We investigate how the asymptotic eigenvalue behaviour of Hille-Tamarkin operators in Banach function spaces depends on the geometry of the spaces involved. It turns out that the relevant properties are cotype p and p-concavity. We prove some eigenvalue estimates for Hille-Tamarkin operators in general Banach function spaces which extend the classical results in Lebesgue spaces. We specialize our results to Lorentz, Orlicz and Zygmund spaces and give applications to Fourier analysis. We are also...

We consider a strongly nonlinear monotone elliptic problem in generalized Orlicz-Musielak spaces. We assume neither a Δ2 nor ∇2-condition for an inhomogeneous and anisotropic N-function but assume it to be log-Hölder continuous with respect to x. We show the existence of weak solutions to the zero Dirichlet boundary value problem. Within the proof the L ∞-truncation method is coupled with a special version of the Minty-Browder trick for non-reflexive and non-separable Banach spaces.

Anisotropic Lipschitz spaces are considered. For these spaces we obtain sharp embeddings in Besov and Lorentz spaces. The methods used are based on estimates of iterative rearrangements. We find a unified approach that arises from the estimation of functions defined as minimum of a given system of functions. The case of L¹-norm is also covered.

We prove norm inequalities between Lorentz and Besov-Lipschitz spaces of fractional smoothness.

In this paper, characterizations of the embeddings between weighted Copson function spaces ${\mathrm{Cop}}_{p_1,q_1}(u_1,v_1)$ and weighted Cesàro function spaces ${\mathrm{Ces}}_{p_2,q_2}(u_2,v_2)$ are given. In particular, two-sided estimates of the optimal constant $c$ in the inequality $$\begin{array}{cc}\hfill d({\int }_0^{\infty }& {\left({\int }_0^{t}f{\left(\tau \right)}^{p_2}{v}_2\left(\tau \right)\mathrm{d}\tau \right)}^{q_2/p_2}{u}_2\left(t\right)\mathrm{d}t{)}^{1/q_2}\hfill \\ & \le c{\left({\int }_0^{\infty }{\left({\int }_t^{\infty }f{\left(\tau \right)}^{p_1}{v}_1\left(\tau \right)\mathrm{d}\tau \right)}^{q_1/p_1}{u}_1\left(t\right)\mathrm{d}t\right)}^{1/q_1},\hfill \end{array}d$$ where $p_1,p_2,q_1,q_2\in (0,\infty )$, $p_2\le q_2$ and $u_1$, $u_2$, $v_1$, $v_2$ are weights on $(0,\infty )$, are obtained. The most innovative part consists of the fact that possibly different parameters $p_1$ and $p_2$ and possibly different inner weights $v_1$ and $v_2$ are allowed. The proof is based on the combination of duality techniques with estimates...

A simple expression is presented that is equivalent to the norm of the Lpv → Lqu embedding of the cone of quasi-concave functions in the case 0 < q < p < ∞. The result is extended to more general cones and the case q = 1 is used to prove a reduction principle which shows that questions of boundedness of operators on these cones may be reduced to the boundedness of related operators on whole spaces. An equivalent norm for the dual of the Lorentz spaceΓp(v) = { f: ( ∫0∞ (f**)pv...

Let $S\subset {ℝ}^{n+1}$ be the graph of the function $\varphi :{[-1,1]}^n\to ℝ$ defined by $\varphi (x_1,\cdots ,x_n)={\sum }_{j=1}^n{\left|x_j\right|}^{{\beta }_j},$ with 1<${\beta }_1\le \cdots \le {\beta }_n,$ and let $\mu$ the measure on ${ℝ}^{n+1}$ induced by the Euclidean area measure on S. In this paper we characterize the set of pairs (p,q) such that the convolution operator with $\mu$ is $L^p$-$L^q$ bounded.

We discuss the existence of entropy solution for the strongly nonlinear unilateral parabolic inequalities associated to the nonlinear parabolic equations ∂u/∂t - div(a(x,t,u,∇u) + Φ(u)) + g(u)M(|∇u|) = μ in Q, in the framework of Orlicz-Sobolev spaces without any restriction on the N-function of the Orlicz spaces, where -div(a(x,t,u,∇u)) is a Leray-Lions operator and $\Phi \in C⁰(ℝ,{ℝ}^N)$. The function g(u)M(|∇u|) is a nonlinear lower order term with natural growth with respect to |∇u|, without satisfying the sign...