zbMATH — the first resource for mathematics

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.

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
Full Text: EuDML
[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.