In this note, we prove sharp necessary conditions for the boundedness of the manifold-adapted Wolff potential between Zygmund spaces on complete noncompact Riemannian manifolds with nonnegative Ricci curvature. The nonlinear homogeneity of the operator determines the natural form of the norm inequality, while the Bishop comparison theorem and localized ball tests determine the admissible Sobolev scaling. The boundedness assumption forces Euclidean lower volume growth, identifies the relation between the source and target integrability exponents, and gives the critical logarithmic constraint for the Zygmund indices. In the linear case, the conclusions reduce to the corresponding Riesz-potential scaling with the expected logarithmic refinement.

We construct a class of digit-defined Cantor sets \(C_{(d_k)}\subset[0,1]\) from a prescribed sequence \((d_k)_{k\ge1}\) by using the digit \(d_k\) to determine the removal ratio at level \(k\). When only finitely many digits are zero, the construction has a uniform separation property after a finite initial stage. This yields compact, perfect, totally disconnected, nowhere dense sets of Lebesgue measure zero. The natural probability measure \(\mu\) is obtained by assigning equal mass to the two children of each cylinder; equivalently, \(\mu\) is the pushforward of the fair Bernoulli measure on \(\{0,1\}^{\mathbb{N}}\) under the coding map. If
\[
\ell_k=\prod_{j=1}^{k}a_j,\qquad a_k=\frac12\Bigl(1-\frac{d_k}{10}\Bigr),
\]
then
\[
\dim_H C_{(d_k)}=\liminf_{k\to\infty}\frac{\log 2}{-\frac1k\sum\limits_{j=1}^{k}\log a_j}.
\]
The associated Cantor function \(F(x)=\mu([0,x])\) is Hölder continuous for every exponent below \(\dim_H C_{(d_k)}\), and no exponent above this dimension is possible. Cartesian products are also treated: under Ahlfors regularity of the factors, Hausdorff dimensions add and the product Cantor function has optimal Hölder threshold equal to the smallest one-dimensional dimension. A sufficient and verifiable condition for Ahlfors regularity is the bounded comparability \(2^{-k}\asymp \ell_k^{\alpha}\), where \(\alpha\) is the limiting similarity dimension. The results identify how the asymptotic distribution of decimal digits controls dimension, singular measure regularity, and product geometry.

We establish necessary and sufficient Boas-type criteria for generalized Lipschitz classes associated with the generalized Dunkl transform on the real line. The criteria are expressed through the growth of weighted spectral moments of \(\mathcal{F}_{D}(f)\) as the spectral radius tends to infinity. A symmetric generalized difference generated by the dual translation operators is used; its transform multiplier is \((1-j_{\alpha}(th))^{n}\), which gives the correct order \(2n\) at the origin and the required lower control away from the origin. Direct theorems are proved for the classes \(B_{\gamma}^{n}\) and \(b_{\gamma}^{n}\), while converse theorems are obtained under the standard one-sign condition on the transform. These results identify the precise connection between decay of the generalized Dunkl transform and \(Q\)-weighted smoothness measured by generalized translations.

We introduce the concept of a generalized sequential Yeh-Feynman integral for functionals defined on Yeh-Wiener space, formulated via stochastic process \(Z_h\) associated with a nonzero function \(h\). Existence theorems and evaluation formulas for generalized sequential Yeh-Feynman integral are established for functionals in the Banach algebra \(\hat{\mathcal S}(L_2(Q))\) and some related functionals. Furthermore, we show that the class of generalized sequential Yeh-Feynman integrable functionals is strictly larger than \(\hat{\mathcal S}(L_2(Q))\). Previous results on sequential Yeh-Feynman integral are recovered as corollaries of our results.

This study investigates a generalized fractional Hill differential equation subject to two-point separated homogeneous boundary conditions. First, we formulate the Green’s function and detail its fundamental characteristics. Subsequently, we derive a Lyapunov-type inequality for this specific boundary value problem. The article illustrates that these results encompass several established findings in existing literature as special cases. Finally, the work highlights potential avenues for future study through a series of open problems.

In this article, we present a new Hilbert-type integral inequality involving variable weight functions and an adjustable parameter. It can be described as a generalization of the Mingzhe integral inequality. Some other integral inequalities are also derived. These results provide a flexible framework for obtaining valuable bounds and facilitating further analytical applications.