Extending recent work for the linear Poisson problem for the Laplacian in the framework of Sobolev-Besov spaces on Lipschitz domains by Jerison and Kenig [16], Fabes, Mendez and Mitrea [9], and Mitrea and Taylor [30], here we take up the task of developing a similar sharp theory for semilinear problems of the type Δu - N(x,u) = F(x), equipped with Dirichlet and Neumann boundary conditions.

Following H. Sato - Y. Okazaky we will prove that: if $X$ is a topological vector space, locally convex and reflexive, and $\mu$ is a gaussian measure on $𝐂(X,X^{\prime })$, then $L^2(\mu )$ is separable.

In this paper we survey some recent results concerning separating polynomials on real Banach spaces. By this we mean a polynomial which separates the origin from the unit sphere of the space, thus providing an analog of the separating quadratic form on Hilbert space.

There are defined sequential moduli in the remainder form for real sequences. Properties of sequence spaces generated by means of the above moduli are investigated.

A new set of sufficient conditions under which every sequence of independent identically distributed functions from a rearrangement invariant (r.i.) space on [0,1] spans there a Hilbertian subspace are given. We apply these results to resolve open problems of N. L. Carothers and S. L. Dilworth, and of M. Sh. Braverman, concerning such sequences in concrete r.i. spaces.

Complete and σ-complete Boolean algebras of projections acting in a Banach space were introduced by W. Bade in the 1950's. A basic fact is that every complete Boolean algebra of projections is necessarily a closed set for the strong operator topology. Here we address the analogous question for σ-complete Boolean algebras: are they always a sequentially closed set for the strong operator topology? For the atomic case the answer is shown to be affirmative. For the general case, we develop criteria...

In this paper, we will characterize sequentially compact sets in a class of generalized Orlicz spaces.

We give an elementary approach which allows us to evaluate Seip's conditions characterizing interpolating and sampling sequences in weighted Bergman spaces of infinite order for a wide class of weights depending on the distance to the boundary of the domain. Our results also give some information on cases not covered by Seip's theory. Moreover, we obtain new criteria for weights to be essential.

The theory of functions plays an important role in harmonic analysis. Because of this, it turns out that some spaces of analytic functions have been studied extensively, such as Hp-spaces, Bergman spaces, etc. One of the major insights that has developed in the study of Hp-spaces is what we call the real atomic characterization of these spaces.

Existence of an optimal shape of a deformable body made from a physically nonlinear material obeying a specific nonlinear generalized Hooke’s law (in fact, the so called deformation theory of plasticity is invoked in this case) is proved. Approximation of the problem by finite elements is also discussed.

Let $n\ge 2$ and $\Omega \subset {ℝ}^n$ be a bounded set. We give a Moser-type inequality for an embedding of the Orlicz-Sobolev space $W_0{L}^{\Phi }\left(\Omega \right)$, where the Young function $\Phi$ behaves like $t^n{log}^{\alpha }\left(t\right)$, $\alpha <n-1$, for $t$ large, into the Zygmund space $Z_0^{\frac{n-1-\alpha }{n}}\left(\Omega \right)$. We also study the same problem for the embedding of the generalized Lorentz-Sobolev space $W_0^m{L}^{\frac{n}{m},q}{log}^{\alpha }L\left(\Omega \right)$, $m<n$, $q\in (1,\infty ]$, $\alpha <\frac{1}{q^{\text{'}}}$, embedded into the Zygmund space $Z_0^{\frac{1}{q^{\text{'}}}-\alpha }\left(\Omega \right)$.

We consider function spaces of Besov, Triebel-Lizorkin, Bessel-potential and Sobolev type on ${ℝ}^d$, equipped with power weights $w\left(x\right)={\left|x\right|}^{\gamma }$, γ > -d. We prove two-weight Sobolev embeddings for these spaces. Moreover, we precisely characterize for which parameters the embeddings hold. The proofs are presented in such a way that they also hold for vector-valued functions.