Let $(A_0,A_1)$ be a regular interpolation couple. Under several different assumptions on a fixed $A^{\beta }$, we show that $A^{\theta }=A_{\theta }$ for every $\theta \in (0,1)$. We also deal with assumptions on ${\overline{A}}^{\beta }$, the closure of $A^{\beta }$ in the dual of ${(A_0^{*},A_1^{*})}_{\beta }$.

We find necessary and sufficient conditions under which the norms of the interpolation spaces ${(N₀,N₁)}_{\theta ,q}$ and ${(X₀,X₁)}_{\theta ,q}$ are equivalent on N, where N is the kernel of a nonzero functional ψ ∈ (X₀ ∩ X₁)* and $N_i$ is the normed space N with the norm inherited from $X_i$ (i = 0,1). Our proof is based on reducing the problem to its partial case studied by Ivanov and Kalton, where ψ is bounded on one of the endpoint spaces. As an application we completely resolve the problem of when the range of the operator $T_{\theta }=S-2^{\theta }I$ (S...

Let $𝔐$ and $\Re$ be algebras of subsets of a set $\Omega$ with $𝔐\subset \Re$, and denote by $E\left(\mu \right)$ the set of all quasi-measure extensions of a given quasi-measure $\mu$ on $𝔐$ to $\Re$. We give some criteria for order boundedness of $E\left(\mu \right)$ in $ba\left(\Re \right)$, in the general case as well as for atomic $\mu$. Order boundedness implies weak compactness of $E\left(\mu \right)$. We show that the converse implication holds under some assumptions on $𝔐$, $\Re$ and $\mu$ or $\mu$ alone, but not in general.

In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X,d,\mu )$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B(x,r)$ converge to $f\left(x\right)$ when $r$ converges to $0$. We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....

We consider the Massera-Schäffer problem for the equation $$-y^{\text{'}}\left(x\right)+q\left(x\right)y\left(x\right)=f\left(x\right),x\in ℝ,$$ where $f\in L_p^{\mathrm{loc}}\left(ℝ\right),$ $p\in [1,\infty )$ and $0\le q\in L_1^{\mathrm{loc}}\left(ℝ\right).$ By a solution of the problem we mean any function $y,$ absolutely continuous and satisfying the above equation almost everywhere in $ℝ.$ Let positive and continuous functions $\mu \left(x\right)$ and $\theta \left(x\right)$ for $x\in ℝ$ be given. Let us introduce the spaces $$\begin{array}{cc}\hfill L_p(ℝ,\mu )& =\left\{f\in L_p^{\mathrm{loc}}{\left(ℝ\right):\parallel f\parallel }_{L_p(ℝ,\mu )}^p={\int }_{-\infty }^{\infty }{\left|\mu \left(x\right)f\left(x\right)\right|}^p\mathrm{d}x<\infty \right\},\\ \hfill L_p(ℝ,\theta )& =\left\{f\in L_p^{\mathrm{loc}}{\left(ℝ\right):\parallel f\parallel }_{L_p(ℝ,\theta )}^p={\int }_{-\infty }^{\infty }{\left|\theta \left(x\right)f\left(x\right)\right|}^p\mathrm{d}x<\infty \right\}.\end{array}$$ We obtain requirements to the functions $\mu$, $\theta$ and $q$ under which (1) for every function $f\in L_p(ℝ,\theta )$ there exists a unique solution $y\in L_p(ℝ,\mu )$ of the above equation; (2) there is an absolute constant...

The notion of $\beta$-normality was introduced and studied by Arhangel’skii, Ludwig in 2001. Recently, almost $\beta$-normal spaces, which is a simultaneous generalization of $\beta$-normal and almost normal spaces, were introduced by Das, Bhat and Tartir. We introduce a new generalization of normality, namely weak $\beta$-normality, in terms of $\theta$-closed sets, which turns out to be a simultaneous generalization of $\beta$-normality and $\theta$-normality. A space $X$ is said to be weakly $\beta$-normal (w$\beta$-normal$)$ if for every...

Let $(X,d,\mu )$ be a metric measure space satisfying the doubling condition and an $L^2$-Poincaré inequality. Consider the nonnegative operator $ℒ$ generalized by a Dirichlet form on $X$. We will show that a solution $u$ to $(-{\partial }_t^2+ℒ)u=0$ on $X\times {ℝ}_{+}$ satisfies an $\alpha$-Carleson condition if and only if $u$ can be represented as the Poisson integral of the operator $ℒ$ with the trace in the generalized Morrey space $L^{2,\alpha }\left(X\right)$, where $\alpha$ is a nonnegative function defined on a class of balls in $X$. This result extends the analogous characterization...

A famous theorem of Carleson says that, given any function $f\in L^p\left(𝕋\right)$, $p\in (1,+\infty )$, its Fourier series $\left(S_{n}f\left(x\right)\right)$ converges for almost every $x\in 𝕋$. Beside this property, the series may diverge at some point, without exceeding $O\left(n^{1/p}\right)$. We define the divergence index at $x$ as the infimum of the positive real numbers $\beta$ such that $S_{n}f\left(x\right)=O\left(n^{\beta }\right)$ and we are interested in the size of the exceptional sets $E_{\beta }$, namely the sets of $x\in 𝕋$ with divergence index equal to $\beta$. We show that quasi-all functions in $L^p\left(𝕋\right)$ have a multifractal behavior with respect to...

Let $G$ be a finite group and $H$ a subgroup. Denote by $D(G;H)$ (or $D\left(G\right)$) the crossed product of $C\left(G\right)$ and $ℂH$ (or $ℂG$) with respect to the adjoint action of the latter on the former. Consider the algebra $\langle D\left(G\right),e\rangle$ generated by $D\left(G\right)$ and $e$, where we regard $E$ as an idempotent operator $e$ on $D\left(G\right)$ for a certain conditional expectation $E$ of $D\left(G\right)$ onto $D(G;H)$. Let us call $\langle D\left(G\right),e\rangle$ the basic construction from the conditional expectation $E:D\left(G\right)\to D(G;H)$. The paper constructs a crossed product algebra $C(G/H\times G)⋊ℂG$, and proves that there is an algebra isomorphism between...