ABSTRACT CAPACITY OF REGIONS AND COMPACT EMBEDDING WITH APPLICATIONS

doi:10.1016/S0252-9602(11)60207-5

Abstract

Abstract:

The weighted Sobolev-Lions type spaces W ^l_p, _γ(Ω; E₀, E) = W ^l_p, _γ (Ω; E) \ L_p_, _γ(Ω; E₀) are two Banach spaces and E₀ is continuously and densely embedded on E. A new concept of capacity of region Ω∈Rⁿ in W ^l_p, _γ (Ω; E₀, E) is introduced. Several conditions in terms of capacity of region Ω and interpolations of E₀ and E are found such that ensure the continuity and compactness of embedding operators. In particular, the most regular class of interpolation spaces E_α between E₀ and E, depending of α and l, are found such that mixed differential operators D^α are bounded and compact from W ^l_p, γ(Ω; E₀, E) to E_α-valued L_p, γ spaces. In applications, the maximal regularity for differential-operator equations with parameters are studied.

Key words: Capacity of regions, embedding theorems, Banach-valued function spaces, differential-operator equations, Semigroups of operators, operator-valued Fourier multipliers, interpolation of Banach spaces

CLC Number:

34G10

Veli Shakhmurov. ABSTRACT CAPACITY OF REGIONS AND COMPACT EMBEDDING WITH APPLICATIONS[J].Acta mathematica scientia,Series B, 2011, 31(1): 49-67.

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