz spaces, functions of bounded variation, Sobolev spaces, etc.). Thus a somewhat unexpected connection between these spaces is established; in particular, from the theorems given below there follows a number of new properties of these spaces. We note that two representatives of the family of spaces under consideration were studied earlier by F. Riesz (Riesz’s lemma, see (¹), p. 85) and by F. John—L. Nirenberg (²).
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We shall need several definitions. Below \(Q_0\) denotes a fixed \(n\)-cube, and \(Q \subset Q_0\) a cube parallel to it. Let \(X\) be a Banach space whose elements are functions on the cube \(I(Q)\), and suppose that the norm in \(X\) has the following property: if \(|I_1(Q)| \le |I_2(Q)|\), \(Q \subset Q_0\), and \(I_2 \in X\), then \(I_1 \in X\) and \(\|I_1\| \le \|I_2\|\). Let \(P\) be the projector \(L_p(Q_0) \to P_{k-1}\), where \(P_{k-1}\) is the space of polynomials of degree \(k - 1\) (consisting only of zero when \(k = 0\)), and let \(P_Q\) be the “transplant” of \(P\) to \(L_p(Q)\) by means of a homothety carrying \(Q_0\) onto \(Q\). Finally, let
\[ I_f^k(Q)=\left\{\frac{1}{m(Q)}\int_Q |f-Pf|^p dx\right\}^{1/p}. \tag{1} \]
Definition. The space \(\mathscr{L}_p^k(X)\) is the linear set of functions \(f \in L_p(Q_0)\) for which \(I_f^k(Q) \in X\). As the norm of \(f\) in this space we take the quantity
\[ \|f\|_{L_p(Q_0)}+\|I_f^k\|_X . \]
- We choose as \(X\) the space of functions \(I(Q)\) having bounded variation in the following sense:
\[ V(I;\Omega)=\sup\left\{\sum m(Q_s)\left|\frac{I(Q_s)}{\varphi(a_s)}\right|^q\right\}^{1/q}<+\infty \tag{2} \]
for every open \(\Omega \subset Q_0\). Here the supremum is taken over all families \(\{Q_s\}\) of pairwise nonoverlapping cubes in \(\Omega\); \(a_s\) is the side length of \(Q_s\), and \(\varphi\) is a majorant, i.e. a positive nondecreasing function on \((0,+\infty)\) satisfying the condition
\[ \sup \frac{\varphi(2\tau)}{\varphi(\tau)}<+\infty . \]
We denote this space by \(V_q^\varphi\); its subspace consisting of those \(I\) for which (2) is absolutely continuous will be denoted by \(v_q^\varphi\).
For the formulation of the first theorem, set
\[ m_Q(f;\tau)=\operatorname{mes}\{x\in Q\mid |(f-P_Q f)(x)|>\tau\} \tag{3} \]
and by \(f_Q^*(\tau)\), \(0<\tau\le m(Q)\), the right-continuous function inverse to (3).
Theorem 1. If \(f\in \mathscr L_p^k(V_q^\varphi)\), \(1\le p,q\le\infty\), then for every \(Q\subset Q_0\) with side \(a\)
\[ f_Q^*(\tau^n)\le cV(I_f^k;Q)\int_\tau^a \frac{\varphi(\tau)}{\tau^{1+n/q}}\,d\tau . \tag{4} \]
From Theorem 1, for \(k=1\) and \(\varphi=1\), we obtain the known results of John—Nirenberg \((^2)\), and for \(k=1\) and \(q=\infty\), the result of S. Spanne \((^3)\).
Corollary 1. If, for some \(r\ge q\), the function
\[ \bar\varphi(\tau)=\tau^\lambda\int_0^\tau \frac{\varphi(u)}{u^{1+a}}\,du,\qquad a=n(q^{-1}-r^{-1}), \]
is defined, then there is a continuous embedding
\[ \mathscr L_p^k(V_q^\varphi)\subset \mathscr L_r^k(V_q^{\bar\varphi}). \]
In particular, if \(p\le r\) and \(\varphi\) is a quasi-degree majorant, i.e. \(\bar\varphi\le\varphi\), then \(\mathscr L_p^k(V_q^\varphi)\) is isomorphic to \(\mathscr L_r^k(V_q^\varphi)\).
Corollary 2. All spaces \(\mathscr L_p^k(V_q^\varphi)\), \(1\le p<q\), are isomorphic.
For the formulation of the second theorem, assume that \(\varphi(\tau)=\tau^s\psi(\tau)\), where \(s\ge0\) is an integer smaller than \(k\), and \(\psi\) is a modulus of continuity. Put
\[ \bar\psi(\tau)=\int_0^\tau \frac{\psi(u)}{u}\,du+\tau\int_\tau^1 \frac{\psi(u)}{u^2}\,du, \tag{5} \]
where, for \(s=k-1\), the second term is omitted.
Theorem 2. If \(f\in \mathscr L_p^k(V_q^\varphi)\), \(\varphi(\tau)=\tau^s\psi(\tau)\), and the function (5) exists, then for every \(\varepsilon>0\)
\[ f=g+h, \]
where \(h\) has support whose measure is less than \(\varepsilon\), and \(g\) belongs to \(C^{s,\bar\psi}\), with
\[ \sup_{|\alpha|=s}\sup_{x\ne y} \frac{\left|D^\alpha(g;x)-D^\alpha(g;y)\right|} {\bar\psi(|x-y|)} =O(\varepsilon^{-1/q})\,V(I_f^k;Q_0). \]
Remark. One can also give an estimate for higher-order differences of the function \(D^\alpha g\) on the set \(Q_0\setminus \operatorname{supp} h\).
- We indicate the connection between \(\mathscr L_p^k(V_q^\varphi)\) and known function spaces.
A. Sobolev spaces.
Theorem 3. The space \(\mathscr L_p^k(V_q^\varphi)\), for \(\varphi(\tau)=\tau^k\), \(1<q\le p\), and \(k>n(q^{-1}-p^{-1})\), is isomorphic to the space \(W_q^k\cap L_p\).
Remark. For \(k=n(q^{-1}-p^{-1})\) only the proper embedding into \(W_q^k\cap L_p\) holds.
For \(n=1\), \(p=\infty\), and \(k=1\), we obtain from Theorem 3 the known lemma of F. Riesz (see \((^1)\), p. 85). From Theorem 1 we now obtain the embedding theorem of S. L. Sobolev \((^4)\) with a certain refinement in the limiting case (cf. \((^5)\)), and from Theorem 2, a certain new property of functions from Sobolev spaces, namely that, after modification on a set of small measure, they coincide with functions from \(C^{k-1,1}\).
B. Lipschitz spaces. In the works \((^{6-9})\) it was shown that if \(\varphi(\tau)=\tau^{s+\alpha}\), \(0\le s<k-1\), \(0<\alpha<1\), or \(\varphi(\tau)=\tau^{s+\alpha}\), \(s=k-1\), \(\alpha=1\), then \(\mathscr L_p^k(V_\infty^\varphi)\) is isomorphic to the space \(C^{s,\alpha}\). It also follows from the results of \((^9)\) that if \(0<s\le k-1\) and \(\alpha=0\), then \(\mathscr L_p^k(V_\infty^\varphi)\) is isomorphic to \(C^{s-1,1-0}\), where the space \(C^{l,1-0}\) is defined by the condition that the second-
differences of step \(h\) of the higher derivatives are majorized by the quantity \(O(|h|)\). For \(q<\infty\) we have the following result.
Theorem 4. If \(\varphi(\tau)=\tau^\alpha,\ 0<\alpha<k\), then for \(p\le q\) the continuous embeddings
\[ L_p\cap B_q^\alpha \subset {\mathscr L}_p^k(V_q^\varphi)\subset H_p^\alpha, \]
hold, and for \(q\le p\) the embeddings
\[ B_p^\alpha\subset {\mathscr L}_p^k(V_q^\varphi)\subset H_q^\alpha\cap L_p. \]
Here \(H_p^\alpha,\ B_p^\alpha\) are the well-known spaces of S. M. Nikol’skii and O. V. Besov (see the survey \({}^{10}\)).
B. Functions of bounded variation. For \(\varphi(\tau)=\tau^{1/q}\) and \(n=1\), the space \({\mathscr L}_\infty(V^\varphi)\) is isomorphic to the space of functions of bounded \(q\)-variation in the sense of Wiener—L. Young. Therefore also for \(n>1\) it is natural to call the functions of the space \({\mathscr L}_\infty(V_q^\varphi)\), where \(\varphi(\tau)=\tau^{1/q}\), functions of bounded \(q\)-variation. The space \({\mathscr L}_1(V_1^\varphi)\) for \(\varphi(\delta)=\tau\) is isomorphic to the space of functions having bounded variation in the sense of Tonelli. More generally, the following holds.
Theorem 5. The space \({\mathscr L}_1^k(V_1^\varphi)\) for \(\varphi(\tau)=\tau^k\) is isomorphic to the space \(BV^k\) of functions in \(L_1(Q_0)\) whose generalized \(k\)-derivatives are Borel measures.
Remark. At the same time, the space \({\mathscr L}_1^k(v_1^\varphi)\), \(\varphi(\tau)=\tau^k\), is isomorphic to the space \(W_1^k\).
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One can choose \(X\) in \({\mathscr L}_p^k(X)\) also in such a way as to obtain any Lipschitz space in the sense of Calderon (see \({}^{11}\)). In particular, one can obtain the spaces \(H_p^\alpha\) of S. M. Nikol’skii or \(B_{p,q}^\alpha\) of O. V. Besov.
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In conclusion we state an interpolation theorem for operators acting from the scale \(L_p\) into the space \({\mathscr L}_p^k(X)\).
Let \(X_0, X_1\) be two spaces of functions of the cube \(I(Q)\), described in item 2. Denote by \(X_s,\ 0<s<1\), the space of those functions \(I(Q)\) for each of which there are a number \(\lambda\) and functions \(I_i\in X_i,\ i=0,1\), such that
\[ |I(Q)|\le \lambda |I_0(Q)|^{1-s}|I_1(Q)|^s,\qquad Q\subset Q_0. \tag{6} \]
We take the lower bound of \(\lambda\) in (6), under \(\|I_i\|_{X_i}\le 1\), as the norm in \(X_s\).
Theorem 6. Let a linear operator \(T\) act continuously from \(L_{p_i}\) to \({\mathscr L}_{p_i}^k(X_i)\), and let its norm be \(a_i,\ i=0,1\). Then \(T\) acts continuously from \(L_p\) to \({\mathscr L}_p^k(X_s)\), where \(1/p=(1-s)/p_0+s/p_1\), and its norm does not exceed \(a_0^{1-s}a_1^s\).
Dnepropetrovsk Chemical-Technological
Institute
Received
23 V 1969
CITED LITERATURE
\({}^{1}\) F. Riesz, B. Sz.-Nagy, Lectures on Functional Analysis, IL, 1964.
\({}^{2}\) F. John, Z. Nirenberg, Comm. Pure Appl. Math., 14, 415 (1961).
\({}^{3}\) S. Spanne, Ann. Scuola Norm. Super. Pisa, 19, No. 4, 593 (1965).
\({}^{4}\) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
\({}^{5}\) V. I. Yudovich, DAN, 138, No. 3, 805 (1961).
\({}^{6}\) S. Campanato, Ann. Scuola Norm. Super. Pisa, 17, 175 (1963).
\({}^{7}\) N. Meyers, Proc. Am. Math. Soc., 15, 717 (1964).
\({}^{8}\) S. Campanato, Ann. Scuola Norm. Super. Pisa, 18, No. 1, 137 (1964).
\({}^{9}\) Yu. A. Brudnyi, Studies in the Theory of Local Best Approximation, dissertation, Dnepropetrovsk, 1965.
\({}^{10}\) S. M. Nikol’skii, UMN, 16, No. 5 (101), 63 (1961).
\({}^{11}\) A. P. Calderon, Studia Math., 24, 113 (1964).