We study mixed norm spaces that arise in connection with embeddings of Sobolev and Besov spaces. We prove Sobolev type inequalities in terms of these mixed norms. Applying these results, we obtain optimal constants in embedding theorems for anisotropic Besov spaces. This gives an extension of the estimate proved by Bourgain, Brezis and Mironescu for isotropic Besov spaces.

Let D be a bounded strictly pseudoconvex domain of ${ℂ}^n$ with smooth boundary. We consider the weighted mixed-norm spaces $A_{\delta ,k}^{p,q}\left(D\right)$ of holomorphic functions with norm $\parallel f{\parallel }_{p,q,\delta ,k}=({\sum }_{\left|\alpha \right|\le k}{ʃ}_0^{r_0}({ʃ}_{\partial D_r}|D^{\alpha }{f|}^{p}d{\sigma }_r{{)}^{q/p}{r}^{\delta q/p-1}dr)}^{1/q}$. We prove that these spaces can be obtained by real interpolation between Bergman-Sobolev spaces $A_{\delta ,k}^p\left(D\right)$ and we give results about real and complex interpolation between them. We apply these results to prove that $A_{\delta ,k}^{p,q}\left(D\right)$ is the intersection of a Besov space $B_s^{p,q}\left(D\right)$ with the space of holomorphic functions on D. Further, we obtain several properties of the mixed-norm...

If $P$ is the Hardy averaging operator - or some of its generalizations, then weighted modular inequalities of the form u (Pf) Cv (f) are established for a general class of functions $\phi$. Modular inequalities for the two- and higher dimensional Hardy averaging operator are also given.

We introduce the notion of the modulus of dentability defined for any point of the unit sphere S(X) of a Banach space X. We calculate effectively this modulus for denting points of the unit ball of the classical interpolation space $L¹+L^{\infty }.$ Moreover, a criterion for denting points of the unit ball in this space is given. We also show that none of denting points of the unit ball of $L¹+L^{\infty }$ is a LUR-point. Consequently, the set of LUR-points of the unit ball of $L¹+L^{\infty }$ is empty.

Let E be a Banach space. Let $L{¹}_{\left(1\right)}({ℝ}^d,E)$ be the Sobolev space of E-valued functions on ${ℝ}^d$ with the norm ${ʃ}_{{ℝ}^d}\parallel f{\parallel }_{E}dx+{ʃ}_{{ℝ}^d}\parallel \nabla f{\parallel }_{E}dx=\parallel f\parallel ₁+\parallel \nabla f\parallel ₁$. It is proved that if $f\in L{¹}_{\left(1\right)}({ℝ}^d,E)$ then there exists a sequence $\left(g_m\right)\subset L_{\left(1\right)}¹({ℝ}^d,E)$ such that $f={\sum }_m{g}_m$; ${\sum }_m(\parallel g_m\parallel ₁+\parallel \nabla g_m\parallel ₁)<\infty$; and $\parallel g_m{\parallel }_{\infty }^{1/d}\parallel g_m\parallel {₁}^{(d-1)/d}\le b\parallel \nabla g_m\parallel ₁$ for m = 1, 2,..., where b is an absolute constant independent of f and E. The result is applied to prove various refinements of the Sobolev type embedding $L_{\left(1\right)}¹({ℝ}^d,E)\hookrightarrow L²({ℝ}^d,E)$. In particular, the embedding into Besov spaces $L{¹}_{\left(1\right)}({ℝ}^d,E)\hookrightarrow B_{p,1}^{\theta (p,d)}({ℝ}^d,E)$ is proved, where $\theta (p,d)=d(p^{-1}+d^{-1}-1)$ for 1 < p ≤ d/(d-1), d=1,2,... The latter embedding in the scalar case is due to Bourgain and Kolyada....

We study the notion of molecules in coorbit spaces. The main result states that if an operator, originally defined on an appropriate space of test functions, maps atoms to molecules, then it can be extended to a bounded operator on coorbit spaces. For time-frequency molecules we recover some boundedness results on modulation spaces, for time-scale molecules we obtain the boundedness on homogeneous Besov spaces.