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Continuous and compact imbeddings of weighted Sobolev spaces. I. (English) Zbl 0676.46030
Summary: We establish some conditions on p, q and the weight functions $$v_ 0$$, $$v_ 1$$, w under which the continuous imbedding $(1.1)\quad W^{1,p}(\Omega;v_ 0,v_ 1)\hookrightarrow L^ q(\Omega;w),$ or the compact imbedding $(1.2)\quad W^{1,p}(\Omega;v_ 0,v_ 1)\hookrightarrow \hookrightarrow L^ q(\Omega;w)$ takes place.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 47B06 Riesz operators; eigenvalue distributions; approximation numbers, $$s$$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
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##### References:
 [1] Guzmán M.: Differentiation of integrals in $$R^{n}$$. Lecture Notes in Mathematics 481, Springer-Verlag, Berlin-Heidelberg-New York 1975. [2] Kufner A.: Weighted Sobolev spaces. John Wiley & Sons, Chichester-New York- Brisbane-Toronto-Singapore 1985. · Zbl 0579.35021 [3] Kufner A., John O., Fučík S.: Function spaces. Academia Praha 1977. [4] Kufner A., Opic B.: How to define reasonably weighted Sobolev spaces. Comment. Math. Univ. Carolinae, 25 (3) (1984), 537. · Zbl 0557.46025 [5] Lizorkin P. I., Otelbaev M.: Imbedding and compactness theorems for Sobolev type spaces with weights I, II. (Russian), Mat. Sb. (N.S.) 108 (150), (1979), No 3, 358-377, MR 80j : : 46054; 1J2 (154) (1980), No 1 (5), 56-85, MR 82i : 46051. [6] Opic B.: Necessary and sufficient conditions for compactness of imbeddings in weighted Sobolev spaces. Časopis Pěst. Mat. · Zbl 0704.46021 [7] Opic B., Gurka P.: $$A_ r$$-condition for two weight functions and compact imbeddings of weighted Sobolev spaces. Czechoslovak Math. J. 38(133) (1988), 611-617. · Zbl 0676.46029 [8] Zajaczkowski W.: On theorem of embedding for weighed Sobolev spaces. Bulletin of the Polish Academy of Sciences Mathematics, 32 (1985), No 3 - 4, 115-121.
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