This paper establishes new fixed-point theorems in the framework of complete p-normed spaces, where p∈(0,1]. By extending the classical Banach, Schauder, and Krasnosel’skii fixed-point theorems, we derive several results for the sum of contraction and compact operators acting on s-convex subsets. The analysis is further generalized to multivalued upper semi-continuous operators by employing Kuratowski and Hausdorff measures of noncompactness. These results lead to new Darbo–Sadovskii-type fixed-point theorems and global versions of Krasnosel’skii’s theorem for multifunctions in p-normed spaces. The theoretical findings are then applied to demonstrate the existence of solutions for nonlinear integral equations formulated in p-normed settings. A section on numerical applications is also provided to illustrate the effectiveness and applicability of the proposed results.

The characterization of compact operators on BK-spaces, which is the basis of this research, makes use of the Hausdorff measure of non-compactness. In this study, the compactness criteria of matrix operators defined on BK-spaces $\ell_p(\mathcal{T})$ and $\ell_{\infty}(\mathcal{T})$ which are the domains of the regular infinite Tetranacci matrix obtained by using the Tetranacci number sequence in $\ell_p$ and $\ell_{\infty}$, respectively, are investigated by using Hausdorff measure of non-compactness and some properties of $\ell_p(\mathcal{T})$ are examined.

In this paper, we introduce a generalized framework for conditional probability in stochastic processes taking values in infinite-dimensional normed spaces. Classical definitions, based on measurability with respect to a conditioning σ-algebra, become inadequate when the available information is restricted to a σ-algebra generated by a finite or countable family of random variables. In such settings, many events of interest are not measurable with respect to the conditioning σ-field, preventing the standard definition of conditional probability. To overcome this limitation, we propose an extension of the coherent conditioning model through the use of Hausdorff measures. The key idea is to exploit the non-equivalence of norms in infinite-dimensional spaces, which gives rise to distinct metric structures and corresponding Hausdorff dimensions for the same events. Conditional probabilities are then defined relative to families of Hausdorff outer measures parameterized by their dimensional exponents. This geometric reformulation allows the notion of conditionality to depend explicitly on the underlying metric and topological properties of the space. The resulting model provides a flexible and coherent framework for analyzing conditioning in infinite-dimensional stochastic systems, with potential implications for Bayesian inference in functional spaces.

Abstract The Generalised Baker–Schmidt Problem (1970) concerns the Hausdorff measure of the set of $\psi$ -approximable points on a non-degenerate manifold. Beresnevich-Dickinson-Velani (in 2006, for the homogeneous setting) and Badziahin-Beresnevich-Velani (in 2013, for the inhomogeneous setting) proved the divergence part of this problem for dual approximation on arbitrary non-degenerate manifolds. The divergence part has also been resolved for the $p$ -adic setting by Datta-Ghosh in 2022, for the inhomogeneous setting. The corresponding convergence counterpart represents a challenging open question. In this paper, we prove the homogeneous $p$ -adic convergence result for hypersurfaces of dimension at least three with some mild regularity condition, as well as for some other classes of manifolds satisfying certain conditions. We provide similar, slightly weaker results for the inhomogeneous setting. We do not restrict to monotonic approximation functions.

Abstract We investigate characterizations of uniformly rectifiable (UR) metric spaces by so-called weak Carleson conditions for flatness coefficients, which measure the extent to which Hausdorff measure on the metric space differs from Hausdorff measure on a normed space. First, we show that UR metric spaces satisfy David and Semmes’s weak constant density condition, a quantitative regularity property that implies most balls in the space support a measure with nearly constant density in a neighborhood of scales and locations. Second, we introduce a metric space variant of Tolsa’s alpha numbers that measure a local normalized $L_{1}$ mass transport cost between the space’s Hausdorff measure and Hausdorff measure on a normed space. We show that a weak Carleson condition for these alpha numbers gives a characterization of metric uniform rectifiability. We derive both results as corollaries of a more general abstract result, which gives a tool for transferring weak Carleson conditions to spaces with very big pieces of spaces with a given weak Carleson condition.

Abstract Let $$F_\rho $$ be the Cauchy transform of the self-similar measure $$\mu =\frac{1}{8}\sum _{j=0}^{7}\mu \circ S_j$$ , where $$S_j(z)=z_j+\rho (z-z_j)$$ with $$ z_{{2l}} = e^{{\frac{{2l}}{4}\pi i}} , $$ $$ z_{{2l + 1}} = \frac{{\sqrt 2 }}{2}e^{{\frac{{2l + 1}}{4}\pi i}} , $$ $$ l = 0,1,2,3, $$ $$ \rho \in (0,\frac{1}{3}) $$ , and K be the attractor of $$\{S_j\}_{j=0}^{7}$$ . In this paper, we derive asymptotic formulas for the Laurent coefficients $$\{a_{4k+1}\}_{k=1}^{\infty }$$ of $$F_\rho $$ in $$|z|>1$$ , and precisely determine the growth rate and the set of accumulation points for these Laurent coefficients. In addition, we give a lower bound for the domain of the starlikeness of $$F_\rho $$ .

This paper is devoted to investigating the geometric properties of solutions to certain degenerate equations, and their nonlocal counterparts, in the context of Poincaré–Einstein manifolds. The operators under consideration arise in the theory of conformal invariants on the visual boundary of Poincaré–Einstein manifolds. Within this framework, we develop a quantitative differentiation theory that includes a quantitative stratification of the singular set and Minkowski-type estimates for the (quantitatively) stratified singular sets of various solutions of PDEs on both the complete manifold and its conformal infinity. All these, together with a new \varepsilon -regularity result for degenerate/singular elliptic operators on Poincaré–Einstein manifolds, lead to uniform Hausdorff measure estimates for the singular sets. Furthermore, the main results in this paper demonstrate a delicate synergy between the geometry of Poincaré–Einstein manifolds and the theory of the corresponding degenerate elliptic operators.

Abstract We aim to develop a q q -analog of recently introduced Motzkin sequence spaces by Erdem et al. [ Motzkin sequence spaces and Motzkin core , Numer. Funct. Anal. Optim. 45 (2024), no. 4–6, 283–303] by using q q -Motzkin numbers and introduce sequence spaces c ( M ( q ) ) c\left({\mathfrak{M}}\left(q)) and c 0 ( M ( q ) ) . {c}_{0}\left({\mathfrak{M}}\left(q)). We investigate some topological properties, compute bases, and obtain their duals. For X ∈ { c ( M ( q ) ) , c 0 ( M ( q ) ) } X\in \{c\left({\mathfrak{M}}\left(q)),{c}_{0}\left({\mathfrak{M}}\left(q))\} and Y ∈ { ℓ ∞ , c , c 0 , ℓ 1 } , Y\in \{{\ell }_{\infty },c,{c}_{0},{\ell }_{1}\}, some results pertaining to characterization of matrix class ( X , Y ) \left(X,Y) is given. We devote the final section to obtain necessary and sufficient conditions for a matrix operator to be compact on the space c 0 ( M ( q ) ) {c}_{0}\left({\mathfrak{M}}\left(q)) via Hausdorff measure of noncompactness.

In this paper, we investigate minimal hypersurfaces in $\mathbb{R}^n$ with respect to the Busemann--Hausdorff measure in a class of Finsler $n$-spaces ($\mathbb{R}^n,\tilde{F}_b=\tilde{\alpha}+\tilde{\beta}$), called Randers spaces, where $\tilde{\alpha}$ is the Euclidean metric and $\tilde{\beta}=b(x)dy^{n}$ is a controlled one-form. We emphasize the fact that $F$ is non-Minkowskian, since $b=b(x)$ is a non-constant function of $x$, which is allowed here. We particularly examine graphs defined on the $xy$-plane that are invariant under one-dimensional isometry groups of $(\mathbb{R}^3,\tilde{F}_b)$. By reducing the minimal graph equation to an ordinary differential equation (ODE), we obtain a new class of explicit examples of minimal surfaces in Finsler geometry.

Hausdorff measure and decay rate of Riesz capacity

Abstract In this paper, we investigate the Hausdorff measure of shrinking target sets in ‐dynamical systems. These sets are dynamically defined in analogy to the classical theory of weighted and multiplicative Diophantine approximation. While the Lebesgue measure and Hausdorff dimension theories for these sets are well‐understood, much remains unknown about the Hausdorff measure theory. We show that the Hausdorff measure of these sets is either zero or full depending upon the convergence or divergence of a certain series, thus providing a rather complete measure theoretic description of these sets.

Consider a compact M subset Rd and l > 0. A maximal distance minimizer problem is to find a connected compact set Σ of the length (one-dimensional Hausdorff measure H1) at most l that minimizes maxy in M dist(y,Σ), where dist stands for the Euclidean distance. We give a survey on the results on the maximal distance minimizers and related problems. Also we fill some natural gaps by showing NP-hardness of the maximal distance minimizing problem, establishing its Γ-convergence, considering the penalized form and discussing uniqueness of a solution. We finish with open questions.

Abstract Since the seminal work of Caffarelli, Kohn and Nirenberg \cite{CKNP} on the partial regularity theory for the three-dimensional incompressible Navier-Stokes equations, where they proved that the one-dimensional parabolic Hausdorff measure of the space-time singular set is zero, extensive research has been devoted to the analysis of the partial regularity and singularity for the Navier-Stokes equations. The current understanding suggests that the dimensional bound on the parabolic Hausdorff measure of space-time singular sets cannot be further sharpened with existing methods. Consequently, scholars turn to investigate an alternative fractal dimension--the box-counting dimension--with the aim of proving that the box-counting dimension of the potential space-time singular set $S$ is at most one. This approach is motivated by the fact that the Hausdorff measure $dim_{H}(S)$ of $S$ is always bounded below by its box-counting dimension $dim_{B}(S)$. Through a series of investigations, the initial bound $dim_{B}(S)\leq\frac{5}{3}$ established in \cite{RSA} has been progressively improved to $dim_{B}(S)\leq\frac{7}{6}$ in \cite{WYI}. Recently, Gong, Wang and Zhang \cite{GWZP} extended the foundational work of \cite{CKNP} on the three-dimensional incompressible Navier-Stokes equations to the Navier-Stokes-Planck-Nernst-Poisson system and proved that the one-dimensional parabolic Hausdorff measure of the space-time singular set is also zero. However, the box-counting dimension of the singular set remains unexplored. In this paper, we investigate the box-counting dimension of the space-time singular set for suitable weak solutions to the Navier-Stokes-Planck-Nernst-Poisson system. By establishing a series of new $\varepsilon$-regularity criteria, we prove that the box-counting dimension of the space-time singular set is at most $\frac{7}{6}$.

This paper explores coherent upper conditional previsions, a class of nonlinear functionals that generalize expectations while preserving consistency properties. The study focuses on their integral representation using the countably additive Möbius transform, which is possible if coherent upper previsions are defined with respect to a monotone set function of bounded variation. In this work, we prove that an integral representation with respect to a countably additive measure is also possible, on the Borel σ-algebra, even when the coherent upper prevision is defined by the Choquet integral with respect to a Hausdorff measure, which is not of bounded variation. It occurs since Hausdorff outer measures are metric measures, and therefore every Borel set is measurable with respect to them. Furthermore, when the conditioning event has a Hausdorff measure in its own Hausdorff dimension equal to zero or infinity, coherent conditional probability is defined via the countably additive Möbius transform of a monotone set function of bounded variation. The paper demonstrates the continuity of coherent conditional previsions induced by Hausdorff measures.

In this paper we show that, given a planar Reifenberg flat domain with small constant and a divergence form operator associated to a real (not necessarily symmetric) uniformly elliptic matrix with Lipschitz coefficients, the Hausdorff dimension of its elliptic measure is at most 1. More precisely, we prove that there exists a subset of the boundary with full elliptic measure and with σ-finite one-dimensional Hausdorff measure. For Reifenberg flat domains, this result extends a previous work of Thomas H. Wolff for the harmonic measure.

Abstract We introduce q-analog of Schröder matrix and discuss its domains in the spaces c and c 0 {c_{0}} . Also, the Schauder basis and α-, β-, γ-duals are given and certain classes of matrix mappings are characterized on these spaces. Moreover, the characterizations of certain compact operators are established via Hausdorff measure of noncompactness.

Abstract We explore the interplay between different definitions of distortion for mappings $$ f:X\rightarrow \mathbb {R}^2 $$ f : X → R 2 , where X is any metric surface, meaning that X is homeomorphic to a domain in $$ \mathbb {R}^2 $$ R 2 and has locally finite 2-dimensional Hausdorff measure. We establish that finite distortion in terms of the familiar analytic definition always implies finite distortion in terms of maximal and minimal stretchings along paths. The converse holds for maps with locally integrable distortion. In particular, we prove the equivalence of various notions of quasiconformality, implying a novel uniformization result for metric surfaces.

Abstract Let ψ : R > 0 → R > 0 be a non-increasing function. Denote by W ( ψ ) the set of ψ-well-approximable points and by E ( ψ ) the set of points x ∈ [ 0 , 1 ] such that for any 0 < ϵ < 1 there exist infinitely many ( p , q ) ∈ Z × N with ( 1 − ϵ ) ψ ( q ) < | x − p q | < ψ ( q ) . In this paper, we investigate the metric properties of the set E ( ψ ) . Specifically, we compute the s-dimensional Hausdorff measure H s ( E ( ψ ) ) of E ( ψ ) for a large class of s ∈ ( 0 , 1 ] . Additionally, we establish that dim H ⁡ E ( ψ 1 ) × ⋯ × E ( ψ n ) = min { dim H ⁡ E ( ψ i ) + n − 1 : 1 ⩽ i ⩽ n } , where ψ i : R > 0 → R > 0 is a non-increasing function satisfying ψ i ( x ) = o ( x − 2 ) for 1 ⩽ i ⩽ n .

This paper demonstrates the controllability of two fractional nonlocal impulsive semilinear differential inclusions with infinite delay, where the linear part is a fractional sectorial operator and the nonlinear term is a multivalued function. The operator families generated by the linear part are not assumed to be compact. The objective is achieved using the properties of fractional sectorial operators and the Hausdorff measure of noncompactness. The results generalise several recent findings, and the method can be used to extend further contributions to cases where the linear term is a fractional sectorial operator and the nonlinear term is a multivalued function, in the presence of instantaneous impulses and infinite delays. The novelty of this work lies in initiating the study of the controllability of a system involving a fractional Caputo derivative under infinite impulses and delays. An example is presented to verify the theoretical developments. Given the wide-ranging applications of fractional calculus in medicine, energy and other scientific fields, this work contributes to those domains. KEYWORDS Caputo derivative, mazur's lemma, mild solutions, multivalued functions, noncompact measure, phase space

Abstract Let { a n } n ∈ N , { b n } n ∈ N be two infinite subsets of positive integers and ψ : N → R > 0 be a positive function. We completely determine the Hausdorff dimensions of the set of all points ( x , y ) ∈ [ 0 , 1 ] 2 which satisfy ‖ a n x ‖ ‖ b n y ‖ < ψ ( n ) for infinitely many n, and the set of all x ∈ [ 0 , 1 ] satisfying ‖ a n x ‖ ‖ b n x ‖ < ψ ( n ) for infinitely many n. This is based on establishing general convergence results for Hausdorff measures of these two sets. We also obtain some results on the set of all x ∈ [ 0 , 1 ] such that max { ‖ a n x ‖ , ‖ b n x ‖ } < ψ ( n ) infinitely often.