Related papers: Lebesgue points of functions in the complex Sobole…

We study the Hardy-Littlewood maximal operator in the Musielak-Orlicz-Sobolev space $W^{1,\varphi}(\mathbb{R}^n)$. Under some natural assumptions on $\varphi$ we show that the maximal function is bounded and continuous in…

The purpose of this article is to define a capacity on certain topological measure spaces $X$ with respect to certain function spaces $V$ consisting of measurable functions. In this general theory we will not fix the space $V$ but we…

We introduce Besov spaces with variable smoothness and integrability by using the continuous version of Calder\`on reproducing formula. We show that our space is well-defined, i.e., independent of the choice of basis functions. We…

Let $T$ be a positive closed current of bidimension (1,1) and unit mass on the complex projective space ${\Bbb P}^n$. We prove that the set $V_\alpha(T)$ of points where $T$ has Lelong number larger than $\alpha$ is contained in a complex…

We show for a certain class of operators $A$ and holomorphic functions $f$ that the functional calculus $A\mapsto f(A)$ is holomorphic. Using this result we are able to prove that fractional Laplacians $(1+\Delta^g)^p$ depend real…

We study the operator $\mathcal{A}$ of multiplication by an independent variable in a matrix Sobolev space $W^2(M)$. In the cases of finite measures on $[a,b]$ with $(2\times 2)$ and $(3\times 3)$ real continuous matrix weights of full rank…

We consider multiplication properties of elements in weighted Fourier Lebesgue and modulation spaces. Especially we extend some results by Pilipovic, Teofanov and Toft (2010).

Polynomial approximation is studied in the Sobolev space $W_p^r(w_{\alpha,\beta})$ that consists of functions whose $r$-th derivatives are in weighted $L^p$ space with the Jacobi weight function $w_{\alpha,\beta}$. This requires…

Let $(M,F)$ be a $C^\infty$ Finsler manifold, $p\geq 1$ a real number, $k$ a positive integer and $H_k^p (M)$ a certain Sobolev space determined by a Finsler structure $F$. Here, it is shown that the set of all real $C^{\infty}$ functions…

Using the theory of quasiconformal mappings, we simplify the proof of the recent result by Taylor and Totik (see IMA Journal of Numerical Analysis 30 (2010) 462--486) on the behavior of the Lebesgue constants for interpolation points on a…

We investigate existence and uniqueness of maximal plurisubharmonic functions on bounded domains with boundary data that are not assumed to be continuous or bounded. The result is applied to approximate (possibly unbounded from above)…

Let $w_0$ be a bounded, $C^3$, strictly plurisubharmonic function defined on $B_1\subset \mathbb{C}^n$. Then $w_0$ has a neighborhood in $L^{\infty}(B_1)$. Suppose that we have a function $\phi$ in this neighborhood with $1-\epsilon \le…

We present an $L_q(L_{p})$-theory for the equation $$ \partial_{t}^{\alpha}u=\phi(\Delta) u +f, \quad t>0,\, x\in \mathbb{R}^d \quad\, ;\, u(0,\cdot)=u_0. $$ Here $p,q>1$, $\alpha\in (0,1)$, $\partial_{t}^{\alpha}$ is the Caputo fractional…

In this paper n-dimensional Sobolev type spaces $ E_{\alpha}^{s,p}(\R^n_+)$ $(\alpha\in \R^n,\;\;\alpha_1> -\frac{1}{2},...,\alpha_n>-\frac{1}{2}, s\in \R, p\in [1,+\infty])$ are defined on $\R^n_+$ by using Fourier-Bessel transform. Some…

We study the mean-value harmonic functions on open subsets of $\mathbb{R}^n$ equipped with weighted Lebesgue measures and norm induced metrics. Our main result is a necessary condition saying that all such functions solve a certain…

Let $X$ be a metric space and $\mu$ an $s$-regular Ahlfors measure. Let $Y$ be a metric space. We prove that for Besov functions $u \in B^r_{q,\infty}(X,\mu;Y)$, every point is a {\it general average Lebesgue point} of $u$ outside a…

Let $p:\R\to(1,\infty)$ be a globally log-H\"older continuous variable exponent and $w:\R\to[0,\infty]$ be a weight. We prove that the Cauchy singular integral operator $S$ is bounded on the weighted variable Lebesgue space…

We extend a recent result of Avelin, Hed, and Persson about approximation of functions $u$ that are plurisubharmonic on a domain $\Omega$ and continuous on $\bar\Omega$, with functions that are plurisubharmonic on (shrinking) neighborhoods…

We study comprehensively local properties of functions in complex Sobolev spaces on a bounded open subset of $\mathbb{C}^n$. The main tool is the corresponding functional capacity for the space which is inspired by the global one due to…

In this note, we present a well-known connection between the Sobolev-Slobodeckij spaces, also known as Fractional Sobolev spaces, and interpolation theory. We show how Sobolev spaces can be equivalently characterized as real and complex…