Abstract In this paper, we study the resonance problem of a class of singular quasilinear parabolic equations with respect to its higher near-eigenvalues. Under a generalized Landesman-Lazer condition, it is proved that the resonance problem admits at least one nontrivial solution in weighted Sobolev spaces. The proof is based upon applying the Galerkin-type technique, the Brouwer’s fixedpoint ...

We present forms of the classical Riesz–Kolmogorov theorem for compactness that are applicable in a wide variety settings. In particular, our theorems apply to classify precompact subsets Lebesgue space $$L^2$$ , Paley–Wiener spaces, weighted Bargmann–Fock and scale Besov–Sobolev spaces holomorphic functions includes Bergman general domains as well Hardy Dirichlet space. criteria characterize c...

1. Provocative example 2. Natural function spaces on the circle S = R/2πZ 3. Topology on C∞(S1) 4. Pointwise convergence of Fourier series 5. Distributions: generalized functions 6. Invariant integration, periodicization 7. Hilbert space theory of Fourier series 8. Levi-Sobolev inequality, Levi-Sobolev imbedding 9. C∞(S1) = limC(S) = limH(S) 10. Distributions, generalized functions, again 11. T...

1. A confusing example 2. Natural function spaces 3. Topology on limits C∞(S1) 4. Distributions (generalized functions) 5. Invariant integration, periodicization 6. Hilbert space theory of Fourier series 7. Completeness in L(S) 8. Sobolev lemma, Sobolev imbedding 9. C∞(S1) = limC(S) = limHs(S) 10. Distributions, generalized functions, again 11. The confusing example explained 12. Appendix: prod...

We approximate functions defined on smooth bounded domains by elements of the eigenspaces Laplacian or Stokes operator in such a way that approximations are and converge both Sobolev Lebesgue spaces. prove an abstract result referred to fractional power spaces positive, self-adjoint, compact-inverse operators Hilbert spaces, then obtain our main using explicit form these for Dirichlet operators...

Fixed point theorems for generalized Lipschitzian semigroups are proved in puniformly convex Banach spaces and in uniformly convex Banach spaces. As applications, its corollaries are given in a Hilbert space, in Lp spaces, in Hardy space Hp , and in Sobolev spaces Hk,p , for 1<p <∞ and k≥ 0.

We represent a bilinear Calder\'on-Zygmund operator at given smoothness level as finite sum of cancellative, complexity zero operators, involving smooth wavelet forms, and continuous paraproduct forms. This representation results in sparse $T(1)$-type bound, which turn yields directly new sharp weighted estimates on Lebesgue Sobolev spaces. Moreover, we apply the theorem to study fractional dif...

Initial-boundary value problems for parabolic equations in unbounded domains with respect to the spatial variables were studied by many authors. As is well known, guarantee uniqueness of solution initial-boundary linear and some nonlinear we need restrictions on solution's behavior as $|x|\to +\infty$ (for example, growth restriction +\infty$, or belonging functional spaces). Note that data-in ...