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On interpolation of bilinear operators by the real method. (English. Russian original) Zbl 0798.46054
Math. Notes 52, No. 1, 641-648 (1992); translation from Mat. Zametki 52, No. 1, 15-24 (1992).
The author studies interpolation properties of bilinear operators on Banach spaces. He obtains interesting results.
One of the questions he studies is the following. Let $$F$$ be an interpolation functor. It is said to interpolate bilinear operators if for arbitrary Banach pairs $$\vec X= (X_ 0,X_ 1)$$, $$\vec Y= (Y_ 0,Y_ 1)$$, $$\vec Z=(Z_ 0,Z_ 1)$$ and for an arbitrary bilinear operator $$B: X_ i\times Y_ i\to Z_ i$$ $$(i=1,2)$$ it follows that $$B: F(\vec X)\times F(\vec Y)\to F(\vec Z)$$.
Several important results are obtained for real interpolation functors.

##### MSC:
 46M35 Abstract interpolation of topological vector spaces 46B70 Interpolation between normed linear spaces
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##### References:
 [1] I. Berg and I. Lefstrem, Interpolation Spaces: An Introduction [Russian translation], Mir, Moscow (1980). [2] S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators [in Russian], Nauka, Moscow (1978). [3] J. Peetre, ?A theory of interpolation of normed spaces,? Notes Math.,39, 1-86 (1969). [4] L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1977). · Zbl 0127.06102 [5] V. J. Ovchinnikov, ?The method of orbits in interpolation theory,? Math. Rept.,1, 349-515 (1984). · Zbl 0875.46007 [6] J.-L. Lions and J. Peetre, ?Sur une classe d’espaces d’interpolation,? Inst. Hautes Etudes Sci. Publ. Math.,19, 5-68 (1964). · Zbl 0148.11403 · doi:10.1007/BF02684796 [7] S. Janson, ?On interpolation of multi-linear operators,? Lect. Notes Math., 1302, 290-302 (1988). · Zbl 0827.46062 [8] R. O’Neil, ?Convolution operators and L(p, q) spaces,? Duke Math. J.,30, 129-142 (1963). [9] E. A. Pavlov, ?On convolution operators in symmetric spaces,? Usp. Mat. Nauk,31, No. 1, 257-258 (1976). · Zbl 0331.44006 [10] E. A. Pavlov, ?On boundedness of convolution operators in symmetric spaces,? Izv. Vuzov. Mat., No. 2, 36-42 (1982). · Zbl 0496.47044 [11] E. A. Pavlov, ?On the Hausdorff-Jung integral inequality in spaces with mixed norms,? Usp. Mat. Nauk,39, No. 2, 183-184 (1984). [12] V. I. Dmitriev, S. G. Krein, and V. I. Ovchinnikov, ?Foundations of the theory of interpolation of linear operators,? in: Intermural Collection of Scientific Papers [in Russian], Yaroslavl’, Izd. Yarosl. Univ., 31-74 (1977). [13] M. Milman, ?Embeddings of Lorentz-Marcinkiewicz spaces with mixed norms,? Anal. Math.,4, 215-223 (1978). · Zbl 0417.46030 · doi:10.1007/BF01908990 [14] H. Trivel, The Theory of Interpolation, Functional Spaces, and Differential Operators [Russian translation], Mir, Moscow (1980). [15] A. V. Bukhvalov, ?The complex method of interpolation in spaces of vector functions and in generalized Besov spaces,? Dokl. Akad. Nauk SSSR,260, No. 2, 265-269 (1981). · Zbl 0493.46062
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