UDC 517.53

MATHEMATICS

A. I. PLOTKIN

ON ISOMETRIC OPERATORS IN SPACES OF SUMMABLE ANALYTIC AND HARMONIC FUNCTIONS

(Presented by Academician V. I. Smirnov on 21 VIII 1968)

Let \(D\) be a bounded domain in the complex plane and let \(\sigma\) be plane Lebesgue measure in \(D\). By \(L_a^p(D)\) \(\bigl(L_h^p(D)\bigr)\) we denote the space of all analytic (respectively, complex-valued harmonic) functions \(f\) for which the norm

\[ \|f\|=\left(\int_D |f|^p\,d\sigma\right)^{1/p} \]

is finite (\(p\ge 1\)), and by \(B(D)\) the space of all analytic functions bounded in \(D\), with the sup-norm.

The following theorem of Kakutani—Chevalley is known \((^1)\):
Let \(D_1\) and \(D_2\) be domains having no \(AB\)-removable boundary points (a boundary point is called \(AB\)-removable if every function in \(B(D)\) admits analytic continuation to some neighborhood of this point). If the algebras \(B(D_1)\) and \(B(D_2)\) are isomorphic, then the domains \(D_1\) and \(D_2\) are conformally equivalent.

The same conclusion follows from the isometry of the \(B\)-spaces \(B(D_1)\) and \(B(D_2)\) \((^2)\).

In the present note we prove analogues of the second of these two theorems for the spaces \(L_a^p(D)\) and \(L_h^p(D)\), and also for the spaces \(E^p(D)\) of V. I. Smirnov \((^3,{}^4)\) (for the definition of \(E^p(D)\) for a multiply connected domain see, for example, \((^5)\)). It turns out that under natural conditions on the domains the isometry of the spaces \(L_a^p(D_1)\) and \(L_a^p(D_2)\) (or of the spaces \(E^p(D_1)\) and \(E^p(D_2)\)), \(p\ne 2\), entails the conformal equivalence of the domains \(D_1\) and \(D_2\). The isometry of the spaces \(L_h^p(D_1)\) and \(L_h^p(D_2)\), however, entails (under somewhat more restrictive conditions on the domains) the congruence of \(D_1\) and \(D_2\).

The basis for the proof of these results is a fairly general Theorem 1, which apparently is also of independent interest. The method used to prove this theorem is a development of the method first applied by Forelli \((^6)\) to describe the general form of isometric operators in the Hardy spaces \(H^p\).

We note that all the theorems formulated below, with obvious changes in the formulations, are also valid for \(0<p<1\).

1. Let \((X_1,\sigma_1)\) and \((X_2,\sigma_2)\) be two spaces with positive normalized measures, and let \(B_0\) be a subalgebra with identity in \(L^\infty(\sigma_1)\). Let \(1\le p<\infty\), \(p\ne 2\), and let \(T\) be a linear operator mapping \(B_0\) into \(L^p(\sigma_2)\).

Lemma 1. Suppose that \(T1=1\), and assume that to each \(f\in B_0\) there correspond two numbers \(a\) and \(\delta\), \(\delta>0\), such that for all \(z\), \(|z|<\delta\),

\[ \int_{X_1} |1+zf|^p\,d\sigma_1 = \int_{X_2} |1+zTf|^p\,d\sigma_2 + a|z|^p . \]

Then: 1) \(a=0\) for all \(f\in B_0\); 2) \(T\) maps \(B_0\) into \(L^\infty(\sigma_2)\); 3) \(T\) is multiplicative on \(B_0\); 4) \(T\) is isometric in the metrics of the spaces \(L^{2k}\), \(k=1,2,\ldots\), and \(L^\infty\).

Lemma 2. Let \(T\) be an isometric operator in the \(L^p\)-metrics, i.e.

\[ \int_{X_1}|f|^p\,d\sigma_1=\int_{X_2}|Tf|^p\,d\sigma_2 \]

for all \(f\in B_0\). Let \(F=T1\), and let \(E\) be such a (measurable) set in \(X_2\) that \(F=0\) \(\sigma_2\)-almost everywhere outside \(E\) and \(F\ne0\) \(\sigma_2\)-almost everywhere on \(E\). Then for every \(f\in B_0\), \(Tf=0\) \(\sigma_2\)-almost everywhere outside \(E\).

With the help of Lemmas 1 and 2 one proves the following.

Theorem 1. Let \(T\) be an isometric operator in the \(L^p\)-metrics \((p\ne2)\). Then \(T\) has the form

\[ Tf=F\varphi(f), \]

where \(F\in L^p(\sigma_2)\) \((F=T1)\) and \(\varphi\) is a homomorphism of the algebra \(B_0\) into \(L^\infty(\sigma_2)\), isometric in the \(L^\infty\)-metrics.

Proof. Let \(F=T1\), \(d\sigma_3=|F|^p d\sigma_2\), and let \(E\) be such a set as in Lemma 2. Put \(T_1f=Tf/F\) on \(E\) and \(T_1f=0\) outside \(E\). Then \(T_1 1=1\) (\(\sigma_3\)-almost everywhere), the measure \(\sigma_3\) is normalized, and, as follows from Lemma 2, \(T_1\) maps \(B_0\) isometrically with the \(L^p(\sigma_1)\)-metric into \(L^p(\sigma_3)\). By Lemma 1, \(T_1\) is a homomorphism of the algebra \(B_0\) into \(L^\infty(\sigma_3)\), isometric in the \(L^\infty(\sigma_1)\) and \(L^\infty(\sigma_3)\) metrics. But, again with the help of Lemma 2, it is easy to show that in this assertion one may replace \(\sigma_3\) by \(\sigma_2\), as was required to prove.

1. Theorem 2. Let \(D_1\) and \(D_2\) be two bounded domains in the plane and \(p\ne2\). If the spaces \(L_a^p(D_1)\) and \(L_a^p(D_2)\) are isometrically isomorphic, then the \(B\)-algebras \(B(D_1)\) and \(B(D_2)\) are algebraically and isometrically isomorphic.

Theorem 3. Let the bounded domains \(D_1\) and \(D_2\) have no \(AB\)-removable boundary points. Then, if the spaces \(L_a^p(D_1)\) and \(L_a^p(D_2)\) \((p\ne2)\) are isometrically isomorphic, the domains \(D_1\) and \(D_2\) are conformally equivalent.

The proof follows from Theorem 2 and the Kakutani–Chevalley theorem formulated above.

Using the more complete formulation of the Kakutani–Chevalley theorem (see \((^1)\)) and narrowing the class of domains under consideration, one can also obtain the general form of isometric mappings of \(L_a^p(D_1)\) onto \(L_a^p(D_2)\).

Theorem 4. Let \(D_1\) and \(D_2\) be Jordan domains and let \(T\) be an isometric isomorphism of \(L_a^p(D_1)\) onto \(L_a^p(D_2)\). Then \(T\) has the form

\[ Tf=b(\tau')^{2/p}f(\tau), \]

where \(b\) is a number, \(|b|=1\), and \(\tau\) is a one-to-one conformal mapping of \(D_2\) onto \(D_1\). Conversely, for any such \(b\) and \(\tau\), the formula written above defines an isometric isomorphism of \(L_a^p(D_1)\) onto \(L_a^p(D_2)\).

1. Let now \(D_1\) and \(D_2\) be finitely connected domains bounded by rectifiable Jordan curves. For the spaces \(E^p(D_1)\) and \(E^p(D_2)\), the analogues of the assertions of § 2 are valid. Namely, the following holds.

Theorem 5. Let \(T\) be an isometric isomorphism of \(E^p(D_1)\) onto \(E^p(D_2)\), \(p\ne2\). Then \(T\) has the form

\[ Tf=b(\tau')^{1/p}f(\tau), \]

where \(b\) is a number, \(|b|=1\), and \(\tau\) is a one-to-one conformal mapping of \(D_2\) onto \(D_1\) such that \((\tau')^{1/p}\) is a single-valued analytic function in \(D_2\). Conversely, for any such \(b\) and \(\tau\), the written formula defines an isometric isomorphism of \(E^p(D_1)\) onto \(E^p(D_2)\).

1. We pass to the consideration of the spaces \(L_h^p(D)\).

Lemma 3. Let \(F\) be a harmonic function in the domain \(D\), \(F\ne0\), and let \(g\) be twice continuously differentiable in \(D\), \(g\ne\mathrm{const}\). Suppose that the function \(Fg^n\), for whatever natural \(n\), is harmoni-

analytic in \(D\). Then either both functions \(F\) and \(g\) are analytic, or both functions \(\overline F\) and \(\overline g\) are analytic.

In what follows we shall consider only domains \(D\) satisfying the following condition:

\((\alpha)\). The linear span \(B(D)\) and \(\overline{B}(D)\) (\(\overline{B}(D)\) is the set of functions complex-conjugate to functions from \(B(D)\)) is dense in \(L_h^p(D)\).

We note that Jordan domains with sufficiently smooth boundary have property \((\alpha)\).

Lemma 4. Let the domains \(D_1\) and \(D_2\) have property \((\alpha)\), and let \(T\) be an isometric isomorphism of \(L_h^p(D_1)\) onto \(L_h^p(D_2)\) \((p \ne 2)\). Then \(T1=\mathrm{const}\).

Theorem 6. If the bounded domains \(D_1\) and \(D_2\) have the same area, have property \((\alpha)\), and have no \(AB\)-removable boundary points, then an arbitrary isometric isomorphism \(T\) of \(L_h^p(D_1)\) onto \(L_h^p(D_2)\) has the form

\[ Tf=bf(\tau), \]

where \(b\) is a number, \(|b|=1\), and \(\tau\) is a congruent transformation of \(D_2\) onto \(D_1\). Conversely, for any such \(b\) and \(\tau\), the formula written above defines an isometric isomorphism of \(L_h^p(D_1)\) onto \(L_h^p(D_2)\).

The author expresses his sincere gratitude to V. P. Khavin for posing the problem and for his attention to the work.

Leningrad State University
named after A. A. Zhdanov

Received
20 VI 1968

REFERENCES CITED

1. S. Kakutani, Lect. func. compl. var., Ann. Asbor. Univ. Mich. Press (1955).
2. M. Nagasava, Kōdai Math. Sem. Repts, 11, 4 (1959).
3. B. I. Smirnov, Zhurn. Leningrad. matem. obshch., 2 (1928).
4. V. I. Smirnov, Izv. AN SSSR, 337 (1932).
5. G. Ts. Tumarkin, S. Ya. Khavinson, UMN, 1 (1958).
6. F. Forelli, Canad. Math. J., 16, 4 (1964).